Perloff

7/17/2019 Perloff

http://slidepdf.com/reader/full/perloff 1/57

STRESS

DISTRIBUTION

WITHIN AND

UNDER

LONG

ELASTIC

EMBANKMENTS

JUNE,

1967

NO.

14

PURDUE

UNIVERSITY

LAFAYETTE

INDIANA

' 7777 ^^S^K

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7/17/2019 Perloff


Research Report

STRESS

BISTRIBUTKM

WITHXH

AND UHDBB

LCK3

ELASTIC BffiABKMEKES

So:

Dr.

G.

A.

Leonards,

Director

Joint

Highway Research

Project

From:

H»

L.

Michael,

Associate

Director

Joint Highway

Research

Project

June

20,

1967

File;

6-6-6

Project:

CE-3S-5?

hed report entitled

Stress Distribution

Within

and

Under

Long

Elastic

Embankments

describes one

phase

of

the

project

concerning

long-tes ormations

of compacted

cohesive

soil embankments.

The

re-

port

was

prepared by

Professor

W.

H.

Perloff,

Research

Engineer,

Mr.

S.

Y.

Baladi, Research

Assistants

and

Professor

M.

E.

Harr.

The

report

presents

diagrams

of

the

distribution

of

stresses

within

and under

long

elastic

embankments continuous with

the

foundation

material.

The si;;

nt

differences

in

the magnitude

and

distribution

of

the stresses

from

those

usually

assumed

are discussed.

Respectfully submitted.,

Harold

L.

Michael

Associate

Director

HLM:i

•

Attachment

Copy: F. L.

Ashbauehez

W. L.

Dolch

W.

H.

Goetz

W.

L.

Greece

G. K.

Hallock

R. H-

Harrell

V.

E.

Harvey

J.

F.

McLaughlin

F.

E.

Mendenhall

R. D.

Miles

J.

C.

Oppenlander

C.

F. Scholer

M. B. Scott

W.

T.

Spencer

F.

W.

Stubbs

E.

R.

J.

Walsh

K.

£» Woods

E.

J. Yoder

7/17/2019 Perloff


Research Report

STRESS

DISTRIBUTION

WITHIN

AND

UNDER

LONG

ELASTIC EMBANKMENTS

by

W.

H. Perloff

,

Associate Professor

of

Soil Mechanics

G.

Y.

Baladi, Graduate Research Assistant

M.

E.

Karr,

Professor

of

Soil

Mechanics

Joint

Highway

Research Project

Project:

CE-36-5F

File

:

6-6-6

Prepared

as Part

of

an

Investigation

Conducted

by

Joint

Highway

Research Project

Engineering

Experiment

Station

Purdue University

in

cooperation

with

the

Indiana

State

Highway

Commission

and

the

U.S. Department of

Transportation

Federal

Highway Administration

Bureau of Public

Roads

The

opinions,

findings

and

conclusions expressed

in

this

publication

are

those

of the

authors

and

not

necessarily

those of

the

Bureau

of

Public

Roads.

Not

Released for

Publication

Subject to Change

Not

Reviewed By

Indiana State Highway Commission

gr

the

Bureau of Public

Roads

Purdue University

Lafayette, Indiana

June

20,

1967

7/17/2019 Perloff


ABSTRACT

The distribution

of

stresses

withia and under long elastic

embankments continuous

with

the underlying material

is

presented.

The

magnitude

and

distribution

of

stress

in

the

foundation

material

in the vicinity of the

embankment is

significantly

different

from

that predicted by

the usual

assumption

of

stress proportional to

embankment

height applied

normal

to the

foundation.

Influence

charts

for

a

variety

of embankment

shapes

are given.

7/17/2019 Perloff


Digitized

by

the

Internet

Archive

in

2011 with

funding

from

LYRASIS members

and

Sloan

Foundation;

Indiana Department of Transportation

7/17/2019 Perloff


BERODUCTIGN

The distribution of

stresses

within

and

under earth

embankments

due

to

the

embankment

weight,

is

of

interest

to civil

engineers

in

a

variety

of

applications,,

Consideration

cf

deformations

within

embank-

ments, analysis

of stability, consolidation

of

underlying compressible

materials, all require determination

of

the

distribution

of

these

stresses

At

the

present

time there is no means

available

by

which

a

closed-

#

form

solution

for such

stresses can be

obtained.

Consequently,

various

approximations

of

the real

problem

have been made,

with

the

objective

of

obtaining at least

an

estimate

of

the

stresses.

The

first

such

effort

was made

by

(BROTHERS

(1920).

He

analyzed the stresses within an homo-

geneous,

isotropic

elastic

half-space

resulting

from

a long embankment

loading. It

was assumed

that the

load

was

applied

normal

to

the

boundary

with

a

magnitude proportional

to

the height of

the embankment. These

results were presented in tabular form by JURGENSCH

(1937)

.

OSTERBERG

(1957)

superimposed

solutions

given

by

NE8MARK

(19^0,

I9U2) to

develop

an

influence

chart for the determination

of

the magnitude

of

vertical

stresses

induced in

an elastic half-space

by

a

long

embankment

loading

with

a

variety

of

cross

sections.

Again,

the

magnitude

of

the

pressure

was

assumed to be proportional

to

the

embankment

height

and

applied

normal

to the

surface

of

the foundation

material.

This normal

loading

approxi-

mation

to

the

actual embankment loading

is

illustrated

in

Figure

la.

TERZAGHI

(19^3)

described an

effort

to

evaluate

the

shear

stresses

transmitted

to

the

foundation

material

by an embankment

made

by

RMDULXC

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WV^WVÂSX^

L

^WAVV/WAWXVAVVI

.V

v

'

v

'->.

(a)

Long

Symmetric

Elastic

Embankment

Continuous With Foundation

^X/AVV/VWAVVAVV/avÂvV^^

(b)

Normal

Loading

Approximation

Fig.

