PILE Foundation Analysis_ASCE William Saul-1980_searchable

8/17/2019 PILE Foundation Analysis_ASCE William Saul-1980_searchable

1/22

PILE FOUND TION N LYSIS

illiam E

aul


2/22

October

8 9

UII EX, bladison

PTLE

FOUNDATION N LPSZS

i l l i a m

E sau1.l Fellow

ASCE

The design of pi le foundatioaa for

s t a t i r

o r dynamic lo cur be

ucom pl i shd through the s t i f f n u s metbod o f

u u l y s utilizing

so i l -

p i l a i n t e r ac t i o n l o d r l r deriuwi from tbe b-prirrg f o d a t l a n

id -

ration although ariy

linear

o r p iec eu is e l l n u r m o d d v i l l su f f ice -

the

Podel a d v w t d

may

b e of f i n i t e l e n gt h o r m d e up of a s e r i e s o f

f i n i t e

length.

due to v ri tion

in

t he s o i l o r

th

pile.

Tbe st ru ct ur al ana lys is of p il e foun tions requir es computation of

disp lac eme t of the p i le cap and forces on and displacements of each of

the p i le s .

A l en gt hy s e t o f r e f r r e n c n t o this problem were givan by

Saul, 1968(1), i n a paper vhich presented th e bas ic framework of t he

material contained herein.

Subsequent ad di ti on s by O'Ueill(2). Murthy

and Shrivastava(3). Pr ll uo b

and

Chandrarakaran(S), Saul(5.6) and

S a d

and 'blf(7. 8) point out shortcomings t o th e orig in al paper and supple-

ment

t

considerably.

For wi kt io n of th e problem and th e computational

techniques have not been f o d acking; se e VesiL(9), Eoules(lO), and

Arya O'Neill and Pfncw(l1).

Imprweteente and discussion of the pile-

s o i l int er ac ti on modeling, hovever,

contin ua t o appear; se e reference:,

( 9 , l l ) for , .general d iscussions

and

smmaries.

Ya'ajor work

i s

being

done by ~e n i f ( 9 ) ,Poulo s(l2) and by h'ovak(13) and th ei r collea goea,

*ere tha references cit ed ar e samples of t he ir exteasive vorks, on pile-

s o i l i n te r ac t io n .

The prinur y t hr us t i n th ese worlu, however, has been

derived from elastic theory although

in

a -re compl ete and complex

sense than the spring foundation idealization. This l e a d s t o a b e t t e r

underatanding of the problem; but as ye t, vi th s o i l pro per tie s normally

ava i lab le ,

t i s

questionable vhether any model

s

b e t t e r s u i t e d f o r

design

and

anr lys f s in pract ice .

In tera ct ion of the p i le cap with

th e s o u has been inve sti gat ed by O',Murka and Dobry(l4).

When t h i s

f a c t o r is t o be included the s t i f fn ess contr ibut ion of the soi l-cap

a9y be di re ct ly added t o t he foundation sti ff ne ss matrix a s computed

herein.

The object ives in t hi s paper are to:

1

Update the material

presented ear l ier(1) . 2 Include inprovements developed since(5,6,8).

Professor, Department

o f

C i v i l

and

Environmental

Engineering

The

Univsr-

s i t y of Wisconsin, Madison 53706.


3/22

3. Prese nt a major development which a llo vs inclus ion of layered s oi ls ,

p i l e s which may vary

i n

s e c t i o n

v i t h l e ng th , o r s h o r t p i l l n g

9).

And,

4

Give exapplr r to i l l us t ra te the coatputat ioaal method.

A DK3ONSTRATXON

Consider a p i l e focmdrtioo conr iscing of a rigid r e i n f o r e d cam

c re t e u p and any rider o f a t t l c h d p i li ng .

Tho p F l u may b e v e r t i c a l

o r ba t t e r ed , shor t o r long ,

a t

t h e same r e f e r m a c e e l m L I o a vit kl t r

c a p o r

a t

severa l levels a d f s im i l r o r d i f f e r e s c m f . ~

sec t ions o r s i zes .

