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1960

The evaluation of the influence of large aggregate on the The evaluation of the influence of large aggregate on the

permeability of single-grained soils permeability of single-grained soils

Edward Clayton Grubbs

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THE EVALUATION OF THE

INFLU:BlJCE OF LARGE AGGREGATE ON THE

PERMEABILITY OF SINGLE-GRAINED SOILS

BY

EIJdARD CLAYTON GRUBBS

A

THESIS

submitted to the faculty of the

SCHOOL OF MINES AND METALLURGY OF THE UNIVERSITY OF MISSOURI

in partial fulfillment of the work required for the

Degree of

MASTER OF SCIENCE IN CIVIL ENGI?-.l"'EERING

Rolla, Missouri

1960

Approved by

CHAPTER

I.

II.

III.

IV•

v.

VI.

VII.

VIII.

TABLE OF CONTENTS

INTRODUCTION ••••••••••••••••••••••••••••••••••• • •• • • ·• •

REVIEM OF LITERATURE••••••••••••••••••••••••••••••••••

}!ATE.RIALS•••••••••••••••••••••••••••••••••••••••••••••

ii

PAGE

l

·4

7

Fine Single-Grain Soil•••••••••••••••••••••••••••••• 7

Large Aggregate••••••••••••••••••••••••••••••••••••• 9

TF~TING APPARATUS•••••••••••••••••••••••••••••••••••••

Type of Test•••••••••••••••••••••••••••••••••••••••• 12

Description 0£ Apparatus•••••••••••••••••••••••••••• 13

PROCEDURE OF TEST••••••••••••••••••••••••••••••••••••• 19

Preparation or Sample••••••••••••••••••••••••••••••• 19

Mixing or the Sand and Aggregate•••••••••••••••••••• 19

Weights••••••••••••••••••••••••••••••••••••••••••••• 20

Flooding of the Sample•••••••••••••••••••••••••••••• 20

Timing of the Test•••••••••••••••••••••••••••••••••• 21

TEST RESULTS•••••••••••••••••••••••••••••••••••••••••• 22

Test Performed•••••••••••••••••••••••••••••••••••••• 22

Einpirical Relations••••••••••••••••••••••••••••••••• 28

Agreement of Empirical Equations with Observed Data.. 39

SUMMARY•••••••••••••••••••••••••••••••••••••••••••••••

CONCIDSIONS ••••••••••••••••••o••••••••••••••••••••••••

APPENDIX. A

APPENDIX B

•••••••••••••••••••••••••••••••••••••••••••••••••••••

•••••••••••••••••••••••••••••••••••••••••••••••••••••

44

46

47

50

53 54

BIBLIOJRAPHY ••••••••••••••••••••••••••••••••••••••••••••••••••• VITA ••••••.••••••••••••••••••••••••••••••••••••• • • •••••• • • ••• • • •

iii

LIST OF FIGURES

FIGURE PAGE

1. Shape and Size of Large Ap~regate •••••••••••••••••••••• 10

2. Falling-Head Permea..~eter Apparatus Completely Assembled. 14

Disassembled Perrneameter ••••••••••••••••••••••••••••••• 16

4. Mold Mounted on Test Tube Shaker••••••••••••••••••••••• 18

5. Percent Aggregate Versus Coefficient of Permeability

of Sand A••••••••••••••••••••••••••••••••••••••••• 23

6. Void Ratio Versus Percent Aggregate for Sand A••••••••• 25

7. Percent Aggregate Versus Coefficient of Permeability

for Sand B • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • .• • • • • • • • 29

8. Percent Aggregate Versus Coefficient of Permeability

for Sand C •••••••••••••••••••••••••••••••••••••••• 30

9. Percent Aggregate Versus Coefficient of Permeability for

Three to Two-Inch Aggregate••••••••••••••••••••••• 31

10. Percent Aggregate Veruus Coefficient of Permeability

for Observed and Calculated Data Using Equation

25 Three-Inch to Two-Inch Aggregate••••••••••••••• 41

iv

LIST OF TABLFS

TABLE PAGE

I. E~xperimental Data for Sand A•••••••••••••••••••••••••••• 26

II. Comparison of Observed Data and Calculated Data for:

Equations 13a, 14 and 15 ••••••••••••••••••••••••••••• 36

III. Comparison of Observed Data and Calculated Data

Using Equation 25 •••••••••••••••••••••••••••••••••••• 42

ACKNCJIJLEOOMENT

The author wishes to express his sincere appreciation for

the contributions made to the research and writing of this thesis

by the following persons:

Prof'essor John B. Heagler, Jr., tor his constant advice

and encouragement during the resea19 ch and writing of

the thesis.

Professor E. W. Carlton, for his valuable suggestions

a.rd complete review of the thesis manuscript.

Mr. Paul Carlton, Head or Research, Missouri River

Division Laboratory, Corps of .Engineers, Cincinnati,

Chio, who suggested the research problem.

Professor C. R. Ren,ington, for his advice concerning

empirical. relations.

Professor J.M. Vickers and Professor J. Kent Roberts

for their aid in preparing illustrations.

v

CHAPTm I

INTRODUCTION

Eaoh year many earth fill struotures are designed and built

with the primary purpose of impounding water. As these structures

are only relatively impervious, engineers must oonsider the control

of seepage in design. In order to predict seepage, the engineer must

have knowledge of the potential £low or water through the earthen

struoture. He is, therefore, taoed with the problem of determining a

coeffioient of permeability £or use in design ca1oulations.

The coeftioient of permeabi1ity is easily enough determined

for fine-grained materials. However, in construction it is very rare

to find proper fill material which does not include quite a high per­

cent ot large aggregate. In most cases the removal of aggregate in

the neighborhood of six inches, plus or minus, would raise the cost

or construction too high to be reasonable. In design therefore the

engineer is not faced solely with the problem of determining the

coefficient ot permeability for the tine material. Rather he is

faced with the problem ot determining an effective coe£ticient of

permeability for the combined stone and fine-grained soil.

The research included herein was per£ormed to study and

evaluate the influence of large aggregates on the coefficient of

permeability of single-grained soi1s. The results of the research

are applicable to a large number of common engineering problems.

Some or these problems are as £ol1ows:

(a) Often small levees are constructed at relatively low

elevations. These levees are designed so as to be completely

inundated at high water stages. Rapid d.rawdown of highwater can be a

main factor in the instability of slopes, especial.ly if the embank­

ments are saturated. The permeability ot the fill material, along

with other data, is necessary for proper design of slopes.

(b) Determinations of seepage losses from small reservoirs

is a problem otten encountered by engineers. An effective value for

the coefficient of permeability would be quite valuable.

(c) In large earth dams, .flow nets can be used to determine

the seepage loss, but a method for proper evaluation of the coefficient

of permeability would be a helpful tool.