I

-Problem Considered

7/17/2019 Perloff


(1936).

It

was

assumed that the embankment

material

was

on

the verge

of

failure

and thus

the shearing

resistance

of

the embankment

was

fully

mobilized

in

order

to

maintain

equilibrium.

Consequently,

TERZ&GHX

(19**3)

suggested that

the magnitude

of the

computed

shear stress

at

the base

was

likely to be lower than

the actual in-situ stresses,

TROLLOPE

(1957),

and

DAVIS and

TAYLOR (I962)

considered

the state

of

stress within

a

granular

embankment

resting

on

a

foundation

which

yielded

an

arbitrary

amount

.

Ho attempt

was

made

to compute the amount

of

founda-

tion movement which would

be

created

by

the

embankment.

FUSS

(i960)

suggested

the

use

of the Schwarz-Christoffel

transforma-

tion

to

map

the

embankment surface into

a

straight

line, thereby

utilizing

the distribution of stresses within

a

semi-infinite

elastic

medium.

How-

ever,

he did

not

carry out

the

suggested procedure

<>

Numerical methods

have been used

to

obtain,

for

particular

cases,

solutions

for

the

stress

distribution

in

an

elastic

embankment

resting

on

elastic

or

rigid

foundations

.

ZIENKIEWICZ (19U7) used

a

finite difference

approach

to

analyze

the

stress

distribution

within

a

triangular gravity

dam

resting on

an elastic foundation

„

This

was

extended

by

ZIENKDMCZ

and

GERSTNER

(1961)

to

the

case

in

which

the foundation

modulus

differed

from

that

of the

dam,

DBJGKALL

and

SCEIVNER

(195U) applied

the

method

of

finite differences to the

solution

of

an

embankment on

a

rigid

foundation

CARLTON

(1962)

used a similar

method

to

study an elastic embankment contin-

uous

with

an

elastic foundation.

The finite

element method

of

numerical

analysis

has been

applied by

ZISNKIEWICZ and

CHBUBG

(1964,

1965)

in

the

study

of

stresses within

but-

tress

dams

resting on elastic foundations .

CLOUGH

and CHOPRA

(1966)

and

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FIHN

(1966),

respectively, have

also

applied the

finite

element method

to

the study

of

a

triangular

dam

on

a

rigid foundation

and

a

rock

slope

con-

tinuous with

its

elastic

foundation.

BROHN

(1962)

and GOODMAU

and

BROWN

(1963}

investigated

the

case

of

a

long elastic slope constructed

incrementally;

it

is not clear

to

what de«

gree

their

results are

influenced

by

the

fact

that compatibility

is

not

satisfied

by

their solution

method

In

each

of

the

cases

approached

by numerical

methods,

the solution

was

either

restricted to a

single

embankment

cross

section

or a

complete

stress

picture

was

not obtained

.

Thus, despite numerous

attempts

to

determine

the

distribution

of

stresses

within

and under

an

embankment,

no

closed-form

solution is presently available

.

It

is

the

objective

of

this paper to

present such

a

solution.

ergblem CONSIDERED

The

problem

considered herein is

the

determination

of

the

distribution

of

stresses

within

and

under

an

embankment

resulting

from

the

self-weight

of

the

embankment.

The embankment

is

shown schematically

in

Figure

lb» It

is

assumed

that

the

embankment and the foundation

material with

which

it

is

continuous, are

composed

of homogeneous,

isotropic,

linear

elastic

material.

Further, the

embankment

is

assumed

to

be

sufficiently long

so

that

plane

strain

conditions

apply.

The

shape

of

the

symmetric cross

section

is

defined

by the

slope angle

a

and

the

ratio of

the

half

-width of

the

top

of

the

embankment, L, to the embankment

height,

H.

The

solution

is

obtained

by

transforming

the

region of

the

embankment

where the

solution is

unknown,

into

a

half-space where

the

solution

can

be

7/17/2019 Perloff


found.

Application of

the

Cauehy

integral

formula

to the

boundary condi-

tions

permits

determination

of

the

stresses.

An

outline of the method is

given in

Appendix 1.

KSSULTS AND DISCUSSION

Vertical Normal

S

tress

A

typical

result

is illustrated in Figure 2.

This figure shows

contours of

the

vertical normal

stress

in

dlmensionless

form,

o^/yH,

for

an

embanlsment with

a

*

U5

,

L/H

«

3?

and

Poissoa°s ratio,

u

®

0.3.

The

contour

lines show

only the effect

of

the

embankment

weight. Thus,

at

depths

below

the

base

of

the

embankment

(y/H

«

0)

the material

is

assumed

weightless.

The

effect

of

the

medium

weight

can

be superimposed

upon

these

values

to give

the

total

stress

acting

at

a

point. The

two

dashed lines

in

Figure

2

show

stress contours, in

terms

of

Oy/VH,

for

the

usual

normal loading

approximation

corresponding to this

embankment. These

indicate that

the vertical normal stresses

produced in

the foundation

mater-

ial below the elastic embankment are generally smaller than computed

for

the

normal

loading

approximation.

The

stress distribution

due

to

the

normal loading

approximation is

independent

of

Poissca

:

s

ratio; the

stresses

due

to

the elastic

embankment

are

dependent

upon

jx.

However,

the vertical stresses are

insensitive to

its magnitude;

changing

n

from

0.3

to

0.5

changes the

vertical

stress

at

a

point

by

less than

five per

cent.

The

effect

of embankment shape

on

the vertical

stress

along vertical

sections

through

the center

line

of

the

embankment and

the toe

of

the

slope,

7/17/2019 Perloff


123456789

/H

Fig.

2-Confours

for

Vertical

Stress

10

7/17/2019 Perloff


is

illustrated in

Figure

3,

for

a

»

1+5°

and

u

*

0.3.

