ABSTRACT
MECHANICAL CHECKING OF NAVY BEANS

by Chris V. Narayan

A stability analysis was developed to compute the
stability modulus EI of navy beans loaded quasi-statically
on end. Critical loads for bean cotyledons were obtained
on an Instron Testing Machine at the point of instability
as signified by the onset of checking or cracking of the
seed coat.

Values of El and elastic modulus, B, were computed
for various moisture contents in the range of 11.5 to
28.2 percent w.b.

Dynamic studies in the form of low velocity impact
of beans by a falling weight and high velocity impact by
a rotating arm were also conduCted. Impact forces to
cause checking were measured, and the corresponding impaCt
energies computed. Comparisons of the energy obtained by
the two types of dynamic tests were made.

The results of the dynamic tests were extrapolated
to field conditions and compared with previously made
field observations on bean harvesting and handling.

The optimum moisture content range for resisting

checking, or cracking of the seed coat, was found to be

Chris V. Narayan

l3.A to 15.6 percent w.b. For beans in this moisture con-
tent range, an impact velocity of 55 fps was required to
cause checking, when the beans were struck with a rotating
arm. Beans at 11.5 percent moisture were found to be the
most susceptible to checking, under both static and

dynamic loading conditions.

Major Professor

Approved & ”J W

Department Chairman

MECHANICAL CHECKING OF NAVY BEANS
Ed,

Chris V. Narayan

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Agricultural Engineering

1969

ACKNOWLEDGMENTS

The author Wishes to express his sincere gratitude
to all who have aided in this study. This is especially
applicable to Dr. B. A. Stout (Agricultural Engineering),
whose guidance, support and patience were invaluable.

The help from Dr. G. E. Mase (Metallurgy, Mechanics
and Materials Science) throughout the Doctoral Program
and that from Dr. W. M. Adams (Crop Science) in developing
the thesis problem served to dissipate many of the prob-
lems that arose along the way.

Of equal importance and equally appreciated is the
approval by Dr. C. W. Hall, Chairman of the Agricultural
Engineering Department, of the assistantship and the

funds which made the study possible.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS
LIST OF TABLES. . . . . . .
LIST OF FIGURES
Chapter
I. INTRODUCTION.

1.1 Objectives . .
1.2 The Thesis Problem

II. BACKGROUND NOTES

2.1 Nutrition . . . . . . . .
2.2 Historical Notes .

III. REVIEW OF LITERATURE

3.1 Mechanics .

3.2 Physical PrOperties of Agricultural

Products. . .
3.3 Dynamic Tests . .
3.“ Elastic Stability.

IV. QUASI-STATIC THEORY

“.1 Elementary Buckling Theories . .
Beam-Column Theory . . . .

Complicating Factors.

Finite Differences .

Finite Difference Solution for a

Bean Cotyledon. . . .

1:31:34":
\J'l-EUJN

V. DYNAMIC THEORY

5.1 Impact by a Falling Weight.
5.2 High Velocity Impact. . .

VI. SUMMARY OF EXPERIMENTAL REQUIREMENTS.
6.1 Quasi-Static Tests

6.2 Low Velocity Impact Tests .
6.3 High Velocity Impact Tests.

iii

Page
ii

vi

FJH

00th

15
18
21

24

26
31
33
35

A8
‘49

A9
53

58
58

59
6O

Chapter Page

VII. APPARATUS. . . . . . . . . . . . 62
7.1 Quasi- Static Tests . . . . . . 62

7.2 Drop Tests . . . . . 6A

7.3 High Velocity Impact Tests. . . . 66

7.“ Auxiliary Equipment . . . . . . 67

7.5 Calibration of Equipment . . . . 69

VIII. EXPERIMENTAL TECHNIQUE . . . . . . . 70
8.1 Quasi- Static Tests . . . . . 7O

8. 2 Low Velocity Impact Tests . . . . 73

8. 3 High Velocity Impact Tests. . . . 74

IX. RESULTS AND DISCUSSION . . . . . . . 76
9.1 Moment of Inertia Factors . . . 76

9.2 Critical Loads and Stability and
Elastic Moduli for Beans Loaded

Quasi-Statically . . . . . . 77
9.3 Dynamic Forces and Energy . . . 90
9.“ Extrapolation of the Results to
Field Conditions . . . . . . . 96
X. SUMMARY AND CONCLUSIONS . . . . . . . 99
10.1 Summary . . . . . . . . . . 99
10.2 Conclusions. . . . . . . . . 100
REFERENCES . . . . . . . . . . . . . . 103

iv

Table

LIST OF TABLES

Average values of inertia factors
B, C and D for bean cotyledons

Critical loads, elastic and stability
moduli for two lines of navy beans.

Impact velocities, rupture forces and
energy for navy beans subjected to drop
tests . . . . . . . . . . .

Impact velocities, rupture forces and
energy for navy beans subjected to free,
high velocity impact . . . . . .

Page

76

78

91

91

Figure

11.
12.

13.

114.
15.
16.
17.
18.

LIST OF FIGURES

Bean loaded on end. . . . . . . .

Simple elastic system buckling under a
critical load. . . . . . . . .

Instability due to offset loading. .

Simply supported beam-column under axial
and lateral loads . . . . . . .

Grid for approximating the derivative of
function y - f(x) at a given point 0 .

Pin-ended column with a grid of A - L/A.

Pin-ended column with variable moment of
inertia. . . . . . . . . .

Bean cotyledon considered as a pin-ended
calm O O O O I O O O O O 0

Impact of a bean by a falling weight. .

Hypothetical acceleration-time_and
velocity-time curves . . . . . .

Impact of two moving bodies. . . . .

Impact of a lightly held bean by a heavy
rotating arm . . . . . . . . .

General layout of the testing area with
the Instron machine and valve-air unit

Location of the test bean . . . . .

The drop test apparatus . . . . . .
Location of the test bean . . . . .
High velocity impact apparatus. . . .

' Location of the test bean . . . . .

vi

Page
26

26
28

32

40
A0

A5

A5
50

50
55

55

63
63
65
65
68
68

Figure

19.
20.

21.

23.

2A.

25.

26.

27.

28.

29.

Page
Schematic of the experimental design. . . 72

Variation of Pc with maturity and moisture
content for ngvy beans. First harvest,
quasi-static loading, line 74 . . . . 79

Variation of P with maturity and moisture
content for navy beans. Second harvest,
quasi-static loading, line 7“ . . . . 80

Variation of Pc with maturity and moisture
content for ngvy beans. First harvest,
quasi-static loading, line 70 . . . . 82

Variation of For with maturity and moisture
content for navy beans. Second harvest,
quasi-static loading, line 70 . . . . 83

Variation of EIM with maturity and moisture
content for navy beans. First harvest . 85

Variation of EIM with maturity and moisture
content for navy beans. Second harvest . 86

Variation of E with maturity and moisture
content for navy beans. First harvest . 88

Variation of E with maturity and moisture
content for navy beans. Second harvest . 89

Comparison of dynamic rupture force FR for
navy beans, determined from low and high
velocity impact tests . . . . . . . 92

Comparison of rupture energy ER for navy

beans, determined from low and high
velocity impact tests . . . . . . . 9A

vii

I. INTRODUCTION

Navy beans, like all agricultural products, suffer
some mechanical damage during harvesting and handling.

For navy beans this damage is manifested in two ways,
internal damage to the cotyledons with no visible external
damage, and visible damage.

Visible damage to the beans varies in severity from
breaking and splitting of the cotyledons to checking, or
cracking of the seed coat. The latter case has become
increasingly important in recent years because of the
large percentage of the bean crop which is pre-cooked and

canned.

1.1 Obgectives

 

This study was undertaken to determine the mechanical

cause

m

of the cracking of the seed coat with a view to

reducing the frequency f its occurrence.

0

1.2 The Thesis Problem

 

The work reported in this thesis may be diVided into
three parts:
1. The measurement of the quasi-static forces

required to cause cracking, in accordance with

a stability analysis deveIOped for bean coty-
ledons.

The measurement of impact forces and energy
required to cause cracking of the seed coat.
Extrapolation of the test results to field con—

ditions.

II. BACKGROUND NOTES

Beans are classified in North America under the

genus Phaseolus. Most of the common, dry edible beans,

 

such as the navy bean (also called white or pea bean),

the lime bean, the great northern, and kidney beans are

ccntained in the varietal classification Phaseolus

 

vulgaris.

The bean is a dicotyledonous seed, varying by
variety from ellipsoidal to kidney shaped. The two coty-
ledons and the embryonic axis, which upon germination produce
the seedling, are encased in a relatively impervious seed
coat. The seed while still in the pod, is nourished
through the hylum, which is the only discontinuity in the
seed coat. The seed coat itself is bicellular in thick—
ness except in the hylar area wnere, for strength pur-
poses, an extra layer of cells exists.

e to function after the seed

(I:

The hylum does not cea
leaves the pod. Hyde (195A) undertook a detailed investi-

gation into the function of the hylum in Leguminosae (a

 

British classification rather broader than Phaseolus) and

 

found that the hylum acted like an hygroscopically acti-

vated valve, discouraging the entry of water, but allowing

the outward flow of moisture and gases from the seed,*

'thus allowing the seed to harden rapidly (a phenomenon
vmhich Maddex (1953), upon encountering it while arti~
:fically drying navy beans, described as "case hardening").
CPhis selectivity on the part of the hylum is probably the
czause of the relative difficulty encountered by researchers,
IBakker-Arkema, et_al. (1966), in rewetting navy beans.

The cellular texture, impermeability and pigmenta-

tzion of the seed coat, the closeness of the cotyledons to
eeach other as well as to the coat, and the germination
eibility of beans vary between and within varieties. Some

c>f these characteristics could, as will be discussed later,

{>1ay an important role in the susceptibility of the seed
tzo physical damage.

2.1 Nutrition
Edible beans, and peas (the two commonly referred to
JCDintly as pulses) have long been an important food supply.
Beans contain not only high percentages of energy com-
FHJunds such as fats and starches but also large amounts of
IIPotein, although no amino acids. In addition beans con-
‘tain the important vitamin thiamin. Thus the ratio of

food value to bulk is very high for beans. For this rea-

Son, as Bracken and Rasmussen (19AM) pointed out the

United States Government called for sharp increases in

g

*When the hylum was blocked Hyde found a reduced
rate of drying.

bean production during both world wars, for supp.y to the

The high protein content of pulses is an important
consideration in the food supply of countries where there
might be a shortage of animal proteins. Finally, beans
are versatile as a food, haVing many modes of consumption.
Beans are presently canned as well as milled into flour.
Recently Bakker-Arkema, et a1. (1969) has reported some

success in processing a pre—cooked bean puree.

2.2 Historical Notes

 

Beans, as Beagle (19A9) pointed out, have had a long
and eventful history as a human foo . He goes so far as
to speculate that because of the relatively large seed
and brief period of germination, the bean may well have
been one of man's earliest cultivated plants. Thompson
(1950) pointed to evidence of the existence of beans and
lentils in the Nile Valley ca. 2000 B.C., while Hutchins
(1947) has written o‘ beans being grown in Switzerland
and Northern Italy as early as the Bronze Age.

Beans have long been cultivated in Latin America,
as well as Asia, and indeed Brazil and Mexico are respec-
tively the world's first and second largest producers of
beans [Anon (1966)].

In the United States the particular type of bean
grown varies with location. Among the bean raising

states, Michigan leads in the production of navy beans.