The p i l e a m y

b m

m hcod in m y

m or

o r

by d i f f e r en t methods a t t h r

samm

t ~ ~r t

varlolu

tima

heir pr in

c i p a l xes may b e a t ny angle and hm any d e g m of f i x i t y b eernem

hinged and r ig id v i t h th e cap.

The cap is ammamed rigid

in

l a t o r c o w

putat ions) , but may be of

.any s h p

and

thichress

variablm

thickness such

s

stepped, and may be coas t r a r t cd 80 that t h e p i l e s

are enbedded a t d i f f e r e n t e l m t i o n . v i t h i a tha cap.

The s o i l mny be

h o m ~ g m w w ,

aryLng

o r layered , i nc lud ing l aye r s o f a r y

w e ~ k

r

n e g l i g i b l e s o l l s .

The pi le

cap m y

be in a m t a c t o r

embedded

i n t h r

top l ay er of s o i l o r e l m t e d

am

a

platfurm.

I n

sunnmv, fo r the

exper

i m m t

t h e r e a r e n e a r ly no cons t r a in t s .

To proceed, a c oo rd ia at m c a t e r f o r t h e p i l a f ou ud at io n

is

assigned

and a s e t of C ar tesian coordinates.

Although these ~ P Ie asmipad

a r b i t r a r i l y , it is

useful

t o choose

an

o r i gi n a t l e a s t v e r t i c a l ly

a l igned v i th the cen te r o f

mass

of the foundation and/or load and have

o ne h o ri z o n t a l a x i s p a r a l l e l t o a n

ads

of syuauetry of found ation or

loads, should one exist .

The demonstration

s

t o load the founda t ioa in e ch of s i x comporr

en t s ,

one

component a t

a

t i m e . T he se s i x a r e t h e t h r e e r e c t i l i n e a r

f or ce s m d t he t h re e r m m ~ t s orresponding to the

ax s

us t es t ab l i shed

as

shown i n

Fig

1 When

w e

of these forces

i

i s

applied, and no w

other , the s i x corresponding components of de fle ct i on , ( ~ 1 ~ ~ 3ectf -

l in ea r and

3

r o t a t l o n i l ,

may

be measured and plot ted .

These forces ~y

be appl ied

by

incre men ts of for ce, i.e., dead loads, o r d i s p l a ce m n t ,

i.e. by jacking, by d e r r l a rg e r incra-ta, s l o w l y o r more

rapidl y , and mono toniul ly o r cycled. The re su l t i ns c am s may be

non l inear v i th the r a t e o f de f l ec t ion inc reas ing v i th h ighor loads.

Creep o r relax atio n, depending on t h e type of load t es t , y e h w a

t ime dependency, 5-e ., a v is coo las t ic m t e r i r l . In

many

c u e s , t he

load-deflection curva v i l l exh ib i t a regima wi th a s l w l y ch an gin g o r

air-st constant s l o p s f o l l w e b;

a regime with a rapidly changing s l o p ,


4/22

a hod

b

Displacrrrotr

Fig 1

~ o ~ d a t i o n

ardr and Dfsplacawntr


5/22

and then anothe r regime v i t h a very stee p, perhaps again constant, s lope.

Although t h is may appear as a typi cal s t r e s sa t r a i n currre wi th an elas -

t ic ,

yield and pl as t i c behavior such

i s most

probably

no t the case and

it

may of ten be di f f i cu l t t o dis t ingu ish t he three zones . The reason

is

t h a t s o i l is o f t e n v i s c a e l a s t i c and t h a t t h e r e a r e

sir.

e f f e c t s ,

edge

ef fe c t s , pos s ib la f r i e t i oa be tween cap nd

soil

s o i l p r e ss ur es d o v e l o p

ing

against

cmbdded caps,

and

y

nu nk r of o ther pos s ib le in f lueac8r

v hi ch co ul d i n h i b i e n p m d u d b i l l t y o f

test

r e s u l t s .

N w e r t h e l u ~ , f

sugges ted that if t h e l o a d

vere

c7eL d a m r a 1 , t i m r s

in

t he neighbor

hood o f magnitude which i s . ~ a a t w l l y xpected, t h a t a near ly cons tant

elo pe t o each cunre would b e found.