The

figure

is

a

composite

diagram

shoving

the

embankment schematically,

and

the

magnitude

of

the vertical

stress

at each

section

as a

function

of

depth. Clearly,

the

L/H

ratio has

a

pronounced

effect on

the distribution

of vertical

stress.

As

L/H

decreases, the

stress decreases.

Furthermore, a

smaller

L/H

ratio produces a

more

rapid

dissipation

of

stress with depth.

The

dashed

lines

show

the vertical normal

stress

for

the normal

load-

ing

approximation

equivalent

in shape

to

the

embankment for which

L/H

a

1.

As

indicated

in

Figure

2,

the vertical

stress

for the

corresponding

elastic

embankment

is

smaller.

Figure k shows

the

distribution

of

vertical normal

stress

along the

base

of

the

embanloaent

for

u

»

0.3,

four

values

of

a,

and

several embank-

ment shapes

shown

schematically

in

the

figure.

The curved solid

lines

represent the distribution

of

stress

against

the

base;

dashed

lines show

the

distribution of

stress assumed

in the

usual normal loading approximation.

The stress distribution is much more uniform under the elastic

embankment

than is

ordinarily assumed. The

difference

becomes

especially

apparent

as

the

L/H

ratio

of the

steeper embankments

decreases.

Moreover,

the magni-

tude of the

stress

under the

central

zone

of the

elastic

embankment

is

less

than

that shown

by

the dashed

curves.

Again the

effect is

enhanced

for

narrow,

steep

embankments

(for

a

s

45°

and

L/H

s

the

vertical

stress is

only

65

per

cent

of

that

usually

assumed)

In

order

to satisfy

equilibrium,

the

areas

under corresponding

dashed

and

solid

curves must be the same.

Hence

the difference between

these curves

becomes

less pronounced,

at

least

near the

central

portion

of

the

embankment,

7/17/2019 Perloff


1

1 1

1

L

1 1

_l

1

my/iWAÂÂV/ANÂy/AW

11

H

1

/

y y

/

-q

s

>^w/sW//&y/s&>ti

/

/

/

l'\

mr

a=

45°

1

/

/

/

///•

1

-

A

^

=

0.3

/

/

z'

//

/

\\\

/0.0/0.5/

/

/

/

\\\

2

///

'V /

2-

/

3

-

/ /

/

/

r

1

3-

0/

\

/

/

/

/

L/H

*

5

W

1

CM

j

4

///

/

/

/

4-

;

5

-

1

/

/

/

/ /

5-

ill 1

-

y/H

/

/

/

/

/

1

6

-

1 /

/

/ /

6-

-

III

III

j

III

/

/

'

1

7

1

/'

/

/

7-

'

1

8

/

/

/

8-

L/H

=

5

9

1

1

/ /

9

'

io-

1

1

'

/

/

10

'

II-

\9

-

liil

1/

1 1

II

1?

L

ill 1

lz

o

.2 .4

.6 .8 1.

l2

o

2

.4

.6

cr./yH

<r_/

y

H

(a)

At

Centerline

(b)

At Toe

of

Slope

Fig.

3

-

Distribution

of

Vertical

Stress

Along

Vertical

Sections

for

Varying

L/H

Ratios

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Fig.

4

-

Distribution

of

Vertical

Normal Stress,

a,

fy-W

On

The Base

of

The

Embankment

For

Varying

oc,

And

L/H

Ratios

7/17/2019 Perloff


as

L/H

increases

o

However

,

near

the

oister

edge

of

the embankment,

the

stresses are

still significantly

larger on

a

proportional

basis, than

indicated by

the normal loading approximation. Thus,

for

embankments

with

moderate

L/H ratios, the

normal loading approximation

leads to

lar-

ger

estimates

of

differential settlement,

assuming one-dimensional

com-

pression.,

than

would be

computed

by

the

method

presented

herein.

Horizontal

Norma

l

Stress

on

iijiii

i

iii

jti

fi

iimia i irnii

fin

i

n an.

1

^t

osnc—

w

t»

Contours

of

horizontal normal

stress,

0/7H,

for

a

»

U5

,

L/H

»

3

and

u

s

0.3

are shown

in Figure

5„

The

dashed lines

are contours

deter-

mined from, the

usual

normal loading

approximation. The

stresses shown are

those

due to

the

embankment

only.

The

figure

shows

that

the

maximum

hori-

zontal stress

occurs

within

the

body

of

the

embankment

and

decreases with

increasing depth.

In

the

foundation

material

in

the

vicinity of

the elastic

embankment,

0L/7H

is

less

than

half of

that

usually

assumed

The

effect

of embankment

shape

on

the horizontal

stress along

vertical

sections

through

the

center

line

of

the

embankment and

the

toe

of

the

slope,

is

illustrated in

Figure

6

for

a

=

k$°

and

|i

**

0.3

°

As

the

embankment

be-

comes

narrower

(L/H

<

1),

the

stress

is

actually

negative

at

some

points

below the

center

line. That

is,

the

embankment

causes

a

reduction in hori-

zontal

stress

at

these points.

The

dashed

line shows

the

stresses

determined

from

the normal

loading

approximation

for L/H

»

1.

The

stress

is

larger than that

due

to

the

elas-

tic embankment at all

depths.

In

fact,

in the

vicinity of the

embankment

it is more than

five times

as

large

under the

center line and twice

as

large

under

the

toe.

In

contrast

to

the

elastic

embankment,

the normal

loading

7/17/2019 Perloff


12

3

4

5

6

7

x/H

Fig. 5-

Contours

for

Horizontal

Stress

8

7/17/2019 Perloff


.2

.

u

4

(a)

At

Centerline

(b),

At

Toe

of

Slope

Fig.