Beagle (1950) speculates that the term "navy" may have
arisen because of the supply of beans to Commodore Perry
on Lake Erie in 1812. Regardless of the accuracy of that
speculation the military may well have played a major role
in the development of bean production. The bean acreage
in the United States did increase sharply during the two
world wars. During the second world war, however, agri-
cultural extension specialists such as Mercer (1945) were
advising farmers to increase not only acreage but mechan-
ization as well. Thus, after the second world war and on
into the fifties not only was mechanization of bean pro-
duction well entrenched, but as Andersen (1960) points

out genetic breeding of disease resistant high yield

lines were well underway. In addition, as Thompson (1950)
reported to the producers, there was an expanding EurOpean
market for dry edible beans; a market which has persisted
to the present.

The increased production in the United States,
accompanied by increased mechanization and bulk handling
led to the problem of mechanical injury to the bean seed;
injury which not only affected market value of the pro—
duct, but also impaired germination. Solorio (1959)
states that the three causes of impaired germination are
mechanical injury, bacteria and insects. Thus there be-
gan in the late forties and through the fifties an in-

creased interest in mechanical injury to seeds, especially

by harvesting and handling equipment. Beagle (19A9) des—
cribed the type of equipment available at that time for
pulling and threshing beans. He discussed many of the
current problems but did not dwell for any length on
injuries. McDow (1949) described the problem of "splits"
occurring with mechanized handling of pea beans. The
term "splits" indicates seeds with the cotyledons split
apart or with at least one-quarter of a cotyledon com-
pletely broken off. He attributed the onset of splits

to poor machine adjustments or low moisture content and
developed the following regression equation for percent

splits (Y) as a function of percent moisture content (x),
Y = 35.5 - 4.15x + 0.1256x2

with the optimum handling moisture content being 16.5
percent w.b.

Harter (1930) and Borthwick (1931) had previously
noted the effect of moisture content on thresher injury
to snap beans and lima beans respectively. They were
mainly interested in germination, and both noted the
production of retarded seedlings ("baldheads") after
mechanical threshing of very dry beans. Borthwick noted
that a common injury was the detachment of the cotyledons.
Toole, gt_al. (1951) working with navy beans found break—
ing damage in threshing tests as high as 20 percent for

low moisture contents. Germination tests proved, through

the c currence of baldheads, that further damage had been
done internally. M.Collum 21953) investigated cotyledon

injury with snap—oeans and found marked varietal differ—

Ul

ences which obscured sid conclusion

(D

The conclusions about moisture content as being an
important factor in seed damage, as well as the increased
production, led to investigations of damage in storage
and during drying. The latter was fairly important since
the harvesting tests results indicated an optimum harvest
moisture of about 18 percent. Maddex (1953) in drying
tests with heated air found a high onset of cracking near
the bottom of the test bin when the moisture content of
the beans dropped below 16 percent. He was able to re-
duce this by decreasing the air temperature (below 1300 C)
and adjusting the RH (to 15 percent), in effect by slowing
the drying rate. Wang (1958) used intermittent applica-
tion of dry air at 1001 F in a test to dry navy beans.

He was interested in seed coat cracking (as opposed to
most of the previous investigators) as well as splitting.

Wang tried to determine the stresses involved in the

(I.

cracking of the :eed coat but his assumption of sphericity
of the pea bean probably accounted for his limited suc-
cess, since the stress distribution over an ellipsoidal
Shaped body and a uniform sphere are very different.

In addition, he did not take the significance of the

discontinuity at the hylum into account.

Brown (1955? was apparently the first to work with
individual navy ceans. He reported forces of twelve to
(6.9 percent to

7'“

orty-two pounds required to break beans

9.2 percent moisture) on the "flat" with various pressure

loeads and two to forty—two pounds to break similar beans

goositioned on edge. His loading rate was in the quasi—

static range. Alkin (1958) working on the susceptibility

zof snap beans to mechanical injury of the seed noted that

t:he coats of resistant varieties adhered more tightly to

tzne cotyledons than did others. He concluded that the

txight seed coat and closely fitting cotyledons must in—
Pribit movement and therefore protect the embryo better.
:Scolorio (1959) struck individual beans by dropping them

lento the path of a rotating (777 rpm) paddle wheel, then

teatamined them for cracks in seed coat (checks) and

"ssplits." At 15.5 percent moisture he found 7.2 percent
Vflisible damage of whioh 6 percent was "checks" and 1.2

H

plits." At 9.7 percent moisture he found 70.3

0)

percent

sibl damage of which there was 27.3 percent

p.»
(D

percent v
V . ' . ,
'Cloecks" and 43.0 percent "splits." Germination tests

IDIHoved that very few of the "splits" produced healthy
seedings. He concluded that when a cotyledon has a

tPansverse crack across it, although the outer end is

ST:iJJ.attached, the food supply in the loosely attached

part is not used. He also found a higher than normal

31r1cidence of "baldheads" in germination tests with the

l_.
O

"checks" which he attributed to minute cracks. Faust

(1955‘ pointed our the possibility of damage (splits)

a

'when pea beans are dropped into deep silos. He attributed

30 to H0 percent of total splits to this.

edless to say this type of work was done on other

in!

N

(b

ggrains as well. ngord (1962), for example, evaluated
‘visible kernel damage to wheat by flails and recommended
21 peripheral speed of 90 fps as an upper limit. Also
tohreshing operations are being continually evaluated and
mrill be as long as it is possible to improve existing
nuachinery, existing genetic lines and existing practices.
(Sioeen (1966), for example, recommended 13 percent moisture
811d 900 rpm cylinder speeds as respective minimum and max—
iJnum values in combining soybeans.

The characteristic features of the above types of

of force and

U)

‘teesting are (a) the leek of measurement
¢Eroergy (except where quoted) wnich cause the mechanical
<ieunage, and (b) the greater interest in splits. Indeed,
‘Nj.th respect to the latter, Perry {1959} quotes the

cifications for Grade No. 1 beans as 2

(D

11- S. D. A. sp
Speaxocent or less "splits, damaged beans, contrasting
<314asses and foreign material." Presumably checking, un-
lfisss so severe as to aCcount for damaged beans, was not
a Specific problem.

As far as splitting and breakage is concerned,

t1'leref'ore, the above tests can be regarded as being

relatively successful in reducing this type of major

damage and los

Ul

, altnough the adherence ~o their recom—
mendations has lei to such practices as rewetting of the
beans prior to major handling and transportation opera-
tions [Thompson (1962), Bakker—Arkema, gt_al. (1966)].
With the increased amount of canning, the checking
of seed ooats has, however, become a serious problem.
Thompson (1950) reported that pea beans were principally
used for canning since they have the tendency to hold
their shape when baked. Perry (1959) pointed out that
the canning industry which uses 90 to 95 percent of the pea
bean crop annually dislikes checked beans because during
the processing the checked beans may split, and also be-
cause checked seed coats may separate from the cotyledons
and float at the top of the cans. French (1962) stated
that cracked seed coats admit oxygen and moisture to the
cotyledon, which is bad for storage, and permit the entry
of bacteria and fungi which causes quality deterioration.
He was very successful in developing a technique for
determining minute cracks in seed coats rapidly, using
Indol Acetate. This chemical enters the crack in the
coat and stains the undersides of the coat a deep blue.
It does not affect the exterior of the coat. For beans
with relatively transparent coats, such as the navy
bean, the results of dumping a handful into a solution of

Indol Acetate is claimed to be remarkably effective in

l2

identifying checked beans. Kannenberg and Allard (1954)
studiei lignin formation in oean seed coats. They found
small amounts of lignin in nonpigmented (white) seed coats.
Noting that the function of lignin is primarily one of

strength and protection they concluded that low lignin

U

content of white seed coats could account in part for
their susceptibility to injury.

Adding to the above the implication of Solorio's
(1959) findings, that the process which causes the check—
ing could also cause internal damage to the cotyledons,

the importance of checking cannot be discounted.

ill. REVI"W OF LITERATURE

In the field of Physical Properties of Agricultural
Products, the investigator is basically attempting to use
the entire field of Theoretical and Applied Mechanics
merely as a tool kit. Hopefully, he simply selects the
tool most applicable to his problem and uses it. This
requires a sound knowledge of mechanics. In addition, he
must be aware of the biological factors which influence
his work. Genetics for example as implied by the findings
of Liu (19A9) working on genetic inheritance of damage
resistance to soybeans, could produce premature obsoles-
cence of a mechanical study. Again, the chemical makeup
of the product studied could have important effects on the
physical properties of that produced, as evidenced by
Kannenberg and Allard (1964) and Dorrell (1968). Although
the Agricultural Engineer in the physical properties field
could not possibly keep up to date in the biological fields,

he should know the tools of his own field very well.

3.1 Mechanics

 

Lazan (1962) listed three approaches for studying
rheological properties: (1) micromechanistic-—solid state
physics, (2) macroanalytical--applied science or engineer-
ing, and (3) ad hoc or simulated testing--the "practical"

13

approach. Of the three approaches, the last two are most
commonly used by engineers. Lazan admits the need for
the practical approach in which one has to work with the
actual object and loading conditions rather than speci-
mens and models. However he does caution:
. . . ad hoc property data are generally not
extendable, in the absence of a more basic under-
standing, to future problems which lie in dif-
ferent regimes of stress and environment.
He notes however that:

Although the ad hoc approach adds relatively
little per unit effort to the store of basic
information, it can often provide engineering
answers in a relatively short time and is
sometimes "the only way out."

The macroanalytical approach involves the determina-
tion of the properties of materials by idealization,
simplified conditions and test specimen geometries. By
idealizing is meant formulating ideal constitutive equa-
tions for the material. The theories of elasticity,
plasticity and viscoelasticity are examples of this ap-
proach. The macroanalytical approach assumes continuity,
isotropy and homegeneity. As Malvern (1965) states these
quantities are difficult to establish with some materials.
However, since by this approach one merely attempts to
model the material behavior as it is observed for a given
range of stress (or strain) then, provided that the model

can predict the behavior of the material for a different

stress regime in this range, the attainment of such a

model can be considered the solution to the problem. The
actual explanation of the behaVior becomes immaterial.

The macroanalytical approach works best when the
material stress—strain behavior is linear, when the load
conditions are quasi—static and when the material is
indeed homogeneous. When these conditions change how-
ever, for non—linear responses for example, or dynamic
loads, or compOSite materials, much more work is required.
When one is faced with all three of these complications,
the ad hoc approach is in most cases "the only way out."
Kerwin (1965) found that he could drastically change
(dampen) the vibration of steel structural members by
using viscoelastic material bonded between them. For his
tests on beams, arches and base members, he had to use
actual members and scale models.

In the study of the physical properties of agri-
cultural products, investigators are often faced with all
three of these complications.

3.2 Physical Properties of
Agricultural Products

 

 

The application of the theory of mechanics to the
testing of agricultural products on a large scale is a
relatively recent development. It is largely based on
the assumption that most agricultural products are
viscoelastic in their load—deformation behavior. Zoerb

(1958) applied this technique in the study of the physical

16

properties of selected grains. He used both the ad hoc
and the analytical approaches. Using core samples of
pea beans at 36.A percent moisture d.b., he plotted
stress-strain curves, and calculated maximum strengths,
for relatively low rates of loading (0.267 ipm). He
evaluated shear strengths of been "slabs" at four mois—
ture contents. For his ad hoc tests, he ran quasi—static
tests on beans on the flat as well as on edge and found
linear relationships between load and deformation at the
low moisture content value of 10.6 percent. To compare
with his static shear tests, he ran impact shear tests
using a pendulum impact tester. The energy required by
the static test for shear failure was higher than that
for the dynamic tests up to about 20 percent moisture.
The situation reversed itself at higher moisture con-
tents. Zoerb also investigated the rheological prop-
erties of pea beans using relaxation tests, in which an
instantaneously applied deformation is held constant and
the load relaxation is measured with time. He proposed
a model of two parallel Maxwell units to represent navy
bean tissue. 0

Perry (1959) extended Zoerb's work on navy beans.
His approach, however, was from the dynamic point of
View and was therefore of the practical or ad hoc type.
His first tests consisted of dropping beans at various

moisture contents and temperatures through three heights

17

of drop, 11.25 ft, 22.5 ft and H5 ft. Maximum visible
damage was found to occur at the “5 ft drop height and

12 percent moisture, w.b. Minimum visible damage
occurred at 18 percent moisture w.b., for all drOp
heights. Beans were dropped individually, in small sam-
ples and poured slowly. Beans were visually examined

for damage and germination tests were also used.