I f

the

f luc tua t i on of load magni

tude

was

expectad t o be wid.

t he c me could be approximated a s bi-

l i n e a r o r p ie ce wi ne l i n e a r .

Thus,

each of t he 6 loading condi t ions

produces

6

def l ect ions d where

d

D /Q hat

is

t he def l ec t ion

13 13

i

j

in

d i r e c t i o n

i

due t o a

uni t

l o ad i n d i r e c t i o n j dii,

is

the measured

d e f l ec t i on a t

i

D

divided by the load

Q

a t

j '6e 6

by

6

mat t ix

of

11 1

t h e s e f l e x i b i l i t y i n f l u en c e c o e f fi c i e n t s

[ d l

where each coluum

b

is

f

produced by one load, is the e t r u c t u r a l flexibility matrix.

It

m y

be

inver t ed to ob ta in the s t r uc tu r a l s t i f fn es s mat r ix IS], IS1 [dl ,

vhere each coef f i c i en t

S

i s

t h e f o r ce a t p o s it i on i duo t o a u n i t dls-

11

placement a t pos i t io n j vi th a l l o ther d i sp lacecnmts equal t o zero .

It

i s t o be observed t h at the slo pe determined upon immediate load-

i n g of a p i l e f ou n da ti on , o r v i t h

a

l i gh t load ing ,

o r

by us ing a di f fer -

e n t ~ounda tion design, 'such a s a scaled-down configuraei on, would be

d i f f c r en t , u s u a ll y s t i ffe< than t he va lue ob tained a s described ear l i e r .

Such a t e s t mu ld

be expensive, t i m consuming, and tru e only fo r tha t

foundat ion in t ha t place. Hovever t h e s i g n i f i c a n c e

i s

i n understanding

t h e n a t u r e o f t h e s t r u c t u r a l e t i f f n e s s n a t r i x s i nc e

it

i s

a r ea l and

h p o r t a n t p r o p er t y o f t h e

foundation

i n fa ct , the foundatfon 's s ignature.

I e la tes fo rces , o r loads , app l i ed to the foundat ion and th e r esu l t ing

d isplacements i n the coord ina tes a l r eady ucab l i sh ed .

Thus,

vhere {Q) ar e loa ds and {A def l ect ions of the foundat ion.

The

compon-

ents o f [S ] a r e c o ns t an t i f t h e s y s t e s

i s

l i n e a r

o r

quas i- l inear a s

descr ibed e ar l i er ; otherwise they are var ia bles vhich may be taken a s

piecewise l inear .

PTLE

BEH.4VIOR

Sinc,e determir . \ t ion of the s t r uc tu ra l s t i f fn es s matr ix

[ S ]

by experi-

ment

has

t h e

limitations

noted a des ign 3l ternatPre

is

necessary and


6/22

provided. I t i s f i r s t usefu l t o cons ider a s im i la r demns t r a t ion con-

ducted on a s ing le pi le .

he

p i l e may be placed i n any mnn er, be of

any shape or materials ,

any

length, be placed n any type of s oi l o r

so i l s , and be f lush v i th the sur face o r ex tend in t o the a i r

For this

d e m n s t r a t i o n t h e p i l e should b e v e r t i c a l .

Coordinate axem are chosen

along the lon gi tudina l cen troi dal ax is and the pr in cip al xxas of bend-

ing. Applying

6' loads , d on g each axis and

a

molnnt about each ,

one a t a

tiw as

shown

in

Fig.

results

i n

fn

dfsplrc8mest vector

{ c ) ~

o r

each

load. Nth arg b vec tor (Cl i

hU

6

coaponeuts, most

w i l l

be zero

s i x

flmmue

abou

a p r inc ipa l

axis

shoul m ut-of-

p h o r s b u d

and

axial c a r p o n a f s .