(d) In highway or railroad fills, it is often desirable to

have a good value for the coef£icient of permeability, so that proper

drainage can be provided.

A complete study 0£ the influence of large aggregate on

permeability would involve the investigation or many different types

of aggregate and tine-grained soils. This would necessitate expensive

equipment and a 1arge expenditure of time. Therefore, to place the

research in the realm of practicality, both ti.me and equipment wise,

it was decided to confine the research to single-grained soils, with

a large majority of the grain-size falling'in the sand category.

It was further decided to make use of limited aggregate sizes.

Tests are performed with three separate single-grained soils

used as a matrix to surround coarse aggregate. Three coarse aggregate

samples are used with two or the single-grain, soils and one coarse

aggregate sample with the third. An empirical equation for use in

approximating the coefficient of permeability is formulated. Use of

the empirical equation should be limited to the sand and the size

aggregate tested. The equation is based on permeability of the

matrix material with no aggregate present and is modi.tied as the

percent aggregate added increases.

CHAPTER II

REVIEW OF LITERATURE

A material is said to be permeable if it contains continuous

voids throughout its mass. Ever; soil and rock satisfies this con­

dition, differing only in the degree of permeability of the various

earth materials. Water passes rapidly through sands, while very

slowly through clays. The passing of water through rock may be so

slow that it will evaporate before an accumulation of water on an

exposed surface can be noticed.

The flow of water through a permeable substance is said to be

governed by the empirical relation:

v = ki (1)

first stated by H. Darcy in 1856. In this equation, v ls the dis­

charge velocity, i is the hydraulic gradient and k is a proportionality

constant known as the coefficient of permeability (1).

This coefficient of permeability constitutes an important

property of soil. Its value depends primarily on the characteristics

of the permeable substance. It is also a function of the unit weight

and viscosity or the fluid. Differences in the unit weight of water,

the only fluid with which the soil engineer is concerned, are

negligible. Variations in temperature cause changes in viscosity.

However, these changes can usually be ignored for the ordinary

temperature range of groundwater. It is customary and justifiable

(1) All references are in bibliography.

for civil engineers to regard the coefficient of permeability as a

property of the soil (2).

The value of the coefficient or permeability norwa.ll.y depends

on the size and shape or the individual grains and on the degree of

consolidation o! the soil. The coefficient of permeability is said

to be independent of the void ratio and density, when comparing soils

of different text1J.ral characteristics. The coefficient of permea­

bility is dependent, however, on the void ratio when the same soil is

considered in different states of' compaction (3).

The constant-head and the falling-head permeameters are

standard laboratory equipment for determining the coefficient or

permeability. A complete discussion of permeability equations, their

derivation and use, is contained in the text "Soil Mechanics

Foundations and Earth Structures" by G. P. Tschebotarioff' (4) and

may be found in most soil mechanics textbooks.

Determinations of the coefficient of permeability for soils

or different predominate size characteristics have been performed and

reported by many investigators (5). However, very little is known

regarding the infiuence that the addition of various amounts of large

aggregate has on the coefficient of permea.bili ty. This problem

occurs in most fills, because of the inability to separate fine

materials from coarse materials during actual construction. It is

interesting to note that throughout the existing literature, only one

study has been .found pertaining to the influence that the addition

of various amounts or large aggregate to finer material has on the

coefficient of permeability. The results of this study indicate that

with the addition of large aggregate to a clay loa..1n, the coefficient

of permeability decreases for the first 20 percent aggregate, then

increases very radical.ly. The rapid increase or the permeability for

percentages of aggregate above 20, is due to the difficulty en­

countered in compacting a gravely clay 1oam soil. Leaving relatively

large void spaces will cause a corresponding increase in permeability

(6).

CHAPTER III

MATERIALS

The basic materials used in testing were tine, single-grained

soils and large aggregates.

~ Single-Grain Soil: When choosing the single-grained soil to be

used in the research, several factors were considered. It was be­

lieved that, by selecting a material and re-using it throughout the

testing series, the variation of test results due to changing

characteristics of the filler material would be mini.mi.zed. There­

£ore, a material that could be re-used a number of times without

seriously effecting its general characteristics was desired. To be

re-used, the single-grained soil would be subjected to a number or cycles of wetting and d.r.fing. It would have to be separated,

preferably rather easily, from the large aggregates. The time to

run ea.ch individual test was also a factor to be considered. The

material should be small enough to give a measurable amount of time,

when conducting the falling-head type permeability test. Too small

of a material would require a much larger head than was available.

The shape of the individual single-grains was considered. A well

rounded grain was preferred over the flat ~onger grain, because of

the uniformity with which the well rounded particles could be

placed into the testing apparatus. A final factor which was con­

sidered was the availability of the material to be used.

From several samples of single-grained materials, a sand was

chosen which best satisfied the before named factors. 1fuen the sand

was exatnined under a microscope, the grains appeared to be all .fairly

well rounded with a minimum ot fractures and pitting. The material

was very easily removed from the large aggregate by washing in a pan.

Drying in an oven at one hundred degrees centigrade had no apparent

effect on the individual grains. The larger size, materials, that

which was retained on a nwnber forty sieve, was removed by sieving

and cast awa:y:.

In order to have several values of the permeability as a

basis for comparing data, the sand was divided by ~ieving into two

parts. Into one container was placed the sand that passed a number

forty sieve but was retained on a number sixty sieve,and that portion

of the sand passing the sixty sieve was placed aside for comparative

purposes. When l.ater it was desired to have a third means of

comparison, another sample was made by mixing in equal portion, by

weight, the material retained on the nwnber sixty sieve with that

which had passed the nu~ber sixty sieve.

The re-use of the material did not cause any inconvenience

or difficulties. Approxi.mate1y eighty pounds of each sampl.e was pre­

pared so that several tests could be run at one time without waiting

for the sample to dcy. The drying of the material in ovens was

accelerated by placing the wet samples in shallow pans. The normal

drying time was twenty-four to thirty hours.

~ Aggregate: In selecting a large aggregate for the research,

two primary !actors were considered.

The shape of the individual aggregate cou1d be a prime factor.

Therefore, a material was used which had as little variation as

possible in its general appearance and shape.

The size of the aggregate was a second factor to be considered.

It was believed, and later verified, that by changing the size of the

particles the permeability would vary somewhat. This change, as will

be explained later, is relatively small, but does establish a

definite trend.

The large aggregate 'Which was selected, was obtained from the

Bray Quarry., approximately f'ive miles south of Rolla, Missouri, just

east off of U. s. Highway 63. The material is a crushed limestone of

local origin. As can be seen .f'rom Figure 1 the aggregate has sharp

corners and is fairly regular in shapeo

The large aggregate was separated by sieving into three size

groups: ©n.e group contained aggregate which passed a three-inch

sieve and was retained on a two-inch sieve. The second group passed

a two-inch sieve and was retained on a one and one-half-inch sieve.