6-Distnbution

of

Horizontal

Stress

Along

Vertical

Sections

for

Varying

L/H

Ratios

7/17/2019 Perloff


approximation

does

not

produce

negative horizontal stress

at

any depth.

The

reason

for

this

difference

becomes

apparent

when

the shear

stresses

transmitted

by

the

embankment

to

the foundation material

are considered.

This

is discussed

below.

The effect

of

Poisson's

ratio

on

the horizontal stress is

illustrated

in Figure

?.

This

figure

shows

the

horizontal stress

along

vertical

sec-

tions through the centerline

and

toe

of

the

embankment

for

a

«

1*5°

and

u

ss

0.5.

The

dashed

line

shows

the

stress due

to

the normal

loading

approxi-

mation

for

L/H

s

1.

Comparison

with

Figure

6

indicates

that

a

change

in

Folsson

e

s ratio

from

0.3

to

0.5

changes the stress

at

shallow

depths

below

the

central

portion

of the embankment

by

as

much

as

a

factor of

three. The

difference decreases

as

the L/H ratio increases

„

The

influence of

u

is

less

pronounced

balow

the

toe than

below

the center

line.

Horizontal

and

Vertical

Shear

Stress

Contours

of

horizontal and vertical shear

stress,

T/7H,

are

shown

in Figure

8,

for

a

=

h5° and

L/H

*

3.

The solid contours are for

Poisson

,,

s

ratio of

0.3*

The long

dashed

contours

are

for

n

«

0«5>

aaad

short

dashed

lines are

for

the

normal

loading approximation.

The

figure indicates

the

existence

of

horizontal shear

stresses

within

the

body

of

the

embankment,

increasing

to

a

value

at the

base near the

toe

of

the

slope,

in

excess

of

0.2

yH.

However, the

maximum

value of horizontal shear

stress

(approxi-

mately

0.3

7H) occurs

below the

base

of

the

embankment.

Like

the horizontal normal stress,

the

shear

stress, t

,

is affected

xy'

markedly

by the magnitude

of

Poisson

8

s

ratio.

However,

the

effect observed

depends

upon the

position

of

the

point considered,

relative

to

the

base

of

7/17/2019 Perloff


y/w/w//s>y/&/Ay//s///>y/A<y/

t<y//c>yA

y/H

4

(q)

At Centerline

(b)At

Toe

of

Slope

Fig.

7—

Distribution

of

Horizontal

Stress

Along

Vertical

Sections

for

Varying

L/H

Ratios

7/17/2019 Perloff


y/H

to

12

13

î

=

0.5

Normal Loading Approximation

10

Fig.

8-

Contours

of

Shear Stress,

~C

:

7/17/2019 Perloff


8

the

embankment.

In the

zone

below the eaibankment

to

a

depth of y/H

equal

approximately

two

or

three,

the

shear stresses

in

the

incompressible

ma-

terial

(n

*

0.5)

are

less than

for the

case

in

which

u

»

0.3.

At

greater

depths, the

reverse is

true.

The

shear

stress determined

from

the normal

loading

approximation

is less

than

that

for

either

u

above

a

depth factor

of approximately three

to

five,

and

more at greater

depths

.

The

magnitude

of this

effect

depends

upon

the

horizontal

location

considered,

as

shown

in the

figure,,

The

normal

loading

approximation

assumes that there

is no

shear stress

at

the

base

of

the embankment. Figures 3

and

9

indicate

that,

for

the elas°

tic

embankment, this assumption

Is

not

reasonable.

Figure

9

shows the

hori-

zontal

shear stress,

T-V7H,

at

the

base

of

the

embankment

for

u

«

0.3*

four

values

of

a,

and a

variety

of

embankment shapes

shown schematically

in

the

figure.

The horizontal shear

stress

is zero at the

centerline,

as

required

by

symmetry,

and

reaches

a

maximum

near the toe

of

the

slops. The

magnitude

of

the

maximum

and

its location

depend

upon

a

and the

embankment

shape

As

L/H

decreases

for

a

given

a,

the vsaximm.

T^/yH

increases,

and

moves closer

to

the toe

of

the slope. The magnitude of the increase

is

slight

for

a

-

15

degrees,

but

becomes

more significant

as

a

increases,

Bote that

a

maximum

i~Ji&

ia

excess

of

Q.k implies that the

horizontal

shear

stress

at the

base

of

a

forty

foot

high

embankment

may be

greater

than

one

ton per

square

foot

(unless

the

shear

strength of

the

material is

such

that

failure is induced).

To

assist

the

designer

in

evaluating the

significance of

these

results

to

his

particular

problem, influence

diagrams

for

vertical normal,

horl»

zontal normal

and

shear stress

distribution

for

a

variety

of

cases

are

7/17/2019 Perloff


Fig

9-

Distribution

ot

Shear Stress,

T,,/jrH

On

The

Base

of The

Embankment For

Varying

a,

And L/H

Ratios

7/17/2019 Perloff


Maximum Shear

Stress

It

is

often

useful to

consider

whether

the

maximum

{i

e.,

principal)

shear

stress,

i^^/rH,

at

any

depth

beneath

the

embankment

exceeds

the

a

available

shear

strength.

Thus,

it

is

desirable

to

know

the

magnitude

and

distribution

of

maximum

shear

stresses

due

to

the embankment.