Checking and splitting increased with increasing drop
height and decreasing moisture content. Some tempera—
ture interaction was evident in the germination tests.

In a second set of tests Perry used a wooden faced bar

to strike beans which were partially confined by a small
movable wooden block. The velocity of the bar was cal-
culated to simulate bean velocity in dropping during com—
mercial handling. A high speed movie camera was used to
measure velocities before and after impact. Energy
balance equations yielded impact energy. Newton's Law
for the bean-block system was used to calculate maximum
force of impact. This impact force was compared with
Zoerb's data on whole beans. Coefficients of restitu-
tion were calculated for the four moisture contents. The
kinetic energy range was 0.006 ft-lbs to 0.017 ft—lbs.
Perry noted in examining the checked beans that the cracks
"seemed to radiate from the hylum," creating a common

check pattern.

18

Perry and Hall (1966) extended the above work to
include auger transport of pea beans in a 21-foot screw
auger at three slopes, horizontal (0 percent damage),

15 degrees (1.1 percent damage), and 28 degrees (1.5
percent damage). They also investigated the deteriora-
tion of bean quality in storage.

After Zoerb's application of mechanics principles
to agricultural product testing, several other researchers
applied viscoelastic principles to quasi-static testing
of various products. Finney (1963) applied linear visco—
elastic theory to the potato, to find material constants.
Mohsenin (1963) developed a tester to perform quasi-static
tests. Mohsenin, et_al. (1963), (1965) extended elastic
and viscoelastic theory to several fruits and vegetables,
as did Timbers (1965) and Arnold (1966).

These tests, however, did not produce data that
could be immediately applied. They are mentioned here
only for the reason that they led to dynamic testing by

the same authors, and by others.

3.3 Dynamic Tests

 

Most of the dynamic tests in this field are ad hoc
or practical tests. Except for large products, they were
performed on the whole product. The most important factor
from the point of view of a literature survey is the

apparatus and the parameters measured.

19

In discussing dynamic testing, Alfrey (19MB)
stresses the importance of energy to cause failure rather
than maximum stress, when investigating impact strength.
For linear elastic and viscoelastic materials, this energy
is proportional to the square of the impact strength
[Kolsky (1952)]. The development of shock waves, varia-
tions in texture and geometry, complicate the relation-
ship.

For these reasons impact testing has been confined
to measuring maximum impact energies. Bilanski (1966)
subjected various types of seeds individually to impact
testing at two loading rates. The slower testing was
done with a pendulum. An energy balance equation was
used to calculate impact energy absorbed by the test seed.
The higher speed testing was done by dropping the seed in
the path of a rotating paddle. Orientation of the seed
was impossible in this case.

Fridley, gt_a1. (1964) impacted peaches with a flat-
plate plunger with a pendulum-type arrangement and com-
puted bruising energy. Mohsenin and Hammerle (1965)
devised a drOp tester to measure bruising energy and im-
pact forces on stationary supported apples struck by a
falling weight. Photocells were used to measure impact
velocity of the weight, and a quartz accelerometer to
measure its acceleration. They plotted volume of bruise

versus impact energy. Wright and Splinter (1968) used

20

a drOp tester similar to the one devised by Mohsenin and
Hammerle to measure impact rupture energy of sweet potato
samples. They compared this to static rupture energy.
For the varieties of sweet potatoes used the rupture
energies (about 2 in.—lbs) were between 4.5 and 3.5 times
smaller than the static. Wright's analysis of the drOp
test was used in this study to design a drOp tester and a
series of low velocity impact tests for determining the
rupture energy of navy beans under dynamic conditions.
The analysis is discussed in section 5.2.

Mitchell and Rounthwaite (1964) impacted individual
grains of wheat by moving them slowly into the path of
a rotating hammer. Hammer speeds up to 120 fps were used
for three moisture contents. A regression equation, not
considering moisture contents was derived between un-
damaged grain and hammer speeds. Clark, et_al. (1967)
used a rotating arm to strike individual cotton seeds.
The seeds were free to move after impact, and were tracked
with a strobosc0pe and exposed photographic film. Energy
balance equations were used again to compute the energy
absorbed by the impacted seeds.

The last two studies discussed have the obvious
advantage of approximating, in the laboratory, the forces
on the seeds in a combine. Lamp (1959) ran threshing

tests on wheat using a modified combine cylinder. He

21

used the restitution equations to calculate energy im-
parted to the grain by the combine cylinder bars, and
attempted to compute threshing forces. Peripheral speeds
of up to 25.0 mph (36.7 fps) were found to be sufficient
for complete threshing. Grain damage was recorded but

not analyzed. Tabiszewski (1968) ran threshing tests on
pea beans using a modified peanut combine. Peripheral
speed of the cylinder was varied between 17 and 37 fps.
The amount of visibly damaged grains was found to increase
with peripheral speed and with decreasing moisture con-
tent. In the range of 10.6 to 12.3 percent moisture w.b.,
the maximum amount of Visible damage was 31 percent.

Other dynamic tests have been devised to measure
different parameters. Diener (1968) used steady state
dynamic tests to measure the complex modulus of cherry
bark. The principles of dynamic viscoelasticity [Ferry
(1961)] were used. The direct application of these re-
sults is, however, not yet possible. Finney and Norris
(1967) have used resonance characteristics of selected

fruits as an indicator of fruit quality with some success.

3.4 Elastic Stability

 

The possibility of structural failure well within
the elastic regime of stress exists for structures of
given geometry. The classical problems of the slender

column and the flag pole [Timoshenko and Gere (1961)]

22

are well known. This stability problem is, in general,
governed by the general fourth order differential equa-

tion of equilibrium

where p is the axial compressive load,

E1 is the flexural stiffness of the section

and q is the transverse load.

When q is absent, the equation becomes homogenous
and the solution is simpler. For a varying cross-
section the problem becomes more complicated although
symmetrical variation lessens the work required.

Moustafa, et_al. (1966) used the methods of sta—
bility to model the wheat plant under axial and lateral
loads. Viscoelastic stability principles and large
deformation theory were used.

For short columns with varying cross-section, the
equilibrium equation is best solved by approximate
methods. The finite difference method is the easiest to
apply. This method is simply one of approximating deriv-
atives with algebraic quantities. For example, the slope
of a curved line between two points y1 and y? at a dis-

tance 2A apart is approximated by

23

This method leads to an Eigenvalue problem with the

Eigenvalues being the parameter of i% required to cause

E

buckling into the fundamental buckling modes. With p

known, the El values can be computed.

IV. QUASI-STATIC THEORY

Preliminary tests involving the loading of indi-
vidual beans with a Valve-Air Loading Unit indicated
that cracking of the seed coat occurred at lowest loads,
when the bean was loaded on end (Figures 1 and 13).

Examination of the above phenomenon in greater
detail indicated that under the end loading conditions
the two cotyledons tended to separate, subjecting the
coat to a tensile stress. Ultimately, this led to failure
of the coat in the form of cracking. Because of the
greater strength of the coat in the hylar region of the
bean, the cracks tended to turn away from this region,
producing the various check patterns often observed on
damaged beans.

It was apparent that under these conditions of end
loading, the failure could be characterized as buckling
on the part of the cotyledons.

In the following sections the principles of stabil-
ity and buckling are discussed, leading to an analysis of

the bean failure under end loading.

24

25

H.1 Elementary Buckling Theories

 

H.1a The Equilibrium
Formulation-—Bifurcation

Theory

Consider the system shown in Figure 2a. It consists
of a rigid bar of length L, pinned at the lower end and
free at the upper end. It is loaded at the free end by
an axial force P, and constrained in the vertical posi-
tion by a coil spring of strength k at the pinned end.
The vertical is the equilibrium position.

The critical load Pcr can be defined as that load
which is Just sufficient to keep the bar in some deformed
position characterized by the angle 0 (Figure 2b). Under
Per the bar neither collapses nor returns to the vertical
position.

Considering the equilibrium conditions for the bar
loaded by Pcr’ spring moment M = k0

. . for equilibrium,

PcrLsin 6 = k6
whence,
_ k6
Pcr ’ L sin 0 (1)

Equation (1) can be linearized by assuming 9 is small, in

which case sin 6 = 6.

Thus, Pcr = % (2)

26

as

Figure 1.——Bean loaded on end.

 

 

 

 

 

 

 

 

 

 

L
k
(a) STABLE (b) BUCKLED
Pigure 2.--Simple elastic system buckling under a critical

load.

27

Under this formulation the bar supposedly remains

vertical under load until this load P reaches the value

 

of Pcr’ at which point the bar bifurcates._ In practice
this need not be so. Other theories have been develOped
to illustrate equation (2) more realistically.

H.1b Equilibrium f

Formulation—- -
Imperfection Theory

 

 

 

There are three types of imperfections:
1. Initial deformation.
ii. Load not perfectly axial--eccentric loading.

iii. The presence of a small lateral load on the

bar.

Of these, the eccentric load imperfection will be
discussed, since, in considering cotyledon buckling, an
end load axially applied to a bean will be slightly off-
set with respect to each individual cotyledon.

Consider now the bar-spring system with the load P
applied a distance e from the geometric axis of the bar
(Figure 3). The buckling criterion now applied is that
when P = Por the deformation 0 will become unbounded.

Now, for equilibrium at any angle 0, for some load-
ing P, the sum of the moments about the pinned end is

zero. Thus,

P (L sin 0 + e cos 6) = k0 (3)

 

 

 

 

 

f

"___J___

(a) STABLE

P"Per

 

 

( b) BUCKLED

Figure 3.--Instability due to offset loading.

29

For small values of 6, equation (3) becomes,

P (L6 + e) k6

whence

_ Pe _ e
e-k—TPL-kP-L W

Applying the buckling criterion to equation (u), it can
be seen that 0 will become unbounded, regardless of the

magnitude of e, when

the same result as given in equation (2).

H.1c Energy Formulation--
The Principle of Stationary
Potential Energy

 

 

 

For any force-deformation system, the above prin-
ciple states: "Of all possible configurations consistent
with the constraints, (or boundary conditions), that one
which Satisfies the equilibrium conditions is the one for
which the total potential energy of the system has a
stationary value."

Thus the energy formulation is also based on equi-
librium.

Consider Figure 2 again. The total potential energy,

n, of the system is that of the load plus that of the

30

spring, and is a function of 6. The potential energy of

the external force P is given by
V = -PL (L ~ cos 0)

The potential energy of the spring, (strain energy), is

given by

k0?

- 1
U ” 2

The total potential energy of the system is

Thus,

—PL (1 - cos 9) + — k62 (5)

:1
ll

1
2

Now, for the configuration for which n is constant,

(dn/de) = O.
dTT_ _ 4
58 — O - -PL sine + k8 (6)
whence,
, ; k0
l L sin 0 ’

which is again the equilibrium equation. Linearizing as
before, by letting 0 be small,

_ — E
P — Por - L ’

which again is the same results as given in equation (2).