The third passed the one and one-half-inch sieve and was retained on

a one-inch sieve.

Ea.ch of these three size groups were divided by weight into

various percentages of the total volume of the mold. These per­

centages, ten, twenty, thirty and forty, were kept separate. F.ach

sand lffien tested with ten percent of the three-inch to two-inch

FIGURE 1

SHAPE AND SIZE OF LARGE AGGREGATE

aggregate would, therefore, use exactly the same particles as the

other size sands. It was believed that this would minimize any

variation due to changes in the shape of the aggregate. After each

test the aggregate was washed and replaced in a container marked

with the weight, percent by volwnen:alilds:siae. It was then allowed to

dry thorough~ before re-use.

_l1 ,·,/",.,

J , .. ~

CHAPTER IV

TESTING APPARATUS

~ !?.! Test: A .falling-head permeameter was used to determine the

coefficient of permeability ot the samples. With this type of

permeameter, a known amount of water passes through the sample in a

measured interval of time, as the head decreases from an initial

value to a final value. The coefficient of permeability is computed

by

K == La At

in which:

K =- the coefficient or permeability in centimeters per second,

L = the length of the sample in centimeters,

a =- the cross-sectional area of the water supply in square centimeters,

A~ the cross-sectional area of the sample in square centimeters,

t = the time interval in seconds,

h1 =- the initial head in centimeters,

h2 = the final head in centimeters

and ln =- the natural logarithum.

The derivation of Equation 2 can be found in practically any soil

mechanics textbook (7).

(2)

For this research, the areas of the samples and water supply,

and the length or the sample are all held constant. By always taking

the initial head as ';JJ5.7 centimeters and the final head as 125.7

centimeters, the equation reduced to a constant divided by the time

interval ( t). This simplified the calculation to:

..} .. .

K= 0.684 (3) t

DescriQtion of ~aratus: The falling-head permeability tests were

performed in a system especially prepared for this research. The

system included a water supp~, a gage stick, connecting hose,

permeameter and drainage facilities.

The permeameter was mar.1ufactured according to a design pre-

pared by the author. The general features are the same as a standard

compaction perirea.rueter, only the dimensions have been increased to

allow a larger size a.ggregat.e to be used. The compaction per:meam.eters

are basically standard proctor molds with special base plates. The

volume of these molds is 0.033 cubic feet. The dimensions of the

permeameter used for this research were increased so that the

effective volume is 0.315 cubic feet.

The water supply was obtained by securing a glass tube, 4.7

centimeters in diameter, at a permanent e1evation. The meter stick

or gage was then secured to the glass tube as can be seen in

Figure 2. The ca.J.ibration of the gage stick 'With the bottom of the

sample was accomplished by the use or an engineers level and a meter

stick, two meters long. The gage stick ~as calibrated, so that to

obtain the correct head it would be necessary only to ad.d 115.7

centimeters to any reading taken on the gage. In all the tests the

initial reading was ninety centimeters. The initial head was thus

205.7 centimeters and the final head was 125.7 centimeters.

F'IGURE 2

FALLilJG-HEAD PERMEAMETER APPARATUS COMPLETELY ASSEJ.vfBLF.J)

The water supply was connected to the top of the permeameter

by a 3/8 inch inside diameter rubber hose as also seen in Figure 2.

The permea.meter consists of a top cover, an extended section,

the mold, the base plate, two porous stones and three gaskets. The

top plate has two openings. One was attached to the water supply by

the hose. The other was used as a bleeder valve. This valve per-

nutted the escape or all air trapped in the system after the top

cover had been sealed. The hooks on the sides were used to bring

pressure against the gasket, to prevent any loss of water at the

joint between the extended section and the top.

The extended section is sho'W?l in Figure 3. This section wa.s

. ..> .

always full of water to make sure that the entire surface area of the

sample was exposed to water. This was accomplished by bleeding orr

any air through the bleeder valve explained above. Inside this

section and resting on the top of the test sample was a porous stone,

one-inch thick and eight inches in diameter. The porous stone was

used to prevent the entering water from digging a hole in the sample.

It also insured an equal distribution of water to the entire surface

area of the sample. The extended section was equipped with two

hooks for connecting to the mold. A rubber gasket was placed between

the end of the mold and extended section to prevent errors caused by

loss of water.

The permeameter mold is 20.6 centimeters in diameter and ldll

hold a sample 26.7 centimeters high. The mold was made from a

section of eight-inch inside diameter steel pipe approximately 3/8 of

an inch thick. The mold also was provided with two hooks to aid in

FIGURE 3

DISASSEMBLED PmMEAMETER

l. BASE PLATE

2. MOLD

J. EXTENDED SECTION

4. POROUS STONES

5. TOP COVER

. .> .• •

securing it to the base plate. Two swival. bolts were used to fasten

the top and extended section securely together. Figure 3 shows the

mold.

The base plate was made of a one-inch thick piece of magnesiwn

alloy with a one-half inch deep, eight inch diameter recess. This

recess provided an inset in which to place a second porous stone. At

the bottom of the recess, circular and radial grooves were cut to

insure rapid removal of the water to the drain. The drain was a

one-half inch diameter hole through the side of the base plate. A

short piece o! stee1 pipe was threaded into the hole and connected to

a rubber hose to guide the waste water into the drain. The base

plate is shown in Figure 3.

When filling the mold with the mixture of sand and aggregate,

it was necessary to get the same compaction for each sample. This

was best accomplished by fastening the mold and base plate securely

to a test tube shakero The test tube shaker and mold are shown in

Figure 4.

FIGURE 4

MOW MOUNTED ON rl'EST TUBE SHAKER

CHAPTER V

PROOF.DURE OF TEST

Pre_pa,ration 2l., Sample: Special attention was given to the preparation

o:f the material to be used in the test. It was desirable, f'or in­

stance, to start with a dried sand to prevent bulld.ng. The sand was

dried a.f'ter every test in an electric~ heated oven at one-hundred

degrees centigrade. The drying often caused some matting together in

small clods of sand. When this occurred, the sand was run through a

Lancaster Counter Batch Mixer which was very successful. in dispersing

these clods.

The large aggregate was selected by hand., weighed and placed

into separate containers. These containers were classified by the

aggregate size., weight and percent volume of the mold. Af'ter the

test the aggregate was washed by hand and returned to its container

for drying. :Each sample was allowed to dry at room temperature

for several days prior to re-use.