Contours

of

^jjQx/yH

are shown

in

Figure K) for the embankment

section

of

Figures

2,

5

and

80

Note

that

the

magnitude of

t

transmitted

from

the

embankment

^^

max

to

the

foundation

material

is approximately

0,25

7H

at

the

base

of

the

em-

bankment in the

vicinity

of

the

toe. However,

the

largest shear

stress,

Oo337H,

occurs

beneath

the

center line

at

Y/H

2=

1.8. It is

also interest-

ing

to

observe that within the embankment,

the

maximum

shear stresses

are

larger

near

the

top

than

in the mid-depth

region,

and

that

they

increase

again

as

depth increases. This

is

believed due

to the

relatively

large

horizontal stresses

which

are

induced

by

the deformation mode

of

the em-

bankment

(cfo

Figure

5)0

Two

contours of

^-^v/yH

for

the

normal loading

approximation

corre-

sponding to the

embankment considered

are

shown

in

Figure 10

as dashed

lines

c

They indicate

a

shear

stress less than

that

produced by

the

elas-

tic

embankment

in a

shallow

zone

below the embankment, but

larger

shear

stresses

at

depth

Data

for

a

variety

of

embankment

shapes,

with

u

0.3»

and

a 9

15°,

30°,

U5

,

60°

and

75°

are

shown

in Figures 11 to

15,

respectively.

In

these

figures

,

the

maximum value

of

t^

at

a

particular depth

is

plotted

as

a

function of

depth

for

various

L/H

ratios

„

The

horizontal location

of

the

point

at

which

this

maximum

value

occurs

is

-also

shown

.

It

can

be

7/17/2019 Perloff


I

2 3

4X/H5

67

8

9

Fig.lO-Contours

for

Max.

Shear

,r

max/YH

7/17/2019 Perloff


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10

observed that

in

Figure

11,

that

as

the

L/H ratio

increases

for

a

-

15°,

peak

t

x

increases

in magnitude,

and

acts at

an

increasing

depth

below

the

embankment.

The

horizontal

location

of

the

maximum shear

stress at

a

particular

depth

moves

from

a

position

near the

toe

of

the

slope

im-

mediately

beneath the

embankment

to the

centerline

of the

embankment

at

a

depth

which

depends upon

the

L/H

ratio.

A

similar

trend

is

shown

in Figure 12

for

a

»

30°.

However,

in

Figure

13

(a

*

k$

),

the

largest

shear stress

occurs near

the

toe

of

the

slope

for

L/H

^ 0.0.

Although

the smallest

L/H

ratio shown

in

Figures

Ik and

15

is

0.5,

the

development

of

large shear stress

near

the toe

of

narrow

steep

embankments

is clearly

indicated.

The dashed

line

in Figure

13

shows the

magnitude of

the maximum

t

as a

function

of

depth

for the

normal loading approximation corresponding

to

the

U5

embankment

for

which

L/H

*>

1.

It

is evident that the

peak mag-

nitudes are nearly the

same

for

the two

cases,

but

it occurs

at approxi-

mately

twice

the depth

in

the

case

of the

normal

loading

approximation.

A

similar effect

is

evident in Figure

10

for L/E

a

3.

Thus

the

influence

of

the

elastic

embankment is more pronounced

nearer

the surface

where

softer soils

might be

expected.

As

a

result

,

it

may be

that current

es=

timates

of

stability,

potential creep

and

other

shear

stress

related

phenomena,

for

soils

at

shallow depths beneath

embankments,

are unccnser-

vatlve.

Relationship of

Results

to

In-situ

Stresses

It

is not immediately

clear

what

relationship

these results

have

to

stresses which

actually exist in

the

field.

In

the

case

of

a

built-up

7/17/2019 Perloff


u

embankment,

it

is

likely

that

the

embankment material

will

exhibit

sig-

nificantly different

mechanical

properties

from the

foundation

material.

For

a

cut-down

slope,

the assumption

of homogeneity

in

the two

zones may

be more

nearly

justified

.

The

non-linearity

in

the mechanical response

of

most natural materials

will

undoubtedly

also

influence

the

results

However,

the

feature which

may

be most

significant,

at

least

in

the

case

of

built-up

embankments,

is

the

fact

that

they

are

constructed

in

layers

rather than

instantaneously.

Thus

when

the topmost

lift is

placed on

an

earth

embankment,

the

upper

material

does

not

undergo strain

due

to

elastic

deformation

of

the

embankment resulting

from

the

stresses imposed

by

the

entire mass. Rather,

the strains

are due

only to

the

increment

of

stress

imposed

by

this layer.

The degree

to

which the

results would

be

changed

is not

clear.

However,

it

is

believed

that

the

results

presented

herein,

provide

a

more

realistic

estimate

of

stress

conditions

than

that

computed

from

the normal

loading

approximation.

Effect of

Results on

Str

ess Bath

Determination

LAMBE

(196U)

has

suggested

that

the stress path

method for predic-

tion

of vertical

settlements

of

cohesive soils is

superior

to

conventional

analyses

in

cases where

compression

is

clearly

not

one-dimensional.

This

approach

involves

three

basic

steps

(LAMES,

I96U):

1.

Estimation

of

the

effective

stress

path

of

an

average

element

in

the compressible

layer, for

the

field

loading.

2.

Performance of a

laboratory compression

test

which

dupli-

cates, insofar

as

practicable,

the

field

effective

stress

path.

3.

Computation

of

settlement

by

multiplying

the

thickness

of

the

layer considered by

the

axial

(vertical) strain

from

the

laboratory

test.

7/17/2019 Perloff


12

Because

the

strains

ia the

laboratory

sample

depend

upon

the

applied

stresses, the

method

requires

a

means

of

correctly

assessing

the

in«situ

stresses

A

comparison

of

the

total stress paths

for several

points

under

the

center

line

of an elastic embankment

{a

»

30

,

L/H

^ nft1|

=

Q„5,

u

«

0.3),

with those computed using

the normal

loading approximation,

is

shown

in

Figure 160

The

dashed

initial stress

line

shows the state

of

stress

in

an

elastic half-space,

for

which

ji

*

0.3,

before

construction

of the

embankment.

The three

points

shown

on the

line

correspond to the stress

states depths

of

0.5,

1*0 and

2.0 times

the

final height of the

embank-

meat, H-.

.

.