3%

1...".

31

A.2 Beam-Column Theory

 

(I)

Consider a simply upported bean as shown in Figure

cted to an axial force P and a distributed

([3

Ha. it is

(I‘

up]
lateral load q7x). Under these loads, the bean will bend
as shown by the dotted line representing the deformed
center line. Taking the x and y directions as shown in

Figure Ma a positive, the radius of curvature, R, can be

(I)

expressed, assuming small deflection, as:

‘ ii 52
5‘“? 5% (7)

ow consider the free body element shown in Figure

“o. For equilibrium, 2M9 = 0

.°. —M + (M + dM) — de + qu %§ - de = o

Dropping powers of differentials higher than one, the

above equation becomes

—M + M + dM - de — de = o
.'. dM - de — de = o
.‘> V:fl_§§l (8)

Substituting for M in (8) using (7),

2
é%'<-EI g_10 _ §_l.= V (8a)

de dx

32

 

 

q(x)

p 1C2":

_ .. -____-_74

'\ ’7

\ /

E DEFLECTION CURVE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y

(a) BEAM COLUMN UNDER LOAD

 

qu

 

 

 

 

 

y '~——-dx -—’i a;—
(b) FREE BODY ELEMENT OF THE DEFLECTED BEAM-‘COLUMN

Figure U.--Simply supported beam—column under axial and
lateral loads.

35

Referring again to the free body element, 2F = O

Y
o“. "V + 'V + dV/ 'f qu 2 O
. 211': _,
. . dx A

Differentiating equation (8a), and substituting for

dV/dx , W
E.
i
g:_ * i:i\ ' iL ’ gl’ — ~ 7

dx‘ ‘Ei dxi’ + dx i? dx) ' q (9) i"

For constant E and 1, and no lateral load, equation (9)

\

takes on its well known form

[T]
H
(Lil
x 4
<
.+
"U
DO:
><wo
I
Q

(9a)

The solution of equation (9a) with the apprOpriate
boundary conditions on y and its derivatives yields the

critical load for the seam-column in question.

I";
(7.
\J I
(1

”.3 COFfipl'

'—
' ‘1
on.
(I)
Q)
(- 7
Cf
(
(.5
U l

(4.3a Short Columns

The Bernoulli—Euler theory, as the Beam-Column
tflieory developed in the previous section is called, has
(Hartain limitations. its accuracy in predicting critical
loads depends on how well the actual boundary conditions
Can be made to suit the theoretical or mathematical bound-

ary conditions. In addition, experience has shown that

3a

good agreement between actual criticol loads and theoreti-
cal cr Euler loads depends on the slenderness ratio of

the column under test. The slenderness ratio is defined
mathematically as L/r, where L is the column length and r is
the minimum radius of gyration of its cross—section. For
steels, depending on how well the boundary conditions can

be satisfied, Timosnenko and Gere report good agreement

with the Euler theory for slenderness ratios greater than
70.

For slenderness ratios lower than 70, empirical
correction factors are applied to the Euler theory to
match the actual results. These factors are often used
for empirical design formulae for columns.

4.3b Columns with Varying
Cross-sections

Variation in the cross—section of the column means
that the moment of inertia, I, of the column will vary,
i..e., I = 1(x).

Now, if the v riation of l is a simple one, e.g.
lJinear, then 1(x) can be based on some reference value,
Io , of the moment of inertia measured at a particular

Seuotion x = x0. In such a case the moment of inertia can

be written as

I(x) = ¢ (x) I0

35

The above expression can then be used with equation (9)

m

and P can be oota: i 9 13° :ge 5,i:q copndirv conditions,

9

by solving the dirt: ential e gation:

b.3c Approximate Solitions

 

ObVicusly, when ciX) is no: a simple function, the
resulting differential eqiation becomes very difficult to
solve. in sioh a case, so approximate solution of the
differential equation, oy a numerical procedure, is often

the best approach. One suzh numerical procedure is the

finite difference method. By ths method, the differential

J

equation is approximated by an algebraic equation at each
of a finite number of chosen points along the column. The
resulting system of eqia‘ions,~ith the appropriate boundary

ll

conditions, is then solved to yield the critical load.

The rocedur' is oitiiiei in the following section.

l‘("
(l

 

14.Ma Approximation
Of‘ Derivatives

 

 

Consider the curve 5 s fix) given in Figure 5.
Suppose the derivatives of y are required for x equal to a
'Value corresponding to the point 0. Two points are chosen
On the x—axis, on either side of the point 0 and labelled

as shown. The "E" and "W" Signs have no mathematical

ufT

36

significance; they serve only to indicate which side of
point 0 the points they label lie. Any two successive
points are separated by a finite distance A on the x—

axis.

Let q be defined as shown. The first derivative of
y(x) at point 0 can be approximated by the following ex—

pression:

I) :(dl)=__g.=yE-yw
dx

 

(y 0 2A ' "'2""'—')‘ . (108.)
The second derivative:

_ y - 2yo + y

(yII)o = E W (10b)

A2

'The third and fourth derivatives at point 0 can be shown

 

 

to be:
. . y - 2y + 2y - y
(leI)Q = EE E w ww (100)
2A3
_ . 57 - “y + 6yo - “y + y
(yIV)0 = EE E u w ww (10d)
A

Thus, a differential equation can be approximated
93' a system of algebraic equations set up at the points
C3hosen (node points), and the system can be solved for

the approximate shape of the curve y(x).

F." ‘- ‘ ‘_‘“"“" “‘“ I

3

‘ II-

37

H.3b Error Involved

 

Consider Figure 5 again. A Taylor series expansion

about point 0 yields

 

 

we 5 A) = 3E = =15) + A f + ‘2‘ W“), + :, (yin), +
y'O-AJ = yw = (y), - A (yr).3 + %; (yu)o - %% (yIII)0 + .
Subtracting the above yields
'yE - 3w) = 25 (yr), + if (yIII)O + .
'. who = ”(E—2; - g}- will), (11a)

Comparing equations (10a) and (11a), it can be seen that
the error involved is of the order of A2.

Similarly, by adding the two equations, it can be
shown that the error involved in the finite difference

appm©Ximaticn of the second d rivative is also of the

(I)

OIKder of A‘.

By expanding y(x + 25) and y(x — 2A), the same order

GIT magnitude of error is obtained for the third and fourth

derivatives.

Thus the accuracy of the approximate solution can
‘be increased by decreasing A, i.e., by using a large
number of node points. However at each node point, one

a1gebraic equation is obtained. The greater the number

5“

38

of node points therefore, the more time consuming the
operation becomes.

An easier method oi increasing the accuracy of the
procedure is by Richardson's extrapolation.

h.uc Richardson's
Extrapolation

 

Recall that the problem at hand is to solve a given
differential equation to find a function y(x), such as
the one shown in Figure 5.

Now, if yC is a finite difference approximation of
y(x) obtained by using a coarse grid spacing AC, and yf
an approximation using a finer grid Af, then by the argu-

ment of the previous section it is known that:

__ ,. -2
y(x) ye + KA C
and also,
y(x) = yf + k& f
Sirme k is unknown, it can be eliminated in the following
manner
k = fi— (y(X) - j;)
‘ c
l
k = V— (Zka) - 37f)

Eh

F! .1...

39

 

 

y(xg 1 l - yc yf
° ' . T7" ' 77* " x2 ‘ x:
c i c f
t , A‘r
' y(x) = , y — ,‘ ,) y
A “A“: C ’\‘ ——\ f
I c I 3
_ _ Ii - . L
Now let h - —— and n. = F—
c in f if

where L is the tOtal length over which the function y(x)

is defined. The function can now be written

’“zf ‘ (n2 )
_ < ) w _ c
y(x) _ nif-nzC ”f nEf-nzC yc (12)

Note that a very good approximation of y(x) would
be obtained even when two relatively coarse grids are used
since the major portion of the error has been eliminated.

A.Hd The Eigenvalue
Problem

Consider the simple pin ended beam-column of Figure
6, with constant E and I. The second order differential

equation of equilibrium is

2
Elu+Py=0 (9)

Apply a very coarse grid, A e % as shown. The node points

beyond the span, a and b are included so that the finite
difference approximation of equation (9) can be made at

points 0 and 3. Values of y are assigned to these

HO

 

 

 

 

 

 

 

WW W. O E SE

Figure 5.--Grid for approximating the derivative of a
function y = f(x) at a given point 0.

 

 

 

 

 

 

 

 

 

 

[A '- ill
V 7
a I 2 3 4 b
P - ._._.,__. _ __ _ __._... — ——-J- .E-u— x
«o—x x—JL—xap—x—«q-«q—A
y E and! CONSTANT

Figure 6.--Pin-ended column with a grid of A = L/U.

Al

imaginary node points a and b by applying the boundary
conditions.

Now for node point 0 the boundary condition is

y, = U

II
C)

similarly, y,

Also, since the column is pin ended, the moments at the

ends are each zero;

 

 

dx3 0 dxz “
ya - 2y0 + y; = o = y, - 2y, + yb
l2 l2
.. ya=-..
and yb = - Y,

Finally owing to symmetry y: y,. Now, applying the
finite difference procedure to point 1 yields, (since

yo = 0):

 

+ —%3 y2 = O (133)

 

CI” _
+ ET yZ '- 0 (13b)

u2

writing equations (13a) and (13b) in matrix form, and

Ecrkz
letting E1 = k,

 

_.—_.k T—

2 2 + k y2

But this system is set up for a solution for y, while the
critical load Pcr is really what is required. Por can be
obtained because the above is an Eigenvalue problem. The
system of equations will have non trivial solutions for
existing values of y if and only if the determinant of

the coefficient matrix is zero. Setting this determinant
equal to zero fixes k and thus fixes Pcr' Thus y is never

solved. Solving for k:

-2 + k l I

2 -2 + k |
u _ uk + k2 — 2 = o
.'. k = 0.586, 3.1ui.

[MSing only the lower value, to get the lowest value of P,

i.
e. Pcr’

_ kEI _ kEIMZ _ 9.37EI
-—7-

“3

The exact solution to this problem is

 

The result from the coarse grid contains an error

If a grid of A = L is used, the result

of only A percent. 2

is

_ 8E1
Pcr - L2

Using Richardson's extrapolation for the results with the

two grids yields

 

_ 9.6EI
Pcr - L2

which is an even more accurate approximation.

U.He Finite Difference
Approximation of the Fourth
Order Differential Equation
with Variable I

Consider the pinned column of Figure 7. The condi-
tions are similar to those of Figure 6 except that the
cross section, and hence I, is not constant. In this case

‘er differential equation given in equation (9a) is the

One that is approximated.

‘2

d2 C12,: 2:
dx2 (El dxz) + P 3x2 = O (9)

AA

Referring to Figure 7, the approximation to this

equation at point 1 is

 

 

 

 

y-2yo+y1 y-2y+y
%s {(31), a — 2(EI) ° 1 2
x2 A2
yl - 2y2 + y yo - 2y1 + Y2
+ (EI)2 b } + P = 0 (9b)
A’- A2

Note that in this approximation E1 is assumed constant
over each node point i A/2.
A.5 Finite Difference Solution
for a Bean Cotyledon

Consider the single cotyledon schematically repre-
sented in Figure 8. It is considered to be simply sup-
ported and end loaded with force P. Let the smaller
principal moment of inertia of the cross-section be IM.

For a finite difference analysis, a grid of A =
L/6 has been applied to Figure 8.

Assume symmetry about point 3. Assume also that
there is a simply supported boundary condition at each
end.