Rw the arial

and tors-

l o r d s a r e exputad

to

r u u l t in o n l y axia l

and

t o r a i o o r l d i r p h ~ t r ,

r espec t fve ly ,

and

t he f l ~ r m - p t o d u c i n g

oad8

in only 2 compaornt d i e

p k r a e n t s each.

he

mat r ix [c] t h m

is

quit s pa rs e, v i t h o d y s l i g h t

f l e x u r a l

coupling.

Once agaio, the soi l -pi lm intmract lon is malio.at

s o

it

would b e w m f u l t o cycle th. load l n t h e neighborhood of magrdtudm

of the l ol ds expmcted

so that

a r e a l i s t i c l i n e a r a p pm xi rm ti on c a n be

achieved betueea load apd daf lec t ion .

Once th i s constant s lop e i s

selecte d, dividing the

mwured

d i sp l ac e wn t s { c ) ~ y th e sagnitude of

t h e l o ad i n d i r s c t i o n

i

t he r e su l t ing d f sp laccmrncs a ra the p i l e

f l e x u r a l i n f l ue n c e c o e f f i c i e n t s g

the def l ecr ion in t h e d i r e c t i o n

i j

t a u n i t f o r ce i n d i r e c ti o n

j

wi th a l l other forces zero.

The pile

f l e x i b i l i t y

catrix s] may

b e i n v e rt e d t o o b t a in t h e p i l e s t i f f n e s s

matr ix [b ] where bWij i s t h e f o rc e i n d i r e c t io n

i

due to a un i t dis -

placement i n di rec t ion j w i t h a l l o t h e r d is pl ac e me nt s

zero

The forn

o f t h e s e p i l e r u t r i c e s v ou ld b e

and thm fo rc es (PIi and displa canmn ts

a c t i ng o n p i l e a r e

re la t ed by

1 l i i and I r I i

[gllCFli

3

GROUP

CTION

Xhen placed in to t he foundation coordinate system, as shovn in

M g 3, th p i l e may b e b a t t e r e d , t h a t i s , placed a t

an

angle Y v i t h


7/22

C

Elevation

v i t h

espect

xis of Pile

Ng P i l e Cap with P i l e


8/22

t he v e r t i c a l r on a ba t t e r s l ope o f l / h i w he re c o t a nge n t

Yi

hi and

i s t h e c l o c k v i s e a n g l e t o t h e d i r e c t i o n o f b a t t e r f ro m t h e

a x i s o f

1

t h e f o un d at i on i n p l an v i w . F u r th e r , t h e p i l e head is l oc at ed a t

c oor d ina t e s

du ,u ,u v i t h r e sp e c t t o t h e u p c o o r d i n a t e s ys te m m d

1 1 2 3

t h e p i l e s p r i n c i p a l a xe s may b e r o ta t e d t o n a n g l e E i v i t h r e s p ac t

t o a c o o r di n a te n p s tc a d e x r l b a d by t h r v e r t i c a l p l a n e co n t a in i n g t h e

b a t t e r e d p i l a vh e r a

ui

i s p e rp e nd i cu l ar t o t h i s

p h

ud

in

t h a

hod

w n t a l p l a n and

U

i pe r pe nd i c u l a r

to

ui and tha

l o n g i t u i h d u s

u; of the p i l e .

Using t h o approprlrto t r a n s f m e i o n m , t h o s t i f f n e s s

mat

o f the p i l o v i t h

rwpmct

t o

the

fauodrtfo eoardhata

system

U

is

T T T

[S Ii

~ ~ l ~ ~ a l ~ ~ ~ l ~ ~ b l ~ ~ ~ l ~ ~ a 1 ~ ~ d l ~4 )

v h e r e

i

CO S E sin^

[ P * ] ~ - [ - . ~ c

and

sins

e inyc osa

C O W si ysina

0 c osy

which m y b e v r i t t e n

T

Is l,

[ c a ~ l ~ [ b l ~

The s t i f f n e s s n a t r i x of t h e f o un d at i on is th e sum of the s t i f f n e ss of a l l