The

open points show

the

stress paths

during

construction

of

an

elastic embankment

continuous with

the foundation

material

„

The

solid points

show

the

stresses

for corresponding embankment

heights

9

de-

termined

by

the conventional

method.

Several features

of

this

comparison

are

especially

noteworthy;

lc

At

relatively

shallow

depths

(y/3

=

0.5),

the

conventional

method

leads

to

a

total

stress path

which

lies

entirely

below

the K line.

That

is, one

would predict

relatively

small

shear°settlements . However, on

the

basis

of

the

elastic

embankment

analysis, the

estimated

shear

induced

settlement

would likely

be

larger,

and compression

settle*,

ment

would

be less,

2

At greater

depth

(y/

H

final

e

2»0)

both

methods

lead

to

stress

paths

which

lie above

the

k

line. The two

paths

are closer, and

the

shear stress

under

the

elastic

em°

bankmeat

is actually

less

than

that due to the normal

loading

approximation.

3.

At intermediate

depths (y/H-.

.

»

1.0)

the

normal

loading

approximation

remains

relatively

close to the

k

line. The

stress

path

due to the elastic

embankment

is

still consid~

erably

steeper..

7/17/2019 Perloff


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7/17/2019 Perloff


13

Because of

the

influence

of the

applied

stress path

oa the measured

laboratory

settlements,

and

therefore on

the computed

field settlements,

it

would

seem

essential

to estimate the

predictive capability of the

method to

the

field

stresses

accurately.

In

the case

considered,

the

stresses

produced

by the

elastic embankment are

significantly

different

from

those

due to the

normal

loading

approximation,

at least

at

shallow

depths

.

The

effect

of

this difference on the

results

predicted

by

the

stress

path

method

is

not obvious,

however

this

question would

appear

to

deserve further

attention,,

7/17/2019 Perloff


Ik

CONCLUSIONS

The analysis

presented

herein

permits

determination

of the

stresses

within and

tinder

long

elastic embankments

which

are

continuous

with

the

underlying

foundation

material,.

The

results

indicate

that the horizontal

distribution

of

vertical

stress is

more nearly uniform than is usually

assumed. Thus,

differential

settlements

computed

using the

normal

loading

approximation

will

be

larger than

those determined

using the

stress

distri-

butions

presented herein

The

horizontal

vertical

shear stresses

created

in

the foundation

mater-

ial by

the

embankment, are

found

to be

significantly

higher

at

shallow depths

for

the

elastic embankment

than

for

the

normal

loading

approximation.

The

influence

diagrams

presented, provide the

designer

with

what

is

believed

to

be a

more

realistic

estimate

of

the vertical

stresses

than

that usually

employed.

ACHJCWIBDGMBHTB

The

research

described

herein

was

supported in

part

by

the

Ohio

Department of

Highways

in

conjunction

with

the

U.

S. Bureau of

Public

Koads,

and

in

part

by

the

Joint

Highway Research

Project,

Purdue

University

in

conjunction

with

the U,

S, Bureau

of

Public

Roads.

The

assistance of

W. L.

DeGroff,

R.

Hoekema,

and B.

Corbett in

the

prepara-

tion of

the

drawings

is

gratefully

acknowledged.

7/17/2019 Perloff


15

REFERENCES

CITED

BROWN,

Co

Bo

8

(1962)

Incremental Analysis

of

Gravitational Stresses

in

Embankments and

Their Effect

Upon

the

Failure

of Earth Structures,

Ph. D«

Dissertations University

of Minnesota „

CARLTON,

To

A., Jr.,

(1962),

The

Distribution of Gravity

Stresses

in

a

Symmetrical Trapezoidal Embankment

and its

Foundation, Ph„

Do

Thesis,

University

of

Texas,

1962.

CAROTRERS, S

D.,

(1920),

Direct

Determination

of

Stresses,

Proceedings,

The

Royal Society of London,

Series

A.,

Vol*

XCVII,

p.

110

CHURCHILL, Be. Vo,

(19*48),

Introduction

to Complex

Variables and

Applications,

McGraw-Hill Book Company,

Inc.,

Mew York.

CLGUGH,

Eo

and CHOPRA

s

A.

K.,

(I966)

Earthquake Stress Analysis

in

Earth

Dams, Journal of the

Engineering

Mechanics Division,

Proceed

ASCE,

Vol.

92,

Ho.

EU2,

April,

1966.

DAVIS,

E

Ho

and TAYLOR,

H.,

The Movement of Bridge

Approaches and

Abutments

on

Soft Foundation

Soils,

Paper

No.

37?

Proceedings

Australian

Research Board,

Part

2,

Vol.

1,

1962.

DINGWALL,

J.

Co

and SCRIVNER,,

?, H.,

(I95U)

Application

of

the

Elastic

Theory

to

Highway

Embankments

by

Use

of

Difference

Equations,

Proceedings,

Highway

Research

Board,

v.

33*

p°

^7U.

FINN,

Wo

Do

L.

s

(i960)

Stresses in

Soil

Masses

Under Various

Boundary

Conditions, Ph. D.

Thesis,

University of

Washington,

i960.

FINN,

Wo

Do L„,

(1966)

Static

and Dynamic

Stresses in

Slopes,

Proceedings,

of

the First

Congress

of the

International

Society of

Rock

Mechanics,

V. II,

p.

167.

GOODMAN,

Lo

E.

and

BROWN, C.

3.,

(1963)

Dead

Load

Stresses and Instability

of Slopes,

Journal

of

the

Soil

Mechanics and

Foundations

Division,

Proceedings

ASCE,

V.

89,

n

SM3,

May

1963.

HENDERSON, F.

Mo,

(i960)

Elliptic

Functions

with

Complex

Arguments,

University

of

Michigan

Fires

s

,

Ann

Arbor

HXLDEBRABD,

?. B.

s

(1956) Introduction

to

Numerical

Analysis,

McGraw-Hill,

New York.