Mixing 21. the Sand ~ Aggregate: The placing of the sand and

aggregate into the mold was accomplished by hand. The mold was

placed on the shaker and sand was scooped into the mold. A.s the sand

was added, the large aggregate was placed with care to insure, as

near as possible, that each particle was complete~ surrounded by a

matrix of fine material.. This became ditticult, if not impossible,

for high percentages of aggregate. An attempt was made to insure an

even distribution of large aggregate throughout the height o! the

,._ '.

sample. This was accomplished by roughly dividing the aggregate into

thirds and placing each third into one-third of the height or the

mold. As the filling of the mold progressed to the point of being one

third full, the mold was vibrated on the electric test tube shaker

for fifteen seconds. The mold was also vibrated for fifteen seconds

a.t two-thirds and at slightly less than completely full.. The vi­

bration insured that no large voids would be 1e.tt. in the sample due to

arching action of the sand and aggregate. It is believed that the

vibrating helped keep useless voids to a minim.um.

Weights: Prior to each test, the weight of the mold was recorded.

The mold was then weighed when filled with the sample. The san,ple

contained a known weight of large aggregate. Thus the percent of

aggregate by weight of the sample was obtainable. Since the volume

of the mold was also known, the void ratio for each test could be

computed. It was interesting to note that as the percentage of

aggregate increased, so did the overall weight increase. The

logical assumption is that the void ratio decreased. This was the

case and will be discussed later.

Flooding of the Sample: After the sample had been weighed, the

permeameter was assembled. To insure accurate results it was

necessary to remove all air voids from the sample. This was

attempted at first by flooding the sample from the top, which led

to uncertainty as to the degree of saturation of the sample. The

successful saturation of the sample was obtained by slowly filling

the apparatus from the bottom. A hose was oonnected to the drain and

to a water supply. By caretully forcing the water up through the

sample, the air was moved upward and allowed to escape through the

bleeder valve in the top of the permeameter. When water began to

bleed from the valve, it was closed and the water supply removed from.

the drain. Immediate~ the water supply connection at the top of the

permeameter was opened. Water was allowed to run through the sample

for approximately f'itteen minutes before any measurements were made.

The bleeder valve was cracked open several times to release any

additional air which was trapped in the system.

Timing 2!. ~ Tests: After sufficient water had passed through the

system, the water supp}Jr was replenished. When the water level

reached ninety centimeters on the gage stick, a stop watch, reading

to the nearest one-tenth of a second, was started. The watch was

stopped when the water level reached ten centimeters on the gage and

the elapsed time was recorded. Two additional tests were performed

and an average or the three observed times was used to compute the

permeability coefficient.

When the test was completed the aggregate and aand were

salvaged for re-use in later tests.

CHAPI'ER VI

TESTS RESULTS

Tests Performed: In performing the tests for the research, it was

necessary first to determine the permeability of the sand with zero

percent aggregate. Tests were performed on three samples of sand

prepared as described previously. The initial permeabilities give a

range o! values from 2. 72 x 10-2 to 0.370 x 10-2 centimeters per

second.

For ease in discussion, the sample of sand which passed the

number forty sieve and was retained on the number sixty sieve will

be called sand A. Likewise, the sample passing the sixty sieve

will be called sand B and the sample prepared by mixing equal

portions of sand A and sand B will be referred to as sand c. Using sand A, tests were performed with all three sizes of

aggregate. The graph in Figure 5 shows the results or these tests.

The addition or aggregate in each case decreased the coefficient of

perneability. The higher the percent aggregate, the greater was the

ef'f'ect on the permeability. It can also be observed that the larger

the aggregate size, the less was the effect on the permeability.

This was probably due to the increase in the distance of the path

which the water must follow. The total · surface area is muoh larger

for a given volume of small aggregate than for the same volwne of

large aggregate. The water 'When passing through the sample reaches

the aggregate and travels around its perimeter. Thus, if more surface

area is placed in the sample the water will travel over a longer path,

· ,

, ·

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taking a longer time. This higher value of time decreases the

per~ability.

The general appearance of the three curves in Figure 5 is

remarkably similar. Each curve is approximately a straight line

until the percent aggregate increases to about thirty percent.

At this point, the curves slope downward rapidly. This indicates

that the voids have been decreased so that the flow of water through

the matrix o! sand has been inaterially hindered.

Placing large aggregate in a matrix of sand should decrease

the void ratio, if proper compaction is accomplished. Figure 6 is a

graph showing the variation of void ratio with the percent aggregate

for sand A and the three aggregate sizes used. The experimental

data for sand A is shown in Table I. Figure 6 is also typical of

the percent aggregate versus void ratio curves obtained tor sands

B and C. It can be seen that as the percent aggregate is increased,

the void ratio decreases, and the curve approaches a straight line.

Some points, at high values of percent aggregate, seem to vary

slightly. This was probably due to difficulty in getting all of the

voids tilled with sand.

The similarity in shape or the curves in Figure S leads

directly to the conclusion that it a smaller sized aggregate were

used in comparable tests, the plot of permeability versus percent

aggregate would fall below the three curves shown. Likewise, if' a

larger stone were used, the plot would fall above the three curves.

There would be, for obvious reasons, some limit as to how far, above

or belcw, these curves could be extrapolated.

EXPERIMEll'TAL DATA FOR SAND A

Specific Gravity of Aggregate 2.65 Specific Gravity of Sand 2.70

Aggregate Percent Weight of Weight of Volume of Volume of Volume of Void Size Aggregate Aggregate Sand {gm) Aggregate Sand Voids Ratio

by Weight (gms) cm3 cm3 cm.3

3 in. to 2 in. 0 14,000 5,180 3,740 0.722

15.8 2328 12,432 879 4,600 3,441 0.629

30.1 4656 10,804 176o 4,020 3,140 0.543

43.0 6984 9,266 2600 .3,430 2,890 0.479

54.1 9312 7,888 3520 2,920 2,480 0 • .385

2 in. to l 1/2 in. 0 14,000 5,180 3,740 0.722

15.6 2328 12,422 879 4,6oo 3,441 o.628

35.6 4656 10,944 1760 4,130 3,030 0.515

42.0 6984 9,656 260o 3640 2,680 0.426

53.2 9312 8,188 3520 3,090 2,310 0.350

11/2 in. to 1 in. 0 14,000 5,180 3,740 0.722

15.9 2328 12,302 879 4,640 3,401 0.616

30.0 4656 10,874 176o 4,100 3,060 0.522

42.5 6984 9,456 2600 3,570 2,750 o.~6

53.6 9312 8,068 3520 3,045 2,365 o.36o

TABLE I

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Three sizes or aggregate were used in the tests with sand B.

A plot of the coefficient of permeability versus the percent aggregate

can be seen in Figure 7. Once again, close similarity can be seen to

exist between the three curves. The large aggregate had less effect

on the permeability than did the smaller aggregate. All these curves

are approximately a straight line until the percent aggregate reaches

about thirty percent. At this point the permeability decreases

rapidly. This pattern seems fairly well developed.