t h e p i l e s n i n t h e f o un da ti on


9/22

Thus,

from Eq. t he l oa ds may be de terminad for a g i v e n d i s p k c w r n t

{ }

of th e foundatiota o r

the

displacemenl: deterrakud for a given load

191

Once the foundat ion di sp lac aw nt s {A} are det arar ined , t he f o r e a

and d i s p ~ ~ t sf i n d iv i d ua l p i l i n g may b e . c u b t d

in

t h e c w r d i ~ ~

ate

sys t e m pa ra L l d

to the

foun t i on e oor d i or t r U

from

or

i n member pri nc ip al

xu

f r m

T

i ~ 1 ~[ c a d

i { ~ ~d

( ~ 1 ~t b - I

i ~ ) I ~ * J ~ ~ C = P I : { A I

13)

whe n e i t he r member p r i n c i p a l

axis

is horizonta l [p] [I] and there-

fore , the rota t ed member s t i f fn es s mat r ix [b] here

Ibl Ipl D I [PI= 0 4

becomes id en ti ca l with [b'] . Elements of

Eq.

10

wi th [b'] [b]

are

presented

in

an al yt ic form i n Appendix I

SOIL-PILE I ~ T I O N

ODELS

l e components of t he pi le s t i f f ne ss mat r ix [b '] , E q s . 2 and 3,

may be obtalned by experiment ,

s

note d , bu t a n a l y t i c node l s us i ng

r e a d i l y a v a i l a b l e s o i l d a t a a r e n ec es sa ry .

Herei n f i v e a na l y t i c pode l s

a r e p rese n te d , a 11 based on t he spr i ng founda t ion i de a l i z a t i on fo r

l a t e r a l l o a d i ng ( f le x ur e ).

It nay

be

assumed th a t i n the ne ightborhood of in te r es t , i .e .,

load magni tude and p i l e dimensions, the so i l ' s rea c t iv e pressure on

t h e p i l e

i s

l i n e a r l y p ropor t i ona l t o t he de f l e c t i on , t hus , de f i n i ng

3

a

value

ks

i n units

of

p r e ss u r e p e r u n i t d e f l e ct i o n su ch a s l b / i n o r

3

N m

which

is

a p r o p e rt y of t h e s o i l .

The pressure then

i s

k x

s i

which is a nal ogous t o a l i ne a r sp r i ng . C ons ide ri ng t he p i l e a s a

b-az t h e r e a c t i v e f o r c e p r uni t l engt h of bean

may be

expressed a s

k D.x where

D

is t h e p ro j e c te d wi d t h of t he beam i n t he d i r r e t l os o f

s r i

bending. ow

k

s n o t

a

pr ima ry s o i l p rope r ty bu t

may

be e x p r e s s~ d n

s

t e n s of e l a s t i c c on st an ts

(15) or adjus ted

f o r

size

and shape (16)

from tes t da ta .

Thus,

i t m y

e t h a t t he p roduct

kSDi=k:

is

nore aean-

i n g f u l t h a n

k

alone.

S i nce t ba spring concept neglec ts shear coupl ing

s

i n t h e i d e a l i z a ci o n , i t a ppea rs t ha t t he re c e r t a i n l y should

b e

considered

an c ge ef

fecr

a s

w e l l as

a

irecc

s pr l ng

e f f e c t . This, hovever,

i s

done


10/22

when t es t s v i th

a u n i t

p l a t e s i z e a r e used as a s t andard to de te rn ine

k and th is value i s then adjusted f or shape and size .

s

The subgrade modulus al so v ar ie s with depth because of confinement

of the so i l , so i l p roper t ies , and poss ib le var i a t ion of so i l wi th depth ,

i.e., la ye ri ng (17).

I t

i s ea s i l y pos tulated and confi rmed by mar su rr

ment th a t the s o i l a t su r f ace about a freestanding p i l e has no ver t i ca l

cons t r a in t

and

there fore, cannot support even

ow

values of hor izontal

pressure.