JURGEIS3QN, L.

(1937)

The

Application

of Theories of

Elasticity

and

Plasticity

to

Foundation

Problems,

Journal of

the

Boston

Society of

Civil

Engineers, July,

193

1

*.

LAMBE, To

Wo,

(196U)

Methods

of

Estimating

Settlement ,

Proceedings

of

Control of

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IS

MUSKHELISHVILI.,

N„

l,

f

(1953^

Some

Basic

Problems

of

the

Mathematical

Theory

of

Elasticity

, P.

Noordhoff,

Groaingen,

—

—

—

3EHMARK, N.

M„

5,

(19^0)

Stress

Distribution

in

Soils,

Proceedings

Purdue

Conference

on Soil Mechanics

and its

Applications,

p,

295,

NEWMARK

S

No

M»

9

(1942)

''Influence

Charts

for

Computation

of

Elastic

Stresses

in Foundations

s

Bulletin.,

University of

Illinois

Engineering

Experiment

Station, No.

32$

«>

OSTEBEERGj

Jo

0,, (1957)

Influence

Value

for Vertical

Stresses

in

a

Semi-

Infinite

Mass Due to

an

Embankment

Loading,

Proceed

ings of

the Fourth

International Conference

on

Soil

Mechanics and

Foundation

Engineering,

Vol= I,

p.

393*

RENDULIC<,

L

os

(193S)

Der

Erddruck im

Strassenbau

und

Bruekenbau,

Forschungsoarbo

Strassenwesen,

Bd„

10,

Voile

u<»

Reich Verlag, Berlin

T3RZAGHE

K.

s

{I9U3)

ôretig

al

Soil

Mechanics,

John

Wiley,

Few

York.

TB5OSB&HK0,

S

c

and

GOCiDIERj,

Jo

N*

s

(1951)

Theory

of

Elasticity, McGraw~

Hill,

New

York.

TROLLOPE,

D.

fi,

5

(1957)

She

Systematic Arching Theory

Applied

to Stability

Analysis of

Embankmente

,

Proceedings 4

th International Conference

on

Soil

Mechanics

and

Foundation Engineering,

V.

2,

p.

382

WYLIE,

Co Ro

S

Jr.,

(1966)

A

dv^g^J

Etogjêering

Mathematics, Third

Edition,

McGraw~I3.il,

New

York.~

ZIENKXEWICZ* 0. Co

j

(I9U7)

Stress Distribution in

Gravity

Dams,

Journal

of

the Institution

of Civil

Engineers,

V.

27,

January

1947,

p» W

ZIEMKIEWICZ

S

Oo Cc

and

CHEUNG* Y*

K*

3

(1964)

Buttress

Dams

on

Complex

Bock

Foundations,

Water Fover, Vol.

16,

No.

5,

May

1964,

p.

193.

ZIENKIEWICZ

a

Oo

Co

and

CHEUNG

9

Y*

K.,

(1965)

Stresses in

Buttress

Dams,

Water Power

,

Vol.

17?

No.

2,

February

1965,

p»

69.

ZIENEIEWICZ, Oo Co

and

GER3TNER,

R.

W

es

,

(I96I)

Foundation

Elasticity

Effects

in

Gravity

Dams,

Proceedings

of

the Institution

of

Civil

Engineers, V.

19»

p.

209

7/17/2019 Perloff


17

APPENDIX

1

SOLUTION

OF

THE

PROBLEM

The method

of solution

is

a

modification of

the

I^KHELISHVTLI

(1953)

method

o

A

brief

outline

of

the solution

is

given

below

„ A

more congolete

discussion

of the

details will

be presented

in

a

forth-

coming paper.

For the plane strain

problem,

the stresses

can

be defined

in

terms of

an Airy stress

function,

U

(x,y)

as

a

x

'

~~

>

dy

-,

2

*

1

-

ja

7J

d

U(x,y)

f. %

where

«

.

a

,

t

are

the

horizontal

normal,

vertical

normal and

shear

x*

y*

xy

stress,

respectively

j

7

is

the unit

weight

of

the material,

^

is

the

Poissoa's

ratio o

For

a

case

in

which

weight is the only

body force

acting^

the requirements of equilibrium

and

compatibility

will

be

satisfied

if

(TIMOSHEMO and

GOCSD23S,

1951)

V

k

J3

far)

-

(2)

2

where

V

is

the

Laplacian

operator.

When

the boundary

conditions

of

the specific

problem

are also satisfied,

the

unique

solution

has

been

obtained.

In

considering

the boundary

conditions associated

with

the

embank«=

7/17/2019 Perloff


IB

form.

Referring

to

the

plane in which

the

embankment

section is

shown

in Figure

lb,

as

the z-plane, a

point

within

the

medium

can be

repre-

sented

by

the

complex

number,

z

«

x

*

iy„

Assuming

that the

stress

•

function is

analytic

within

the

medium,

the

function

can

be

written as;

U(x,y)

-

Re

[

z*(z)

*

%(z)]

where z

a

x

-

iy,

<t>

and

3&

are

single-valued

analytic functions

through-

out the

z-plane « The functions *

and

X

are

determined from

the

condi~

tions that the

normal and tangential

stresses

on

the boundary are equal

to

zero,

Substituting Equation

3

into

Equations

1,

and

expressing

the

stresses

in

terms

of

the boundary

tractions leads

to:

zV'(z)

(U)

x

.(.)

+

p

(£&)]

,

?

(A.-]

where

a bar

indicates the

complex

conjugate

of the

quantity, 9 is

the

angle

between the

slope

and the

x-axis

measured in

a

clockwise

direction,

u

is

2toisson°s

ratio,

N

and

T

are the normal

and tangential components,

respectivelyj

of

the

boundary

traction.