The curves for sand A resemble generally the curves for sand

B. For :further comparison, sand C was used. Tests were run using

only aggregate which passed the three-inch screen and was retained on

the two-inch screen. The results of these tests can be seen in Figure

8. Once again, the permeability decreases very rapidly above thirty

percent aggregate. Under thirty percent aggregate, the coefficient

of permeability varies as a straight line with the percent aggregate.

Empirical Relations: Figure 8 shows only the three-inch to two-inch

aggregate plotted for sands A, B and c. It is indicated from this

graph that some relationship ~ exist between the three curves. This

relationship would depend on the initial permeability and on the

percent aggregate. It would also appear that three equations could

be developed from the empirica1 data a~ hand, which would satisfy

each or the curves of Figure 9. These equations could be of aid in

determining if a relationship exists between the three curves.

From seven separate sets of data, it has been apparent that

the plot of percent aggregate versus the coefficient of permeability

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H. ~~-2 - - '. -+-, ; -- i .... ~-,-- --- : ...... :.- ; _ t~-d.:_ ~i--.~-i---- -.~· ----. :- - ~ \--- . '+-~.1--,i.--J-I -~~;-- --~--r--~- ·- : ~+E.ER_µRTi . . E~-r oq ~+--t- ---; : . :-~ i • ~ j • :· . I : . I ! i : . I I . ; I I ~· ,· . l . I 1, I '. ; · : . , . j : l 1 £0!~~ ; · I . \ j' . : : 1 : ~

·----.. ... .......... . . ... - -- - ·. - -- ---- ·-·:- -1.-- -i- ---.-.-+- ---1-- -.-~.--.- - . . nr-" • -·- ----. - +---,--_ - . ·r-·-. --- -- ----t--.·----~.---·-r---: •. -·-·- _

___ ; __ ~.-oc~ - L+_: : E¢. ~~Eilit PF iP .ftM._f_~Bt _~L~YjKL.~-·- -t-L--f---. _.: I • . 2 . I - • . • • I I . I I . I l I . I I • i •

: i x10- .1. _ l . . :_ - - i ·'. : . i ; : ! - I . ' . : ' . . . ' i . : . •. 1 I ' ;. · :

-- ·-------~ ---~-- i -~ - ; __ J.:_ _; __ : . H.. _lQ_~.--..:...--1-- -· ~---;---' --r:-- .-- --rl-- :--· . I . I : - I I l . , . I I • ' I . • . • I . i : I . I l I ' ! I . I . ' ! : I t l t I . I . i . ;

r- -r---1- ---~ ---- -- --t--+~- r~-- ; , : ; ; ; : l : ; ; : , _ ; : - 1--:r- -t~---1' ----r-··---1- - -~- - --L o.~- ---.L. __ I_·· --·-· i-----:-~- + · : _! __ ~:----~----L~'.-+ · · -+---+--t-...:..r-__ _J__:_ __ ---L-~

. { I ' I ' • . . ' ' . . - . •. . · 1 · . I . i . I :x10- . I i . ' : ! ; : . i i : I . . I .. . . . I . . ! I

!-----~ ---+---~--- ------~-·----~-1'--·--+---t---~+-~~-t-~-+- -,:--·h 1··- ;~-+ · ~ -+-,,--1~-~---~1; ~,. :--•· ~~L1--.-r~~ I ; . I i . l 1 . : : , : ! . : . I ; I : ·, i · : . · · : . · 1. · i · ! ~--------·-r-- ---;--t-~---,----+- • -...--4--------1--.:,..--;- . I ,-. --i-----~---1 : : : ! l . I : l . i l I 1 : '. I I ; I l : I . : ! i . :

~

. - 7---+-- ~--- --r--+,:--- -:-- -+---t- ---i- -i---~--- ~----, ----;---+-+--+ +·---t-------- 11---t-~--- -.-- -1-- -i---~ I : : : : : • I I l I j • ; I

1 ~:-- ;-· : --r .:- r-- r ~1~-bi~T:A~REGfT£-ffr-: 4f +- -4{5- ; i :--~r-~~-.

r---: -- -:-~--- 11 : ·,1 ·-r--- -~---;---:--r--~· -;- -~t,_--f-1 11 11 ___ 1 . • , . , ] . , +-. . I t • • FtGU 0 E a, · 1 1 j- · · ~ -.---- --+- . . -,----. --r---·-----i--~ . . ,- -· ·-· .. 0 • ----r·----- - --,---- 1-·-c"'-- --L-- - --- -- - -

L_)· : .. : l: !> 1> l _ :: :.i I · ! -~- · · t. -L .. ! i : :_ -i : ; <. i :

has appeared as a straight line for values up to thirty percent

aggregate. Therefore, it may be suggested that the first portion of

the· data can be represented by the simple equation:

where:

Kx = Ko + BX (4)

Kx = coefficient of permeability at any percent aggregate,

K0

= coefficient of permeability at zero percent aggregate,

X = percent aggregate

and B = slope of the straight line.

As X increases above .30 percent, however, Equation 4 does not at all

represent the observed data. This is to say that the calculated

value Kx is much larger than the observed value Ko, so that the

residuals Kx - Ko are quite large for that portion of the curve over

X = 30 and very small for values of X less than 30%. According to

Dr. Joseph Lipka (8) in such cases a modification of the simple

0 •) ~ .

equation by the addition of one or more terms may cause the calculated

curve and the observed curve to fit approximately throughout. Such a

term was found for the three to two-inch aggregate in sand A.

Equation 4 now becomes:

Kx = Ko+ BX+ R (5) where R is the residual or difference between the observed and the

calculated straight line values.

Having chosen a form for the approximate equation, it is

now necessary to determine the constants. In general there are

three methods dependent upon the degree of accuracy required, i.e.,

the method of selected points, the method of avera~es, or the

method of Least Squares. The method of sel.ected points involves

the least amount of work and produces less accurate values.

The method of averages results in better

values and involves more calculations. The method of Least Squares

gives the best possible values for the constants, but the work of

determining these val.ues is quite laborious. A combination ot the

method of averages and selected points was used in evaluating the

constant B. As w.i.11 be pointed out below I the percent error from

ti·, ''""\ ' .... . ,\ ' .

the difference in calculated values and observed values are less than

one percent, justifying the use or the less complex method.

To evaluate the constant B ot Equation 4, the observed points

were plotted as in Figure 9. A smooth curve was drawn approximately

through the observed points. In the first thirty percent aggregate

there were three observed points. Three additional points were

chosen f'rom the smooth curve and the coordinates of theee six points

were placed in the data. Th~s.e , da.t'.a:'l.w:~ then divided into half', and

two equations involving the unlmown value B were written in the form:

where:

L Kx = N,~0 + BL X

Kx = the sum of the first three values of the K coordinates

N = number or coordinates summed

and X = sum of first three values of the X coor-

(6)

dinates. The second equation involves the next three terms in the

same manner. The two equations were then solved sinru.ltaneously and

the value . ot Ko and B were found to. be.:

B ~ 0.312 x l0-4 Ko= 2.72 x 10-2

where Ko is the K intercept, and B is the slope or the straight line

portion of the curve. E;quation 4 now can be wri-tten as:

Kx = 2.72 x 10-2 - 0.312 x 10-4x (7)

and Equation 5 as:

or

Kx = 2. 72 x 10-2 - 0.312 x 10-4x + R

The value of R cou1d be of the form

R = c~DX

R =- cxD.