Bowever, prw.nce of a

cap

may p rm r id r t h e c o o r t r a i n t o r

accounc f o r t h e c a p m i l f r i c t i o n . The rubgradr;nodulua for an over-

coruolldatd c o h e s i a s o i l a p p u r r t o approach e oasc aa r d u e with

depth once out of

ran-

o f s u r f a c s e f f e c t s , i.a.

k

D coastant. Raw

a i

e w r ,

n

granuLu

o r nonmlLy

10

co ha iv a soi l. , th o rub+e

modu

1

in ruses vith

depth and uny be a s s c ud t o 'do so l i n u r l y s o

tha t

k @ vhera 2 is d e pt h o f s o i l and t h e soil p a r t o r

h s the

s

d t s

of 1 b / h e 4 o r N/= .

Solut ion of the beam equat ion int roduces p a r w c e r s and

9

vhera

1

which hap.

d t s

of p e t un it lengt h, i.e., in.-1 o r

mn

.

Note that

thene parameters

are

directional, i .e. , ~ b e d i f f e r e n t v i t h r e s p ec t t o

each p d c i p a l a d s i f Ii and or D ar e no t equal .

I f o r I.n

where L is th e eubedded l eng th of the p i l e , the p i l e aay be considered

u being long o r in f in i t e in l eng th ; which

means

t h a t l a t e r a l de f le c -

t io n has been ef fec t iv el y damped to ne gl i gib le above th e pi l e t i p .

For

the

case

of

a

long p i l e four models f o r t he f l exura l s t i f fne s s coef f i -

c i t r a re p sented ' in Appendix X I

They

are:

Al s i n g l e l a y e r o f s o i l v i t h k a constant.

s

A2. s i n g l e l a y er o f s o i l v i t h k -2 i.e. increas ing l i nea r ly wi th

s

depth.

81.

A two-layer system where

kS=O

in

the top laye r because of an ele-

vated cap (platform), neg l igi ble or poor so i l , o r lack of confinement

to develop n e f f e c ti v e l a t e r a l s o i l p r es su re .

The lov er laye r has

k A constant,

BZ. Wo-la yer system where k

0

i n t h e t o p la y e r, s i m i la r t o 8 1 and

s

k-WZ i n t h e l o v e r 1 e v e l ~ Z - O t t h e t o p of t h e second l ay e r.

When o r JlLca l a t e ra l d i sp lac emat oE th e p i l e t i p may occur

and th e s t i f f ne ss co eff i cie nts which apply fo r long pi le s become le ss

usefu l. The smalle'r BL o r 9L th e nore pronounced the e ffe ct.

I n

addi t ion to shor t o r in te rned ia te l eng th p i l es , non l inear var i a t ion of

9


11/22

ks

v i t h d e pt h, l a y e ri n g o r s t r a t a o f t h e

soil,

o r v a r i at i o n s i n t h e p i l e

sec t ion p roper t i e s o r naa ter i al s a lonq

i t s

length protride application

f o r

a

f i f t h model based on

a

layered system.

Model C is based on a

beam

o r spr ing foundat ion element

of

f i d t e l en gth v f t h

lateral

kin-

matic degrees of

freedam

(2

r ec t i l ine ar d i sp laceemnts

and

2

r o t a t i o o r l

d i sp lacements a t each

end , see

A p d i x 111

t

is assumui f o r rl

C that fo r each sectioa o r cf t

kaDIis

cons tan t md the

p i l e

fs

p r i s r n t i c ; hovevet, erch e l 5 r any h m d i f f a r a a t s o i l

and

p i l e

proper t i es , i nc lud ing

k s 4 aad th m

any

he

here

f r o m one to

r

fo rg e number of segments.

A

pile of

¶ 9 p a t t S

w i l l have 4 P t l ) ?at

e ra 1 d e g r m of freedom i n d i s p h c r i w a t ,

Since

the egrees of

freedom, comprnsiarr

and

totsioa, arm

not

coupled v i t h the f 3 m

t h e y a r e

included

k t - although they could err r i ly be included a t thim

stage. In th e computatioa t he 8 by stiff s matrix shavn r

Appendix III

i s

computed fo r each segment and the element sti ff ne ss

m trices summai

to

produce a 4 -) square s t i f fn es s mat r ix o f the

p l l e .

This matr ix +a than condensed

to

e l imina te

all

degrees

of

f r e e

don except those o f in te r es t fb ], a t t h e p i l o head.