The finite difference procedure is as follows:
Boundary conditions:

yo=ys=o

ya a ~y1

yb ' -y5

“5

 

 

 

 

 

 

 

 

 

 

 

 

a O l 2
‘\o\ "'__’_
p\ L\ x
‘N:\/
y7‘nfl’2
I ’2
~—x-~—x-~—x-~—x—a
Io In 1:2
E CONSTANT
Iiigure 7.-—Pin—ended column with variable moment of inertia.
A L _J
a O l 2 3 4 5 1 b

 

 

 

 

 

 

 

P $9

A1,. 31,, c1... 1,. CIM 31,, A1,,
E CONSTANT, x =L/6

 

 

 

 

W1HUre 8.——Boan cotyledon considered as a pin-ended column.

A6

Symmetry Conditions:
y1 = V,
y, = y“
IWoment of Inertia:

.Let I = I

a M
I0 = 16 = AIM
I1 = I5 = BIM

I2 = I“ = CIM

Substituting these values into equation (9b) and
letting (PA2)/(EIM) = k, the following equations are
obtained.

Node Point 1.

Y1 (23 + C - 2k) - y2 (2B + 20 + k) + cy3 = 0
Node Point 2.

yl (—2B - 2C + k) + y2 (B + AC + 2 — 2k)

+ y3 (-2c — 2 + k) = o
NOde Point 3.

y1 (20) + y2 (—2c - u — 2c + 2k)

+ y3 (20 + A - 2k) = O

In matrix form these equations can be expressed as

a single system as,

A7

 

(2B + c — 2k) {-23 - 2c -k) + c y1
(-2B - 2C + k) \B + AC + 2 - 2k) (—2C - 2 + k) y2
1 2c 2(— 2c — 2 + k) 2(c + 2 — k) y,
1.. _i._ is

 

 

 

At this point the determinant of the coefficient
rnatrix is set equal to zero to obtain the values of k.
Iiowever B and C will be determined for the beans at the
three stages of maturity used, then substituted in
(directly to save much of the bookkeeping required. With
“these values, the coefficient matrix will be simplified
and.its determinant easier to compute.

For extrapolation purposes a second, coarser grid,
in.which A = %-, will be used.

For this grid, the boundary conditions and symmetry

conditions are the same as for the finer grid. The in—

ertias will be,

F1
“
H

Again using equation (9b) and letting (pA2)/EIM be
k, the finite difference equations become:

Node Point 1.

yl (up + 2 — 2k) + y, (-2D - 2 + k) = o

A8
Node Point 2.

y, (-AD — A + 2k) + y; 42D + A - 2k) = 0

[\
.9.
f. I

£(2D + 1 — k) (-2D -

(-2D — 2 + k) 2(2D +

-—

l\_)
l
>7
V

 

Again, values of D will be determined and sub-
stituted before the coefficient matrix will be set equal
to zero.

From both grids, the values of Pc the lowest value

r,

in each case will be computed. Extrapolation will give

the final value of P
cr

P 2
But k = CPA

 

P will be computed as

a

wnere K will be known from the calculations.
Now the actual value of For will be measured on the
IInstron Testing Machine. Thus the value of ElM can be

computed.

V. DYNAMIC THEORY

As with most agricultural materials, beans are more
often damaged by dynamic rather than quasi—static loads.
Thus, while the preceding analysis serves to determine
some of the mechanical properties of the bean, it could
not answer many of the questions on damage to beans under
actual harvesting and handling conditions.

The following analyses consider two types of impact
conditions, a low velocity impact on a supported bean, by
a falling weight, and a high velocity impact of a bean
which is free to move after impact. In both cases, the
loading is on end as in the static analysis (Figures 9

and 12).

5.1 Impact by a Falling Weight

 

Suppose a weight W, falling under gravity with a
constant acceleration g were to strike a stationary bean
(Figure 9) with an impact velocity Vi’ and were to com-
press the bean from its original length L to a length y
at which it ruptures.

The energy, at impact, of the system is:

H
OQIEI

V2+WL

E.=—- i

l

l\)

”9

SO

 

Figure 9.--Impact of a bean by a falling weight.

 

 

ACCELERATION

 

 

 

 

 

V;

>.

t

(D

C)

.l

I“

>'
r*—— t "H
TIME

Figure lO.--Hypothetical acceleration-time and velocity—

time curves [After Wright (1968)].

51

The energy at the rupture point, y, is

f
sipated during impact is

Where V is the velocity of W at y. Then the energy dis—

_ i K 2 _ 2 _
E — 2 g (Vi Vf ) + mg (L y)

Now suppose, as Wright (1968) suggested, the de-
celeration of the weight W is linear during impact up to
the point of rupture. Then the decrease in velocity is
parabolic.

Using an analysis similar to that of Wright, hypo-
thetical impact curves can be drawn for acceleration and
velocity (Figure 10). At time t = O, the weight is re-
leased from a height h. At point A impact occurs; the
bean is of length L. The deceleration of the weight is
linear up to point B, at which point the bean has been
reduced to length y.

The displacement of the weight during the impact
from A to B is (L—y) and can be computed from the area
under the velocity—time curve between A and B.

The change in velocity of the weight as it tra-
verses the impact length (L-y) can be computed from the
constant deceleration of the weight and the time of im-

pact, i.e. the time between points A and C.

52

These calculations are as follows:
From Figure 10, the required area under the velocity
curve is made up of a rectangle EFGH and half of a parabola

EIF.

Area EFGH = Vft = (Vi - AV)t

_ ‘ _ 2
Area BIF — ( ) (Vi - Vf)t — § (AV)t

cum

Adding these two equations yields the displacement of W

during impact.

_ l
(L - Y) — t<Vi -§AV)

Thus the energy dissipated during impact is

_ ‘w . 2 2 1
ER — 5% (Vi — (Vi — AV) ) + Wt(Vi — § AV)
_ w 2. . 1 ,
— 5% (2Vi AV - AV ) + Wt(Vi — 3 AV)
E = 3L AV(2V — AV) + Wt(V — l AV)
B 2g i i 3

Where t is the duration of impact up to rupture and AV is
the change in velocity of the falling weight during t.
Now let ai be the constant rate of deceleration of
the weight during impact up to the point of rupture. Be-
yond this rupture point the deceleration will no longer
be linear, but this is of no consequence here. Letting t

again be the duration of impact, then it can be measured

53

as the horizontal distance from the initiation of impact
to the point of rupture on the deceleration curve. The
change in velocity AV is the area under the deceleration
curve up to the point of rupture.

Thus the rupture energy ER of the bean can be com-
and

puted with suitable instrumentation to measure V a

i’ i

t.
The maximum force transmitted to the bean by the
weight to cause rupture can either be computed or, with

suitable instrumentation, be measured directly.

5.2 High Velocity Impact

 

When two deformable bodies in motion collide along
a common line of action there is an exchange of energy
consistent with the law of conservation of energy. During
the first period of the impact the bodies come closer into
contact with one another through a compression or deforma-
tion experienced by each, resulting in a fitting together
of the two surfaces over a finite area. Because of the
elastic properties of the bodies, a mutual force is called
into action between them and tends to separate them. Lord
Kelvin (1912), citing Newton's work in this area, states
that provided the impact is not so violent as to destroy
either body, the relative velocity of separation after the
impact bears a proportion to their previous relative
velocity of approach, which is constant for the same two

bodies. This proportion, always less than unity, approaches

SN

unity the harder and more elastic the bodies are. The
proportion, denoted by e is called the coefficient of
restitution. If two bodies (Figure 11) moving with

velocities V and Vi1 respectively, collide and separate

i
at velocities Vf and VI,1 respectively, then by Newton's
definition.
Vfl— Vf
Coefficient of restitution e = -—___——T
V. - V.
i 1

Now consider the case shown in Figure 12 where a
mass on a rotating arm is about to strike a stationary

bean. The following assumptions and definitions will be

made.

Vai = initial velocity (peripheral) of the mass
on the arm.

Vaf = final velocity after impact of the same
mass

Vbi and be = initial and final velocities of

the bean.
m = mass of the bean.
e = coefficient of restitution between the bean

and the steel mass.
Assuming that the impact does not slow down the rotating

arm, the arguments used by Lamp (1959) are valid here.

Vai = Vaf = Va

9v, 55 9, v;

(a) APPROACH

00

(b) IMPACT

<::E>—’Jvf (::23}+-V¢
(c) SEPARATION

Figure ll.-—Impact of two moving bodies.

 

Figure l2.——Impact of a lightly held bean by a heavy
rotating arm.

_m

5H
"’3

tux
of

and

and

thus

56

Assume that the bean is originally at rest.

Vbi = 0

u
<

Let be b

By the restitution equation

V - V

af bf = "8(Va1 ' Vbi)

Applying the above assumptions the restitution equation

becomes

Vb = Va (1 + e)

If the bean were hard and perfectly elastic it would re—
turn all its internally absorbed energy; its coefficient
of restitution would be unity. In this case the final

velocity of the bean would be

1 -
Vb - 2 Va
1 _ _
Vb — Vb — Va(l e)
1 -
and Vb + Vb - Va(3 + e)

'. (vb‘)2 - (Vb)2 = va2(1-e) (3 + e)

and the energy absorbed by the bean and not returned is

thus

57

E = % m{(vb1)2 _ (Vb)2}
E = i m V 2(l - e)(3 + e)
2 a

In his research on beans, Perry (1959) determined
e for bean-steel impact. He gives a value of 0.57 which
he states is quite independent of the moisture content of
the bean within the range of ll percent to 18 percent
moisture, w.b. Using Perry's value for e then, the energy

absorbed by the bean is
s - % m<o.u3> (3.57) Va2

If Va is Just sufficient to cause cracking of the seed
coat of the bean then B can be considered to be the

rupture energy ER
' E - 1 m<1 535) v 2
o o R 2' o a

. _ 2
. . ER 0.767mVa

where Va is the peripheral velocity of the rotating arm
of Figure 12, and m is the mass of the impacted bean.

Thus rupture energy can be quite easily obtained for
the high velocity tests. As in the low velocity tests, im-
pact force can be measured directly with suitable instru—

mentation.

VI. SUMMARY OF EXPERIMENTAL REQUIREMENTS

Because of the differences in the quasi-static and
dytiamic formulations, as well as those in the experi-
menital instrumentation required, three sets of tests are

required.
6.1 Quasi—Static Tests

6. la Type of Analysis

 

Stability of end-loaded beans under the influence

Of‘ an applied load sufficient to cause buckling.

6..lb Working Equations

 

(25 + c - 2k) (-2B - 2c - k) c

1. (-2b — 2c + k) (a + no - 2k + 2) (-2c - 2 + k) 0

2c 2(-2c + k — 2) 2(c + 2 — k)

hfliere B and C are the fractions relating the moments of
iJiertia of the cotyledon at 1/6 L, and 1/3 L along the

cotyledon to the moment of inertia, 1M of the center, and
1

 

where Pcr is the critical applied load

58

59

A is the grid, A = L/6

L is the length of the cotyledon and E is the modu—

1118. of elasticity of the bean.

(2D + 1 — k) (—2D - 2 + k) 0

(-2d — 2 + k) 2(2D - 2 — 2k)

wfiuexre D is the fraction relating the moments of inertia

eat «%-L and l L along the cotyledon and k is defined as

2
before except that A = L/U.

FEirially the extrapolation equation will be used.

hfldezre the f and c refer to fine and coarse grids.