(8)

(9)

(10)

To evaluate R, the curve was extended to pass through the observed

points. Two additional points were then picked from the curve in a

manner similar to that described above. The four X coordinates

were then p1aced into :Equation 7 and the values of Kx' calculated.

The difterence between the observed and the calculated straight line

values, i.e., the residuals were found. A plot of X versus log R

indicated that the equation of the form

R = c~Dl

does not apply. However, a plot o! the log X versus the log R

approaches a straight line. Thererore, the equation of the form

R = cxD,

will give a relationship which is useable. F,quation 8 now becomes

Kx = 2. 72 x 10-2 - .312 x 10-4x + cxD.

To evaluate the constants C and D the logarithum of ~uation 10

gives, for use in the method of averages,

E log R =- N log C + D I; log X

(11)

(12)

By using N equal two, and summing halt of the data, two equations

can be written and solved simu1taneously. Performing this operation

we rind that: c = -6.47 x 10-13,

and D = 5.8.

. · "'\ ... ·· __

'-- .......

Thus F,quation ll becomes:

Kx = 2.72 x 10-2 - 0.312 x 10-4 x - 6.47 x 10-13 x5·8 (13)

Rewri-ting it in terms cf the initial permeability ~uation 13

becomes:

Kx = Ko - 0.312 x 10-4 x -6.47 x 10-lJ x5·8 (JJa)

A check to compare calculated values with observed values will show

that the calculated values agree within 1.1% or the observed values.

The complete derivation of Equation 13 may be found in Appendix A.

The same procedure was followed in finding am empirical

equation for sand Band sand C, with three to two-inch aggregate.

For sand B

Kx = K0 - 1.02 x 10-5 X -1.3 x 10-12 x5.17 (14)

and for sand C

(15)

Table 2 shows the observed and calculated values of the

coefficient of' permeability, together with the percent differences,

for all three equations. No deviations larger than two and tour­

tenths percent were observed, thus validating the empirical

equations.

(\""':' .. ·, . ·, \ '-· ....

COMPARISON OF OBSERVED DATA AND CALCULATED DATA

For sand A, Equation 13a:

Percent K x 102 cmjsec. K x 102 cm/sec. Percent Aggregate Observed Ca1cul.ated Difference Difference

0 2.72 2.72 0 0

15.8 2.67 2.67 0 0

30.1 2.63 2.60 .03 1.1

43.0 2.38 2.39 .01 0.4

54.1 1.88 1.90 .02 1.1

For sand B, Equation 14:

0 0.370 0.370 0 0

15.8 0.352 0.354 .002 o.6

29.s 0.340 0.334 .006 1.8

42.1 0.21l7 0.294 .007 2.4

54.0 0.198 0.199 .001 0.5

For sani C, Equation 15:

0 0.834 0.834 0 0

15.7 0.795 0.794 0.001 0.1

29.3 0.760 0.746 0.014 1.8

43.0 0.607 0.615 0.008 1.3

53.0 0.355 0.3'55 0 0

TABLE II

The similarity in form of Equation lJa, 14 and 15 is

immediately apparent, i.e.:

and

Kx = Ko - B1X - C1xDl

Kx = K0 - B~ - C2XD2

Kx = Ko - B3X - C3xD3

(16)

(17)

(18)

These three equations can be combined in one sing1e equation giving

the coefficient of permeability at any point, if a relation can be

found between the initial permeability and the constants B, C and D.

Such an equation would be of the form:

(19)

The relation of Bi, B2 and~ was considered first in order

to determine the best possible value for £1 (K0 ). No true straight

line relation was found, however, it was observed that a straight

line could be approximated on regular coordinate paper. To get the

best equation of the form

(20)

where E and F are both constants, the method of Least Squares was

used. This method will give the str~ght line relationship with the

minimum amount o! deviation. This method is also described by Dr.

Lipka (9). The calculations are found in Appendix B. F,quatian 20

C ,,t ·

l ' \,

was found to be:

r1 (K0 ) = 0.112 x 10-4 + o.834 x 10-3 Ko (21)

The relation of C1, C2 and CJ was determined to find the best

value for f2 (Ko). Once again no true straight line relationship

could be obtained, but a plot of the values or K0 versus C approxi­

mate a straight line. The best possible value .for the r2 (Ko) was

obtained by the method or Least Squares as previously shown.

Evaluation or r2 (Ko) was round to be:

£2 (Ko) = G + HKo ::s -1.36 x l0-12 + 0.264 x 10-10 Ko (22)

where G and Hare both constants.

The third set of constants, D1, D2 and D3, was found to

approximate a straight line on logarithmic paper. The relationship

was then:

where J and L are constants. The method or averages was used to

evaluate the constants. These calculations are also shown in

Appendix B. The value of t3 (Ko) was .found to be:

f3 (Ko)= JK0 L ::s 7.0 Ko 0.0523 (24)

By substituting F,quations 21, 22 and 24 into F,quation 19 it becomes:

Kx =- Ko - ( 0.112 x l0-4 + 0.834 x 10-3 Ko) X - ( 1.36 x

1012 - 0.264 x 10-lO Ko ) x 7.0 K0 9.!P5?3 (25)

where Kx ~ coefficient of permeability in centimeters per second at any percent aggregate,

K0 = the initial coefficient or permeabil.ity with no aggregate added, centimeters per second,

X ~ the percent aggregate by weight in the samp1e.

~1 ·"" ~. l '.). ·- ··~

39

F,qua.tion 25 should be recognized as onlJ' an approximation o!

the influence that the addition or one size or aggregate to fine­

grained soils has on the coefficient of permeability. It should also

be remembered that the equation was derived for aggregate or uniform

size. This research also considered only a limited range or initial

permeabilities. The validity of Equation 25 outside this range is

beyond the scope of this research.

Agreement 2! Flllpirical F.guations with Observed ~: The empirical

relations above agree within certain limitations with the observed

data. As stated before, Equations 13a, 14 and 15 agree veey closely

to observed data. It is in combining these equations that significant

variations can be seen.