The large matrix

map be

r e t a in e d i f l a t e r c om pu ta ti on f o r

stress

resu l tan ts , i .e ., d is -

placements, she ar, moments, a re desi red d o n g th e pile . The proccduta

o u t l in e d f o r m d e l C

is

best accompiished thmugh.use of a computer.

The f a c t o r

6

used n expressions

f o r

the p i l e s t i f f n e s s c o e f f i -

c i e n t s

b

i n f l e xu r e

s

a measure of t h e c o n n e c t iv i t y o f t h e p i l e

v

t o. t h e p i l e c a p .

Thus, 61

1.0 where 6 0 f o r a pinned o r hinged

condi t ion

aud

6

1.0 for a f ixe d condi t ion. I f the connection fs

seai- r igid, i .e . , 0

<

6 (1.0, the

d u e

f

6 may be

estimated.

With

m ~ d e l t h e l ay er ed o r f i n i t e l e ng t h p i l e , 6 should be formulated

a mul t ip l i e r t o the coef f i c i en t s coutputed ear l i e r . These mul t ip l i e r s

a r e 6 fo r b 44s bnS6, bPLS a d b Z4; fo r bVli and

b 2Z

t is 0 5 ( 1 ~ ) .

The long i tud inal member s t i f fn es s coe ff ic ient s , b j fa xia l and

b 66-torsion

have been sugguted ( I ) in t h e form

vhcre hE/L and JC/L a r e axial

and

t o r s i a n d member s t i f fnmss , respect lo aly,

txis

been def ined ear l i e r ,

and

t h e c o e f f i c f e n t s

kL

and

k

are emperical

T

p a r t i c i p a t i o n f a c t o r s .

I f t h e p i l e were f ix ed a t i t s end only, these

fac to r s vou ld

e

1.0; however, they

nay

d if fe r considerably. O Neill(2)

n o t e s t h a t may b e much

larger, larger

than the 2.0 pre vio usly sugges-

ted.

The f a c t o r kL

may

b e l e s s

o r

greater than 1.0.

I f t he p i l e t i p


12/22

did not move and re l i ed on f r i c t io n fo r bear ing

k

v o d d b e about-=

kvement of the pi le t i p , however, ac t s to decrease \.

Vorks vhich

may be con sulted inc lu de tho se by Novak(l8), Pouloa(19).

nd

m l p h

.Pd Wroth(20) fo r di scuss ion or alternative c i l c u k t i o n o f b Vj 3. F or

rmving t i p o r l a ye ra d syst em be s t e va l ur t i oo of t h e l ong i t ud i mf

s t i f f n e s s ap p ea rs t o b r e m u n c ia l .

I n

r

k y e r d s ystem t he relatiara-

sh ip between lo rd om th. p i l e

nd

d i r r p l a c u p t a t thm t op of t h e

l a y er , where d i s p k w n d u d a s r ig id body

- of

t h e p i l e

b r u u s e o f soil

shear

a d horten ing of th e pF1. due

to

a h t i c OIF

.

pre sri oo, may be denoted by bi j i f o r lay-r

1.

The s t i f f n e u c o a ff i-

c i c n t o f t he p i l e i n n l a ye rs i e t he n ca.lcul.tad f rom

1 1

1

1-- -...

(17)

ill b;3i

b33,1 b33,2

A

similar c a l c u l a t i on c a n be

d e

o d et er mi n e t h e t o r s i o n a l s t i f f n e s s

c oe f f i c i e n t , bqb6 , i n a l a ye re d syst em.

COMPUTATIONS

A

10 inch

MI

pi pe p i l e was

used in

computations t o compare pi l e

s t i f f n e s s p r o pe r ti e s ruing t h e f f n i t e l e n g t h p i l e , Hodel

C

and the long

p i l e . t b d e l

Al

Results

a r e g i ven

bdov

(ft . ) b l l (k/ in.) bi4(in -klrad ian) bi5(k/rad. or in.-k/in.)