6 . 1c Quantities
232__Be Measured

 

These are: B, C, D, IM, L and Pcr'

61.1d Objectives
Computation of EIM and E. Compilation of the

cJPitical loads for beans of various moisture contents.
6.2 Low Velocity Impact Tests

6.2a Type of Analysis

Energy balance analysis of a weight falling under

gravity and impinging upon a stationary, supported bean.

60

 

 

 

6..2b Working Equations
E = 3”— AV(2V — AV) + Wt(V — l AV)
B 2g 1 i 3
‘wrnere W is the falling weight,
g is the acceleration due to gravity,
AV is the change in velocity of the weight
during impact,
ER is the rupture energy of the bean.
Vi is the impact velocity
t is the duration of impact up to rupture.
6 .2c Quantities
rTo Be Measured
These are: Vi’ AV, t, assuming that W and g are
1&riown.
53. 2d Objective

 

Computation of rupture energy, ER’ for beans of

'Vtarious moisture contents, and compilation of the maximum

ianact force F

6.3a

R’ which can be measured directly.

6.3 High Velocity Impact Tests

 

Typg of Analysis

 

Energy balance analysis using the known restitution

properties of beans.

61

 

 

 

6. 3b Working Equations
m = , 2
ER 0.77 m Va
where ER is the rupture energy,
m is the mass of the impacted bean,
Va is the peripheral velocity of the impacting
arm, and,
0.77 is a constant arrived at taking the coeffi-
cient of restitution of beans into account.
6 . 3c Quantities
‘Ico Be Measured
Va, m.
6 . 3d Objective

 

Computation of E for various moisture contents,

R

aliCi compilation of the maximum force F which can be

R

me asured directly .

VII. APPARATUS

For preliminary qualitative testing a Valve—Air
Ilriit of the type developed by Mohsenin (1963) was used.
CFrie applied quasi—static loads were measured by a Sanborn

ESCD—pound strain gage load cell. The unit is shown in
I?fi_gure 13 (left).

7.1 Quasi-Static Tests
As the theory of Section 5.2 was being deveIOped,
tide experimental procedure for determining the required

\faidables was designed around an Instron Table Model Test-

iiqg Machine. Figure 13 shows the general layout of the

zapparatus. The cross—head of the Instron was slightly
Inodified so that the test beans could be loaded with a
3/8 inch diameter cylindrical probe. In this way the

‘test bean could be positioned in the upright position

Vvith a small disc as shown in Figure 14. The cross—head

vvould then be lowered manually just enough to hold the

Ibean upright for the start of the test. Test loads and

Cieformations were recorded on the Instron chart in the

Iiormal way.

During each test, the bean was kept under constant

Observation with the help of a Bausch & Lomb stereo

62

63

 

Figure l3.——General layout of the testing area with
the Instron machine (right) and the
Valve—Air unit (left).

 

Figure lA.--Location of the test bean.

614
mi (2 roscope .

The first appearance of a crack in the seed

cc>£1t signified the end of the test.

7.2 Drop Tests

For these tests a Drop Tester was designed and con-

stzzructed. Figure 15 shows the complete test set up. The

axoroaratus consists of a hollow cylindrical drop weight

(2)* to which the probe (3) used in the static tests is

afizizached. The path of the falling weight is guided by a

23—cinch diameter plexiglass tube which is slit down one
ssixie to allow the passage of a cable connected to an

exacelerometer in the drop cylinder. An electromagnet (l),

Ibcnnered by a d-c supply (6), was used to raise the weight
to the required drop height. A Sigma 8P3 Photorelay, (5),

eunergized by a Sigma 8L3 Light Source, (A), was used to

‘tzrigger the OscilloscOpe as well as to indicate the time

tuaken for the cylinder to go by. The bean to be impacted

Vvas positioned on a load sensing device which consists of
at quartz load cell sandwiched between two thick mild steel
Ciiscs (Figure 16).

The impact deceleration of the falling weight was
Ineasured by a Piezotron Model 818 quartz accelerometer,

“filich requires no charge amplifier. The impact force was

Ineasured by a Kistler Model 912 quartz load cell. The

Esignal from the load cell was fed into a Kistler Model

‘

l *The numbers in parentheses refer to those in Figure
5.

 

Figure 15.--The drop test apparatus.

c

mfimthNH

Figure

Electromagnet

Drop weight

Probe

Light source for photorelay
Photorelay

D—C power supply

Trigger control box

Load cell assembly

 

l6.-—Location of the test bean.

’3
O\

503M7‘ charge amplifier (nOt shown). The control box

{7) : 2w; in Fig-:e ;5 contains circuitry for de—energizing
the e;e-:romagnet, and a 15 ;olt battery whose output
goassed through the normally open points of the photorelay,

\Vhen the light was interrupted by the falling weight, into

conannel No. i of the oscilloscope.

-J
.3
l)
O
(n
()
I...
l-J
I’—
(
U
( )
U
T1
(D
C
(1]
(D
Q.)
2
DJ
0]
n)
F3
(D
x.
('1
"3
O
:3
H“
><
*3
<4
'U
(D
U‘I
J:
KO

0

(I

'~torage model, set to trigger on the battery input into

<ohannel No. 1. The accelerometer and amplified load cell

«outputs were fed into channels No. 3 and No. 2 respec-
‘tively. The length of the battery trace on channel No. 1

:indicated the time taken by the 2-inch long cylinder to
{bass the photorelay, thus providing a measure of the impact
\Jelocity. A Hewlett Packard Oscilloscope camera was used,
vvith a special adapter, to photograph the stored traces
:for some tests. Because of the storage capability how-

eever, the measurements were mostly taken directly from the

roatterns stored on the screen.

5 the impact apparatus developed by

{—r

For these tes
.Btcrkhardt (1969) was used. The apparatus consists of a
5 lop electric motor powering a flywheel shaft through a
VEtriable speed belt drive. An aluminum beam impact arm,
mOunted on a shaft in line with the driven shaft could be

erlgaged with the latter by means of an electric clutch.

67

True system was designed such that the impact arm attains
the Speed of the driven Shaft in about 270 degrees of motion.

At the extremity of the impact arm, which is 18
ixocfloes long, a Kistler Model 901A load washer mounted
toeioween two steel discs (Figures 17, 18) is used to mea-
SLirwe the impact force. The test bean was held upright
toe13ween two thin strips of tape as shown. After impact,
tfloez bean was caught in a well padded catch box.

Impact takes place after.about 280 degrees of arm
renoation. About 10 degrees after impact, a cam Operated
svuitch disengages the electric clutch and energizes a.
brwake which stops the arm. The total arm rotation in a
teest is a little more than two revolutions. Thus the
Siignal cable is run through the arm and out through a
hcble near the shaft and thence to a Kistler Model 503 M15
Ctiarge amplifier. No slip rings are used.

The amplified signal is fed into the Tektronix
fStorage Oscilloscope. The latter is triggered by the

<3utput of a Piezotron accelerometer mounted on the impact

arm, near the load washer.

7.A Auxiliary Equipment
A Gillings-Hamco thin-sectioning machine was used to
section beans at the required positions for computing
values of the inertia factors B, C and D.
Moisture contents were determined by oven drying at

10140 C for A8 hours. For this, a Freas Model 625 forced

68

 

Figure 17.-—High velocity impact apparatus.

5 hp electric switch
Variable speed pulley
Tachometer

Flywheel

Electric clutch

Electric brake

Cam actuated limit switch
Brake release switch
Impact arm

\OWNOU’IJZ'UUNH

 

Figure l8.-—Location of the test bean.

69

(draft precision drying oven was used. Weighings were made
on a Seederer—Kohlbusch precision balance.

For the high velocity impact tests, some rewetting
of the test beans was necessary. For this, a small con—
ditioning chamber was constructed, and its internal en-
vironment was controlled by an Aminco-Aire conditioner.
The conditions within the chamber were monitoried with a

Hygrodynamics Model 51-3001 hygrometer indicator.

7.5 Calibration of Equipment

The Instron load cell as well as the dynamic load
cells were calibrated in place before and after each test
run using the Instron calibration weights.

The accelerometer calibration was checked by bolting
it to a Wilcoxin Model SR—AA quartz accelerometer and
vibrating the two in the range of 1000 to 2000 hz. on a
small shaker. The Wilcoxin accelerometer had previously
(5/1/68) been sent back to the manufacturer for recalibra-

tion.

VIII. EXPERIMENTAL TECHNIQUE

in September, 1968, a series of quasi—static tests
were conducted on two lines of navy beans which were
developed by the ichigan State University CrOp Science
Department. These lines were designated No. 70 and No. 7”.
Dynamic tests were conducted on a commercial variety,

Seafarer navy beans, during September and October of 1968.
8.1 Quasi—Static Tests

8.1a Harvest

Four harvests, two for each line were made on four
separate dates. The period between the first harvest of
line 74 and that of line 70 was one day. The same applied
to the second harvests. The period between the first and
second haTV’SE for each line was two weeks.

Because of staggered planting dates the first set
of harvests yielded beans with green, yellow and white
(ripe) pods, making a wide variety of moisture contents
available. Between the time of actual harvest and the
start of a test the beans were kept in the pods in a
refrigerator. The second set of harvests yielded only

yellow pods and ripe beans.

70

71

8.1b Test Procedure
(Figure 19)

 

 

For each line and each pod color, nine beans were
selected and tested as soon as possible after harvesting
and shelling. The nine beans were considered as three
3-bean samples. Each 3-oean sample was tested at one of
three Instron cross-head speeds. These speeds were 0.05,
0.1, and 0.2 ipm. The use of three cross-head speeds was
necessitated by the possibility that time dependent pro-
perties of the beans could affect the critical loads in
the quasi-static range.

The rest of the shelled beans, in their labelled
containers, were left out to dry in the laboratory. After
72 hours, another set of samples was chosen and tested.
Another test was conducted after 1AA hours.

Each 3—bean sample, after being tested at its
designated cross-head speed was sealed immediately in a
small plastic bag and was used at the end of the entire
test for that particular line, color and time, for moisture
content determination.

For each bean, a test consisted of loading the bean
on end at its designated loading rate until a crack ap-
peared in the seed coat. At this point the loading was
stopped. The load thus obtained was considered the criti-
cal load for the bean, and therefore, twice the critical

load, PC for each cotyledon. The strength of the coat

r,
was considered as part of the cotyledon stiffness.

72

Harvest Date

 

 

5(1) (D "S Q___.._..._.

 

 

 

 

 

 

70 74
Y D D Y
e r r e
l y y l
l , l
o Drfl//,/ o
w w
0 72 114
(f) (2|) (1)
9 Beans 9 Beans 9 Beans

0.05 0.1 0.2

\1/

9 Beans - Moisture Sample

SWWWQ

Line

Color

 

Repeat
for the
Others

Hours
After
Harvest

Set

Beans
per Set

Cross-
head
Speed,
ipm

Figure l9.—-Schematic of the experimental design.

73

8.1c Moment of
Inertia Factors

 

For each line of beans, considering the mature
beans only, three samples of five beans each were sec—
tioned at distances of %, %, % and g from one end; L
being the length of each bean. The major and minor dia-
meters of each section was measured. The values were
averaged over the 15 measurements for each position of
sectioning. The average moment of inertia was computed
for each position of sectioning of each line. The ratios
B, C and D were then computed for the cotyledons. Thus

for the beans used for the Instron tests only the lengths

and the diameters of the center were measured.

8.1d Follow-up Tests

 

Impact tests were conducted during September and
October of 1968 on a commercial line of Seafarer beans.
For comparison purposes, Instron tests, using only one
loading rate, 0.2 ipm, and two moisture contents were made
on beans of this line also. For each moisture content a

sample size of ten beans was used.