In order to evaluate &iuation 20, it was necessary to assume a

straight line relationship existed. This assumption contributes some

error. Evaluating Equation 21 for known values of Ko, we find that

when Ko :a 2. 72 x 10-2 centimeters per second, the value of !1 (Ko) is

0.339 x 10-4. It can be seen that from F.quati.on 13a the corresponding

value is 0.312 x 10-4. The percent difference in the £1 (Ko) :tor

Equation l3a is approximately eight percent. Eight percent difference

in the £1 (Ko) term, however, will cause a difference or onzy tour

tenths or one percent in the value or Kx at thirty percent aggregate.

That is to say that when X =- 30 using' Equation 13a,

Kx = 2.72 x 10-2 - .09 x 10-2

or Kx = 2. 63 x 10-2 cm/ sec.

When Equation 25 is used tor X :a 30, and Ko = 2.72 x 10-2 cm per second,

.,.,. ,.--~

r:_ ;L (. J _t ·: ~

Kx 2 2.72 x 10-2 - .101 x 10-2 - .03 x 10-2

or Kx ::s 2. 62 x 10-2 cm/ second.

The difference between the resulting value of Kx is 1.5 percent.

The evaluation of Equation 20 for Ko= 0.834 x 10-2 cm per

second, gives a value of n (Ko) :s 0.182 x 10-4. When compared to

its corresponding value of 0.255 x 10-4 in Equation 15, a much higher

percent difference can be seen. However, when Equation 15 is solved

for X = 30 and compared to the corresponding solution of Equation 25,

a difference of only 2.8% is observed in values of Kx• Likewise,

when X z .30 a comparison of Ec:J..uations 14 and 25 show a difference of

only 3.7% in values of Kx•

In order to evaluate :Equations 22 and 23, it was also

necessary to assume that straight line relations existed. In both

cases the percent difference in values found for individual curves,

closel,y agree with corresponding values found by use of F,quation 25.

In order to show clearly the effect of the combining of equations,

Figure 10 shows the observed curves plotted with Equation 25 for their

respective values of Ko· Tab1e 3 lists the observed values of Kx, the

calculated values of Kx using :Equation 25, the differences and the

percent differences.

Both Figure 10 and Table 3 show clearly the limitations of

F.quation ·25. The agreement between calculated values and observed

values for sand A is excellent. Sand B agrees quite well, having no

discrepancies between observed values and calcu1ated. vaJ.ues in excess

of ten percent. Sand C, however, does not agree as closely as sand B.

The agreement is very good up to X = 35. From here on, as the percent

COMPARISON OF OBSERVED DATA AND CALCULATED DATA USUJG mUATIOM 25

For Sand A

Percent K x 10··2 cm/sec. K x 10 ··2 cm/sec. Aggregate Observed Calcul.ated Difference

0 2.72 2.72 0

15.8 2.67 2.67 0

30.1 2.6.3 2.59 .04

43.0 2 • .38 2.38 0

54.1 1.88 1.90 .02

For Sand B

0 0.370 0.370 0.000

15.8 0.352 0.347 0.005

29.s 0.340 0.321. 0.01.9

42.l 0.287 0.272 0.015

54.0 0.198 0.179 0.019

For Sand C

0 0.834 o.a34 0.000

15.7 0.795 0.806 o.on

29.3 0.760 0.772 0.012

4.3.0 o.6o7 o.668 0.061

5.3.0 0 • .355 0.469 0.114

TABLE III

1~ (-·, l :· .' . .

Percent Difference

0

0

1.5

0

1.1

0

1.4

5.6

5.2

9.6

0

1.4

1.6

10.0

24.3

43

aggregate increases, the discrepancies between observed values and

c~lculated values increase rapidly. At 43 percent aggregate the

discrepancy is ten percent, while at 53 percent aggregate the

discrepancy reaches 24.3 percent.

Thus it oan be seen that ~uation 25 is only an approxi.mation

ot the influence that the addition or various percentages of one

size or large aggregate has on the coef'ficient of permeability. This

approximation is, when compared to observed data, accurate to within

ten percent for values up to 43 percent aggregate. It can also be

shown that up to fifty percent aggregate, F,quation 25 is an approxi­

mation to within 17 percent of observed values. For values above 43

percent aggregate, however 1 F,quation 25 shou1d be used with caution.

It should also be remembered that Equation 25 was derived

for aggregate which was of uniform. size, i.e., two to three

inch, and f'or a limited range ot values tor the coefficient of

permeability. Ir a mixture of' slightly larger and smaller size

aggregate were used, it is believed that very little additional

error would result from applying F,quation 25.

CHAPTER VII

SUMMARY

The work -which is presented in this study, has been per.f'ono.ed

in order to evaluate the influence o! large aggregate on the permea­

bility of a single-grained soil.

In this research certain noteworthy results were obte.ined,

which are summarized as .follows:

1. The addition of large aggregate to single-grained soi1s ·

decreases the penneability of the soil.

2. The size of the aggregate is a factor as to the magnitude

of the influence of large aggregate on the coefficient of permea­

bility. The infiuence of smaller aggregates was greater than the

influence of larger aggregates.

3. As the percent aggregate increased, the void ratio

decreased approximately as a straight line relationship.

4. The decrease in the coefficient or perm.eabil.ity was

linear for the first thirty percent aggregate. Above thirty percent

aggregate, the permeabil.ity decreased. quite rapidly.

5. When considering only the two to three inch aggregate in

three separate matrices of .fine-grained soils, three equations were

found which satisfied observed data. · These equations all were of

the form

Kx :a Ko - BX - CXD,

and differed only in the constants B, C and D.

,'! !'"--~

! ! I .'

6. The data :for the two to three-inch aggregate 'Were · . .combined

with partial success, into one equation. This equation,

Kx = Ko - (0.112 x 10-4 + 0.834 x 10-3 Kc,) - (l.36 x 10-12

-0.264 x 10-lO Ko) x(?.O y~0.0523)

is a good approximation up to values o:f forty-three percent aggre-

gate. Forty-three percent aggregate can be used with an error of no

more than ten percent of the actual observed value.

7. The results are limited to cases in which the fine material

forms a matrix for the large aggregate.

CHAPTER .VIII

CONCLUSIONS

This research has provided a highly satisfactory method ot

predicting the effective coefficient of permeability for the mixtures

of large aggregate and single-grained soils tested. The results are

applicable for sing1e-grained soils with zero percent aggregate

having a coefficient of permeability between the va1ues of 2.72 x 10-2

and 0.370 x 10-2 centimeters per second and tor aggregate between

three to two inches in diameter.

It is believed that the method used could be extended to in­

clude a much larger range ot values for the coefficient of permea­

bility by additional research. A wider range of values would

possibly modify the constants somewhat, but probably would ·not effect

the general form of the equation developed by this study.

APPENDIX A

The Derivation of F,quation 13, using Sand A and three to two-inch aggregates.