Evaluation of

the

stresses is

then

accomplished

by

a two

step

trans-

format

ion.

o

First,

the

boundary of

the

z-plane

is

transformed

into

the

straight

line

boundary

of

an

auxiliary

plane, the t-plane,

t

<»

?*

+

itrj,

by

the

application

of

the

Schwarz-Christoffel

transformation

(CHURCHITI.,

19^8):

t

x

z

«

f(t)

«

R

f

l^^-^jt]

a

dX

*

S

(5)

>-0

7/17/2019 Perloff


19

where

P.

and

S

are

constants,

p

is the

modulus

;

X

is a

dummy

variable

and

n

it/a. Note

that

when

n

a

2,

corresponding

to

a

slope angle

of

90°,

the

integral expression

in

Equation

5

is

aa

elliptic

integral of

the

second

kind, for which tables

or

charts

(HENBERSCK,

i960)

are available

For

those

cases where n

is

larger than

2,

the integral can

be

evaluated

numerically

on the computer

.

In this

analysis,

Equation

5

was evaluated

for

all

values

of

n

on

the

XEM

709U

digital computer by

a Simpson

°s

rule

integration

(KILDEBHAMD,

1956)

Because Equation

5

represents a

conforms transformation,

straight

lines

€«»

constant and

t) -

constant in

the t-plane,

correspond

to

orthogonal

curvilinear

coordinates

in

the

z-pXane,,

This is

illustrated

in

Figure

A„la.

Thus, the

boundary conditions

in terms

of

*(z)

and

7C(z)

can

be

written as

functions

of

* [f(t>]

-

{t)

and

3tff(t)J

-

7t(*)«

Then the boundary

tractions becomes

n

*

it

=

o=$ct)

*

ytsr

*

g(i

£

n

)

**

f

f

c*>]

m

4$f

*iw*

*[*&}*«*&

(6)

where

&(t)

«

'(t),

Wp(t)

•

X

8

(*)

a^

f

e

(t)

is

the

integrand

of

Equation

5<

Recognizing

>}

s

and

t

»

£

in

Equation

6,

and

rearranging

leads tos

sfo)

1+

(l-2u)

fTTT

Im

Jf(f),

:

(7)

7/17/2019 Perloff


20

In

this

form,

Equation

7

expresses

the

effect

in the t~plane

of

the

geometric

shape

of the

embankment.

This

effect

can

be

visualized as

a

fictitious

loading

applied

to the

boundary

of

the

t-plane

The

real

and

imaginary

parts

of

this

fictitious

boundary

loading

are

shown in Figures A,

lb

and

A.lc,

respectively

Ifeving

expressed

the

desired

functions

of the boundary,

it

is

necessary

to

determine

them inside

the

boundary.

This

is

accomplished

by

the application of

the

Cauchy

integral

formula.

This formula

states

that

for

a

given

function

g(C)

along

a

closed

contour, C,

which

satis-

fies

certain conditions, the

value

of the function

at

an

interior

point,

t, is

(WYLIE,

1966):

g(t)

2ai

(8)

If

the point

t

is

exterior to the

closed

contour,

then

the

integral

expression

equals

zero.

By

this means,

the desired

functions

can be

evaluated inside

the

boundary.

Knowing

ty(t)

and ^(t),

and

substitu-

ting Equation

3

into

Equations

1,

leads

to

the

determination of

the

stresses

a

-

2

Re

3v

<JKt)

-Re

[fW

-f#

*#t)]

+

f^jrlm

[f(t)

o

y

-

2

Re

f

(t)

+

Re

[

fW

-$$

<£(t)

7

Im [f(t)]

(9)

v

-*

7/17/2019 Perloff


MM

(a)

Geometrical

Transformation

stffi

,,,,

1

U

rrr

rr>

e

(b)

Transformation

of

Fictitious

Stresses on

Boundary

-Real

Part

(c)

Transformation

of.

Fictitious

Stresses

on

Boundary-

Imag.

Part

Fig.

A.I-

Graphical

Representation

of

Transformation

Procedure

7/17/2019 Perloff


21

AEEESSD2X

2

This

Appendix

contains

influence

diagrams

for

vertical

normal

stress

and

horisontal normal stress

along

vertical

sections for

\t.

e

0.3,

a

e

15

,

30°,

h%

,

60

and

75°,

and

various

eE&ankment

shapes.

Influence

diagrams

for

horizontal and

vertical shear

stress

for

a

=

^5°,

ne0,3

and various

ejBbaasioasafc shapes are

also

given*

The diagrams

indicate

the stress

due

to

the

embankment

weight alone.

Stresses

due

to

the weight of material

underlying the embankment

aaist be

superimposed

to

obtain

the total stress, The

stresses

are

expressed

in

disaensioslsss

form

as

JyH

9

o_/?H or

'^/?'

H

'•

2ite

coordinates

are

also

ia

dlmeaaionless

form. For convenience

in

the

seaa-logarithsie plot,

the

depth

is

measured

from

the tog

of ths

embankment, and

designated y/K.

This

is

ia

contrast

to

the

discussion

ia

the body of the paper

where

the

vertical distances

are

measured

from the

base

of

the embankment and de-

signated

y/H.

Each

of the four

diagrams in

a

given

figure

refers

to

a

particular

vertical section,

shoaa

schematically on

the

dis;gram.

The upper left

diagram

indicates

stresses

along a vertical

section

midday

between the

centerlins

and

the

tot?

of

the

slope;

the

lower left

diagram indicates

stresses

aloisg

a

vertical

section

through

the

toe

of the

slope;

the lower

right

diagram

indicates

stresses

along a

vertical

section

located

a

dis-

tance

from

centerline

equal

to

1.5

times

the

distance

from the

centerline

to the toe

of

the

slope.

7/17/2019 Perloff


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