8.2 Low Velocity Impact Tests
Beans of the Seafarer variety were used for these
tests. Because of weather problems during harvest, only
one harvest was made. By allowing the beans to dry in
the pods in the laboratory, however, four moisture con—

tents were obtained for testing. In addition, beans that

74

had been allowed to dry down to 11 percent moisture were
re-conditioned to 17 percent to see whether this re-
conditioning had any effect on the impact resistance of
the beans. This was done primarily because much of the
high velocity impaCt testing was done on rewetted beans.
For each test, at the given moisture content, a
drop height was found, by trial and error, at which
cracking of the coat took place. Ten beans were then
selected and tested. Acceleration, impact force and im-
paCt velocity were measured for each bean and the results
averaged. Impact energy to cause rupture was computed.
At the higher moisture contents, because the check-
ing was not severe, the tested beans were then used for
moisture content determination. At the very dry condition
(11 percent) however there was complete splitting of some
of the test beans. For this case ten other beans, kept
under identical conditions but not tested, had to be used

to determine the moisture content.

8.3 High Velocity Impact Tests
Because of the difficulty in relating the two pre-
XLiously described tests, high velocity tests were per-
fkormed. By the time the tests were designed only very
(try beans were available for testing. Accordingly, in
Enidition to the tests on the dry beans, rewetting was
Carided out using the Aminco-Aire Unit in order to attain

two other moisture content conditions.

75

The tests were carried out in a manner to those for
low velocity impact. For each moisture content, the
velocity of the impact arm required to initiate seed coat
cracking was found by trial and error. Then a ten-bean
sample was tested and the impact force measured. Checks
were made to ensure that the tape supporting the test
bean was not affecting the test results by impacting the 5
tape only. In all cases, the tape broke free from its

attachments without eliciting a response from the load

 

cell.

For the very dry beans, even the lowest velocities
at which the apparatus could operate were sufficient to
cause severe checking and splitting of the beans. Thus
tests were carried out at three moisture levels only. As
was the case with the dry beans of the low velocity im-
pact tests, dry beans, because of the severe checking were
not used to determine moisture content. Instead, ten
beans which had been stored under the same conditions but

Mniich were not themselves impacted, were used to determine

tfloe moisture content.

IX. RESULTS AND DISCUSSION

9.1 Moment of Inertia Factors

 

The dimensions measured for the three 5-bean samples
were averaged and the moment of inertia was then computed
for each position or station along the bean. This value
was then divided by two, assuming equal sized cotyledons,

and the parallel Axis theorem was then used to compute

 

the moment of inertia of each cotyledon.

The value of the factors B, C and D were then com-
puted. These are presented in Table l. The beans used
for this determination were at 13.5 percent w.b. moisture
level which was found to be the best moisture content for
sectioning the beans without tearing or shattering. The
assumption made here is that the bean will maintain its
shape, even though it changes slightly in size, as the

moisture content changes.

TABLE l.--Average values of inertia factors B, C and D for
bean cotyledons.

 

 

Station Moment of inertia Ratio
along of cotyledon I/I
bean in.“ x 105 M
L/6 1.17 B = 0.37
L/3 2.67 C = 0.8“
L/A 1.53 D = 0.A8

Center 3.18 1.00

 

76

77

Thus these values of B, C, and D were used for all
the computations involved in the quasi—static data reduc-
tion.

9.2 Critical Loads and Stability and

Elastic Moduli for Beans
Loaded Quasi-Statically

 

 

For each moisture content, no effect of rate of load-
ing was evident. This is in agreement with Zoerb's (1958) ;

conclusion that there is very little viscoelastic effect

 

within the range of loading rates used in this experiment. U
Thus the results for each line, maturity (pod color)

and moisture content, i.e., each 9-bean sample shown in

Figure 19, were averaged. Using the measured value of

critical load Por (half the load which caused the seed

coat to crack), the major dimensions of the beans, and

the values given in Table l, the maximum stability modulus

EIM and the values of the apparent elastic modulus E for

the cotyledon was then computed for each moisture content.

These values are presented in Table 2 and illus-

trated in Figures 20 through 27.

9.2a Critical Loads Pcr

 

Figures 20 and 21 show the effect of moisture con-
tent and maturity on the critical load Pcr' For line 7A
the load increases rapidly with decreasing moisture con-
tent, peaks in the range of 12.5 - 13.5 percent moisture

range but as can be seen in Figure 21, these differences

78

TABLE 2.——Critical loads, elastic and stability moduli
for two lines of navy beans.

 

 

Harvest Hours of Matur— Lire Moisture Pcr EIM E _
No. drying ity* 1 percent lbs lb-in.2 psiiclo “

1 ~6 G 74 20.1 7.7 0.221 0.98

Y 74 18.2 11.5 0.342 1.54

R 74 16.1 16.8 0.531 1.90

72 G 74 13.5 18.1 0.474 2.84

Y 74 13.2 20.0 0.544 2.64

R 74 13.5 18.8 0.621 2.68

144 G 74 12.0 16.8 0.414 2.26

Y 74 11.8 21.9 0.547 2.16

R 74 12.0 19.3 0.606 2.13

4 ~2 Y 74 16.2 17.3 0.487 1.66

R 74 13.4 19.3 0.636 1.67

48 Y 74 12.8 19.2 0.552 1.86

R 74 12.2 19.8 0.650 1.76

144 Y 74 10.8 16.9 0.464 1.48

R 74 10.8 16.7 0.508 1.35

2 70 G 70 28.2 2.8 0.115 0.23

Y 70 26.3 3.6 0.167 0.26

R 70 26.0 3.8 0.165 0.30

72 G 70 14.0 18.0 0.409 1.59

Y 70 13.5 18.5 0.449 1.42

R 70 13.5 18.2 0.454 1.51

144 G 70 11.8 18.8 0.413 1.42

Y 70 11.7 18.4 0.465 1.45

R 70 11.7 18.8 0.466 1.48

3 70 Y 70 16.4 15.4 0.346 1.50

R 70 13.2 19.8 0.502 1.49

48 Y 70 13.5 17.0 0.379 1.43

R 70 12.5 19.0 0.482 1.43

144 Y 70 10.5 17.0 0.329 1.40

R 70 10.5 17.5 0.406 1.31

 

*The letters G, Y and R designate beans
yellow and ripe pods, respectively.

from green,

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['1

81

tend to disappear as the beans dry out further, for the
yellow and ripe beans. The beans from the green pods be-
come much weaker at the low moisture contents. The
similarity of results shown in Figures 20 and 21 for the
low moisture range indicates that the extra two weeks of
maturing on the plant had little effect on the critical
load.

The beans of line 70 were much wetter than those of
line 74 at the first harvest, and at the high moisture
contents, yielded a different result. Figures 22 and 23
show the variation of For for the two harvests. The
critical loads start from a much lower point but eventually
attain approximately the same maximum value, 19 - 20 lbs,
in the 12.5 - 13.5 percent moisture range. Thus lower
rates of increase of Pcr were found.

It is possible, however, that had the harvest been
made one week later that the curves may have been more
similar to those for line 74, i.e., an extra week of
maturing on the plant may be critical at that stage. The
second harvest yielded beans of which those in yellow pods
were found to be weaker than on the first harvest, at
about 13 percent moisture. This was the case also, but
to a lesser extent with the beans of line 74.

For both lines of beans the optimum moisture con-
tent for withstanding axially compressive loads was 13

percent w.b. At this moisture level the critical load

"53 7 ‘ 1.4.!‘2‘...

82

 

L
l5 IO

20
MOISTURE CONTENT, PERCENT w.b.

 

 

.7
.4
s -3
20
0°- .
is
uI-3u1 ‘
.1
31:19.
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III
0’71 ..
i 1 l l l l l 1 l T
o m N It °
N - Q

'scn ”a ‘cvm woman

urity and moisture
st harvest,
line 70.

PS4

D's-z."
r"

-fi‘inf.

I“

‘72:--

83

.0s eeaa .weaesoa
momma whooom .mcmmb mwm:
o h

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whopmfloe U.

c a
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84

to produce cracking of the seed coat was between 18 - 20
pounds for a single cotyledon, i.e., 36 - 40 pounds for
a bean.

In most cases the cracks on the seed coat started
at one extremity of the bean and progressed around the
hylum. This is because the coat is thickest near the
hylum and less susceptible to cracking there. For the
very wet beans however (26 percent w.b.), the tissue in
the center of the hylum appeared to be very soft and the
cracks started there.

For each moisture level some beans were loaded on
the side as a check. At the lowest moisture level, 11
percent, the side loaded beans were found to break at
loads very little lower than those required to crack the
coats of end loaded beans. Thus, when very dry,beans be-
come susceptible to mechanical damage from both side and
end loads.

9.2b Stability and
Elastic Moduli

 

 

The variation in E1 values with moisture content,

M
line and maturity are shown in Figures 24 and 25. This
variation is very similar to that for critical loads,
except that at the lower moisture contents there is a
definite ranking according to maturity, with the ripe
beans exhibiting the highest values and those from green

pods, the lowest values. Also, line 74 beans were found

85

 

 

 

j<3
.4
K' -
V’
I“ .
‘é
o _.2
_(3
N
(I)
(no _ID
00 N
CDIL
o. _
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53
”45..” 4h
mm-
<9>m _
a II
o m. 1
L I l l l l I ll L
0. O Q 0 Q o
9 <0 V; N

0
ouxz'm's‘l‘wla ‘smnoow All'IIQVIS

MOISTURE CONTENT, PERCENT w.b.

24.——T

Figure

I! “I

.0 .. u u
I: 7.713

 

.mm .mcmop z>mc mom p.6ucoo
Hiflwl

esopmflofi bum mu. emu npflz sz we so pm

.n.) kzmomma .FZMHZOO NEH—.902
o. 0. ON
d

 

 

C)

0.
N

C) C)
«5 v‘
on raw: 's1‘”13 ‘smnaow All‘llBVlS

0.
a)

87

to have higher EI values in the optimum moisture content

M
range of 12.5 - 13.5 percent w.b.

This difference between lines becomes very apparent
upon examination of the curves for elastic modulus E,
shown in Figures 26 and 27.

The optimum moisture content in this case was found
to be about 14 percent w.b., at which point the average
E value for line 74 beans is 27,500 psi., almost double
the 15,000 psi. value for line 70.

This difference in E values arises out of the dif-
ferences in moment of inertia caused by small differences
in major dimensions of the two lines.

The relative shapes of the two curves are similar
to those of RIM and For except that the ranking according
to maturity is no longer definite. The decrease at low
moisture content is preserved.

0f the two quantities EI and E, the former is the

M
more significant measure of the beans ability to with-
stand a buckling type of failure while the latter is a
material constant which would be more useful for com-
puting deformations of flat-loaded beans.

Because of the close mathematical relationship be-

tween Pcr and EI the similarity of these two sets of

M:
curves was expected. Any differences in shape would be

due to small differences in the lengths of the test beans.

 

.pmo>nm£ pmswm .mnmmm meme Low pcopcoo
pa,

)

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O. 0N. m. w.b. ON n.NN mN nNN On
4 _ fl _ d — _ _
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9
8

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“3 u mo cowumflpm>nl.nm mammau

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m. cm mu 0»

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moon mazm - m
moon 3044m> u > 100.0

 

,-OI"ISd 'Sfl‘IflOOW OllSV‘IB lNBHVddV

90

9.2 c Follow up Tests

Two tests were run each on lO-beam samples of Sea-
farer variety navy beans. Two moisture contents, 15.0
and 13.3 percent w.b. were used. Only critical loads Pcr
were measured and these averaged out to be 17.1 lbs. at '
15.0 percent w.b., and 18.5 1bs.at 13.5 percent w.b.