Deviation of F.quation 13:

Percent Ko cm/sec. Kxl -R = Kxl-Aggregate X Observed Ko Log R

* 0 2.72 x 10-2

s.o 2.70 x 10-2

*15.8 23.8 2.67 x 10-2 s.09_x_10-2

20.0 2.66 x 10-2

25.0 2.64 x 10-2

*JO.l 75.1 2.23 x 10-2 7.93-x-10-'2

35.0 2.56 x 10-2 2.61 x 10-2 0.05 x 10!"2 6.699-10

*43.0 2.38 x 10-2 2.59 x 10-2 0.21 x 10-2 1·.l~-10 4.021-10

48.0 2.20 x 10-2 2.57 x 10-2 0.37 x 10-2 7.568-10

*54.1 1.88 x 10-2 2.55 x 10-2 0.67 x 10-2 .7..822-10 5.394-10

*Observed values, other values obtain from Curve Figure 5

Ko :a A+ BX

L.K0 ::a NA+ BLX

(1) 8.09 x 10-2 = 3A + B (23.8)

(2) 7.93 x l0-2 :a 3A + B (75.1)

B = -0.312 x 10-4 and A= 2.72 x 10-2

Therefore:

Kx1 = 2. 72 x 10-2 - 0.312 x 10-4x

Log x

1.544

!·231 'j.177

1.681

!·1.3l 3.414

(1) Kx1 = 2.72 x 10-2 - 0.312 x l0-4 (35) = 2.61 x 10-2

(2) Kx.1 s 2.72 x 10-2 - 0.312 x 10-4 (43) = 2.59 x 10-2

(3) Kx1 = 2.72 x 10-2 - 0.312 x 10-4 (48) :: 2.57 x 10-2

(4) Kx1 = 2.72 x 10-2 - 0.312 x 10-4 (54.1) = 2.55 x 10-2

R = cxD

log R :a log C + D log X

or

~ log R = N log C + D L log X

(1) 4.021-10 = 2 log C + D (3.177)

(2) 5·394-10: 2 log C + D(3.414)

Therefore D = 5.8

and C = 6.47 x 10-13

Then

R 2 6.47 x 10-13 X 5.8

Equation_ 13 is thus

Kx ~ 2.72 x 10-2 - 0.312 x 10-4 X -6.47 x 10-13 x_5.8

/~ /'- -\ ,. : : - ~ .i

APPENDIX B

Derivations ot Equation 21 and

Equation 2.4

Derivation of Equation 21: f1 (K0 ) = E + F Ko

Using the method of Least Squares:

let Ko= E + F (Bx)

Solve simultaneously tor values of E and F:

(1) (E + F Ko1 - B1) + (E + F Ko2 - B2) + (E + F Ko3 - B3) = 0

(2) Ko1 (E + F Ko1 - B1) + Ko2 {E + F Ko2 - B2) + Ko3

(E + F K03 - B3) ~ 0

where: E and K are constants

Ko1: 2.72 x 10-2

Ko2 = 0.834 x 10-2

K0J 3 0.370 x 10-2

B1 = O.Jl2 x 10-4

B2 = 0.355 x 10-4

83 - 0.102 x 10-4

Then {l) and (2) become

(1) 3 E - 0.699 x 10-4 F - 3.92.4 x l0-2 = 0

(2) -0.669 x l0-4 E + 0.167 x 10-8 F + 1.101 x l0-6 ~ 0

hence:

E = -1.34 x 10-2

Ko =-1.34 x 10-2 - 1200 (~)

or

-r1 (Ko)=- 0.112 x 10-4 + o.s.34 x 10-3 Ko

then

£1 (K0 ) ~ -0.112 x l0-4 - 0.834 x 10-3 Ko

,.. .... .,,. \ . ' ... '- ........

Derivation ot Equation 24:

r3 (Ko)=- J Kc,L

Ko f3 (r~) Log Ko

2.72 x 10-2 5.80 s.4346-1.0

o.s34 x 10-2 5.51 7.9213-10

0.370 x 10-2 5.17 1·2.6~2=.l.Q 5.4894-10

Using the Method of Averages:

E f3 (Ko) = N log J + L ~ log K0

hence two equations:

(1) 0.7634 = l log J + L (8.4346-10)

(2) 1.4547 = 2 log J -~ L (5.4894-10)

Solving simu...ltaneously £or J and L:

J = 7.00

L ~ 0.0523

Equation 24 thus becomes:

f) (Ko) ;: J KoL .,. 7.00 Ko0.0523 •

Log f.3 (Ko)

0.7634

0.7412

Q..1ll5_ 1.4547

BIBLIOGRAPHY

1. Marvls, F. T. and E. F. Wilsey (1936). A Study of the Permeability or Sand. University of Iowa Studies. Studies in :Ehgineering.

2. Peck., Ralph B., Walter E. Hanson and T. H. Thomburn (1955). Foundation Engineering. John Wiley and Sons, Inc., New York. Page 51-58., Bulletin 7, Page 6-7.

3. Spangler, Merlin G. (1951). Soil Engineering. International Textbook Company., Scranton, Pennsyl.vania. Page 125-133·

4. Tschebotariofr., G. P. (1957). Soil Mechanics, Foundations and Earth Structures. McGraw-Hill Book Company, Inc., New York, Page 76-79.

5. Seeley, Elwyn E. (1960). Data Book for Civil Engineers, Volume 1, Design. Third edition, John Wiley and Sons, Inc., Page 9-33·

6. Kawakami, Fuseayoshi (1954). Mixing Coarse Materials into Impervious Zones ot Earth Dams. The Technology Reports of the Tohoku University, Serl.dai, Japan. VolU,"ne 18, No. 2, Page 168-177-

7. Tschebotarioff, G. P. (1957). Soil Mechanics, Foundations and Earth Structures. McGraw-Hill Book Company, Inc., New York, Page 76-79.

8. Lipka, Joseph, Ph.D (1918). Graphical and Y~chanical Compu­tation. John Wiley and Sons, Inc., Page 120-139.

9. Lipka, Joseph., Ph.D (1918). Graphical and Mechanical Compu­tation. John Wiley and Sons, Inc., Page 120-125.

t\ C' ... ·:...

r:. l:. '- --

VITA

F,dward Clayton Grubbs was born January 14, 1931. at Salina,

Kansas. He received his elementary and high school education in

Houston, Texas.

Upon graduation from high school in 1950, he enl.isted in

the United States Navy and served for tour years. His services

in the Navy included sea duty on a surface craft and on submarines.

He enrolled at the University or Texas in June, 1954 and

received a Bachelor or Science Degree in Civil Engineering in

August, 1957. At this time he was appointed an Instructor in the

Civil Engineering Department at the University of Missouri School

of ~.ines and Metallurgy and was enrolled as a graduate student in

Civil Engineering.

On December 23, 1952, he was married to Martha Lou

strother of Houston, Texas. He has two daughters, Lou Rae

born in 1955 and Linda bon1 in 1958.

'_'.,/