These results are in good agreement with those for the _m

beans tested the previous year (September, 1967). 7

9.3 Dynamic Forces and Energy

The dynamic rupture force FR and energy ER are
presented in Tables 3 and 4 for the drop tests and high
velocity tests, respectively. They are the values re-
quired to cause cracking for the whole bean.

As expected, the dynamic forces to cause cracking
were lower than the quasi-static forces at corresponding
moisture contents. Because of the fact that for the
drop tests the beans were restrained at their bases, the
drop test rupture forces were very much lower than those
for the high velocity impact (Figure 28).

As can be seen in Figure 28, the variation of
dynamic rupture force with moisture content was somewhat
similar to that for critical quasi-static loads, with
lower forces required to cause skin rupture at both the
high and low extremities of the moisture content range.
The optimum moisture content, deduced from Figure 28, was

found to be about 14.5 percent w.b., for both dynamic

91

TABLE 3.~-Impact velocities, rupture forces and energy
for navy beans subjected to drop tests.

 

 

Moisture Drop height Velocity Energy ER Force FR

percent w.b. in. fps ft-lbs lbs
16.9 8.5 6.6 0.050 9.0
15.2 7.5 5.5 0.055 10.1 my
13.5 6.0 3.7 0.04m 10.0 E
11.5 3.0 2.4 0.015 7.2

I

15.5 7.5 5.4 0.053 10.0

(rewetted)

 

TABLE 4.--Impact velocities, rupture forces and energy for
navy beans subjected to free, high
velocity impact.

 

 

Moisture Axéigfiin Velocity Energy ER Force FR

percent w.b. lbs x 10- fps ft-lbs lbs
17.4 0.595 47.1 0.032 21.4
15.6 0.577 55.0 0.042 23.6

13.4 0.560 55.0 0.041 23.2

 

FR: Lbs.

RUPTURE FORCE,

N-hangNJh

IO

8
6
4
2
O

h‘I tilil‘

92

, , ’ {—— _ — E T‘fl
“0” HIGH VELOCITY IMPACT TEST .
t

 

- O/Y DROP TEST

 

l l 1 1 l L 1

I7 I6 l5 I4 I3 l2 II
MOISTURE CONTENT, PERCENT w.b.

e 28.—-Comparison of dynamic rupture force F“ for
navy beans, determined from low and high
velocity impact tests.

93

tests. This value is somewhat lower than the 17.5 percent
w.b. predicted by Solorio (1959).

The maximum rupture forces at this optimum moisture
content would be 10.“ lbs and 23.8 lbs for the drop test
and the high velocity impact test respectively. The maxi-
mum impact velocities were 6.60 and 5H.97 fps. The mini-

mum impact velocities required to cause rupture were 2.uu

3

fps with the drop tester for beans at 11.5 percent moisture

and “7.12 fps for the rotating arm for beans at 13.“ per-
cent moisture.

The values of forces and velocities given above are
those sufficient to cause cracking of the seed coats of
more than 50 percent of the impacted beans.

Figure 29 gives the variation of rupture energy ER
with moisture content for the two dynamic tests. Since
the strain energy required to cause rupture under dynamic
conditions should be invariable, there should be close
agreement between the rupture energies measured by the
two tests. It does appear, from Figure 29, that there is
reasonable agreement at the lower moisture contents (12.5
percent difference at 13.6 percent moisture). There are,
however, large differences in the two results at higher
moisture contents (25 percent difference at 16 percent
moisture). The two most likely causes of these differ-

ences are (a) strain rate or viscoelastic effect

II Iii-iv“-

9U

   
     

 

 

0.06 -
DROP TEST if"
. 0.05— I
m I
D I.
T'
I 0.04- /,”°’—~“‘n
“If / K~IIICI-I VELOCITY
// IMPACT TEST
" D
25 0.03 -
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u:
i!
u:
m 0.02 I“
a:
:3
F-
% 0 o.
a: .
0 Ti 1 1 l L J 4 1
l7 l6 I5 I4 l3 l2 II

MOISTURE CONTENT, PERCENT w.b.

I".I.I"_l.ll’(‘. :“I.--(fornpar1son of‘ rupture energy I9.” I‘or navy beans,
determined from low and high velocity impact tests.

95

exhibited by the beans at high moisture contents, and
(b) inaccuracies in the drop test apparatus.
Viscoelastic behavior of the beans under load, al-
though negligible under quasi-static conditions, could
occur under dynamic conditions. Such behavior would be
manifest by a greater resistance to deformation under
high loading rates, and would result in lower strain m

energy in the high velocity test.

L‘TT—T‘FZT 1T3?

The second reason for the discrepancy shown in
Figure 29 lies in the nature of the drop test apparatus.
In a drop test, unlike the quasi-static or high velocity
test, the load is still being applied after cracking has
occurred. When the beans are dry and thus have a high
EIM value, this is no problem since the bean withstands
this load with no further damage. When the beans are
wet however, (16 percent moisture and above), the checked
beans continue to deform under the continued load. In I
this case there is some difficulty in determining from
the deceleration curve the exact point at which cracking
of the coat occurred. Hence an important source of pos-
sible error.

A smaller diameter drop tube, with a shorter and
lighter drop weight would probably have yielded results in
closer agreement with those of the rotating arm tester.

As was the case with impact rupture forces, the

rupture energy decreased with decreasing moisture

96

content, in the range of 11.5 to 14.5 percent moisture.
This extends Tabiszewski's (i968) findings in bean thresh-
ing tests in which visible damage increased, at a given
combine cylinder speed, with decreasing moisture content
in the range of 10.6 to 12.3 percent moisture.

9.“ Extrapolation of the Results
to Field Conditions

——
n 'u- A.
o

The results of the quasi-static tests cannot be

2;"“'" . .9; -_

immediately applied to actual conditions since the dis-
tribution of forces between individual beans stored in
deep silos or in sacks is not yet known. The quasi-static
tests have, however, demonstrated that beans above 11
percent moisture content w.b. are most susceptible to
checking, or cracking of the seed coat, when the load is
applied on end. This fact was used in the dynamic tests
to determine the minimum impact forces and energy required
to cause checking. On the basis of the results obtained
from the dynamic tests, deductions concerning bean damage

can be made for two field conditions.

9.Ha Threshing

From the results of the high velocity tests, as
shown in Table U, the impact velocity between individual
beans and a rigid structure such as a cylinder bar must
be below 55 fps when the moisture content of the beans is
between 13.4 and 15.6 percent w.b., and below M7 fps when

the beans are at 17.u percent w.b. For beans whose

moisture content is of the order of 11.5 percent w.b.,

the limiting velocity of such impacts can be computed
using the rupture energy obtained at this moisture content
by the drop test (0.015 ft—lbs), and the restitution equa-
tion developed in section 5.2. This procedure is valid
here since there is reasonable agreement between the two
dynamic tests at the lower moisture levels. The computa-

tion yields 30 fps as the limiting impact velocity be-

} ('3 I’thgl 'As A 1

tween a cylinder bar and beans at 11.5 percent moisture. ;
Lamp (1959), in a threshing study, concluded that a

combine cylinder speed of 37 fps was sufficient for com—

plete threshing of grains. While this is a safe speed for

beans above 13.4 percent moisture, harvesting of beans at

11.5 percent moisture would require a compromise between

checking of the beans on the one hand, caused by speeds

above 30 fps, and threshing losses on the other, caused

by speeds below 37 fps.

9.ub Handling
The limiting velocities discussed above also apply

to the case where moving beans impact rigid, stationary
surfaces. Such conditions arise when beans are poured
into deep silos. Perry (1959) found that beans, dropped
from rest, attained velocities of about 25 fps after a
free fall of 11 ft, and about 46 fps after a free fall of

H5 ft.

98

Combining Perry's results with those of the two
dynamic tests conducted in this study, it may be concluded
that beans may be safely dropped into silos 45 ft deep,
when the moisture content is between 13.4 and 15.6 per-
cent w.b. When the moisture content of the beans is
higher (17.4 percent w.b.), or lower (11.5 percent), there

will be cracking caused by the impact after the 45 ft drop. s

X. SUMMARY AND CONCLUSIONS

10.1 Summary

Three types of tests were conducted on navy beans
in order to determine the loads and energies required to
produce checking or cracking of the seed coat. 8

A series of quasi-static tests were run on two
varietal lines of beans at three stages of maturity.

The beans were loaded individually on end and the criti—
cal loads required to cause cotyledon buckling were
measured on an Instron Testing Machine. With the criti-
cal loads a finite difference stability analysis was used
to compute elastic and stability moduli for the test beans.

In order to increase the range of test moisture con-
tents the beans were harvested twice at high moisture con-
tents and allowed to dry in the laboratory for periods of
O, 48, 72 and 144 hours before testing.

An optimum moisture content range of 12.5 to 13.5
percent moisture w.b. was found for which the critical
loads to cause cotyledon buckling was between 18 and 20
lbs.

At very low moisture contents, 10.5 and 11 percent

w.b., not only did the critical loads decrease but the

txeans became susceptible to damage from side loads as well.

99

100

Two series of dynamic tests, a drOp test and a high
velocity impact test were conducted on individual beans
which were restrained for the former test and free to move
in the latter.

Forces, FR’ required to cause checking were measured
with dynamic load cells, and the accompanying energy ER
absorption by the impacted beans was computed.

An optimum moisture content of 14.5 percent was

-‘r . ;~'_‘|‘_‘..

found at which the dynamic loads required to cause check-

ing were at their maximum values for both sets of tests.

10.2 Conclusions

1. In the range of 11.5 to 28 percent moisture con-
tent w.b., no discernible differences arise in the criti-
cal loads required to cause cotyledon buckling of end
loaded beans when the rate of loading is varied from 0.05
to 0.2 ipm.

2. At moisture contents above 11.5 percent w.b.,
checking of bean seed coats is more likely to be caused
by end loads causing outward buckling of the cotyledons
and consequent tensile rupture of the seed coat.

3. The Optimum moisture content at which beans can
resist buckling under end loads is 13 percent w.b.

4. Beans, when very dry (below 11.5 percent w.b.)
become very susceptible to mechanical damage to both
seed coat and cotyledons caused by static end and side

loads and by dynamic loads.

101

5. For dynamic load conditions similar to those
exiSting in a combine cylinder, the optimum moisture con—
tent for lowest incidence of Checking is 14.5 percent
w.b. At this moisture content the velocity of the rigid
body impacting the bean should be kept below 50 fps, and
the strain energy imparted to the bean should be less
than 0.04 ft—lbs.

6. For beans below 11.5 percent moisture content
impact velocities as low as 30 fps will cause checking
and splitting.

7. In theory a simple drop test with an instru-
mented falling weight impacting a stationary bean should
be able to give the energy required to rupture the seed
coat. This energy could then be used with the simple
restitution equations to obtain limiting velocities for
high velocity, free impacts. Complications arise however
when the drop weight is heavy enough to cause continuing
deformation of the bean cotyledons after checking has
occurred. This places a serious limitation on a drop test
apparatus especially when the beans are at high moisture
levels.

8. Beans may be safely poured into deep silos (45
ft), or threshed at the cylinder periperal speed found to
be the best for seed separation, when their moisture con-

tent is in the range of 13 - 15 percent w.b. At a

‘T‘T’lfia”: 37'. 1.51““.- E

102

moisture content of 11.5 percent or lower however check-
ing will occur in both cases, and some compromise will

have to be made between combine efficiency and checking.

‘ _. A. A;
+-

r-

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