2 51.31

9,821 614.2

95.10 69,494 2196.3

6 114 -02 157,355 3461.2

10 117.04 212,505

3598.2

1 8 125.83 215,886

3683.3

24 126. 06 216,786 3696.9

i n f i n i t e 126.08 216,802 3697.0

2 4

P i l e p r o p e r t i e s a r e :

A

16.1 in.

I x =

I

211.9 in.

,

x

=

D

10.75 in.,

E =

30,000 hi

nd G

12,000 hi

So il md ulu s ks 0.2

cc

. -

I .

I

I

kc i .

It c a n be

seen

t h a t t h e r e

is

a r a p i d ch an ge a f t e r 6 f t . v i t h t h e .

s t i f f n e s s c o e f f i ~ i e n t s pp ro ac hi ng t h e v a l u e f o r I nf ln Ct e.

Note tha t

\

0.0171/in., thua

&

f i e l d s

I 2

15.35 ft.

as

long pi le .

I f

t h e above p i l e

i s

t oppe d v i t h a 2 f t . c a n t i l e v e r , i.e. a 26 f t -

p i l e v i t h 24 f t . embedded and 2 f t . i n

air

t he e oe f f i c i mnt s are:

I

i l

=

78.82 klin ., b14 211,508 in-k /rad, and

bi5 =

3257.1 k/rad.

Obviously, the condit io n of

t he t op l a ye r

i s

of major importance.

A

problem vas solved, see Fig. 4 , t o i l lu s t r a t e the method of

computation. The pi le pro per t ies

are:

11

i

.

.-

-*.

. - .-.-..---

.


13/22

Fig Example rob lem

1 2


14/22

4

I

in .

1

in 1

u in

1

u

i n

~ ~ ~ - 2 )

P i l e No. L in.1 1 2

-

400 30 20 1.32 30- 3 16.10 211.9

2 400 0

0

-7.68 0

0 113.09 1017.9

3 400 -20

-20

1.32

300.

16.76 294.7

i 4

P i l e No.

Ix in.

1

D x i n . )

D in- ) Elcei) G k s i ) Mater ial

211.9

10.75 10.75

30,000

1 2 , 0 0 0 S t - l P i p

2 1017.9 12

1 2

1,500

300 Timber Pole

3

100.6 10.224 10.03

30 000

12,000 BPlO

The soil subgrad.

d U ; L W

k,

=

0.1

kci

A

follObLiOO Qd of

jQIT = [40.k, 2O.k. 600.k. 0 , 500 in-k, 01

w a r

used.

Resul ts arm

as

follows:

a l l

units

i n k ip s and

inch-

1

P i l e

stiffness v h r

. :.\ - -

i l e

b l

-

i 2

5

b4

-

;5

-

i 6

-

i 5 b;4

-

74.90

74.96 1207.5

182,302

182,302

12,714

2614.0 -2616.0

2

57.00 57.00

424.1

64,291

64,291

4,071

1353.6 -1353.6

3 58.99 78.40

1257.0 230,558

102,426

ll.859

1736.1 -3006.3

2. The foundation s t i f m a matrix,

308.06

-

26.93 276.97

SYn

- '

---

435.18 74.24 2704.86

11

- [

497.66 2600.15 -1699.42 1,277,722

i

327.69. -9410.49 -9449.02 -1,202,452 1,905,765

756.74 421.86 6912.82

-

285,315 189,44 8 327,523

3. The

fmmdation

displacemmts ,

{ } ~

t-0.5132, 0.4991, 0.3741, 0.009045, 0.010398, -0.0054891

I

4

The

pi le displacmnents in member coordinates,

{XI: -

[-0.2441, 0.4742, 0.1749, 0.014099. 0.004483, -0.0010861

{XI

- [-0.5931, 0.5686, 0.3741, 0.009015, 0.010398. -0.0051891

1x1: -

-0.8944. -0.2291, 0.1899, -0.003017, 0.013032. -0.0064121

5.

The

pi le force s i n member coordin ates.

T

{rI l

-

1-6-58, -1.31,

211.2,

1330.8. 179.2. -11.81

{z}:

=

1-19.72. 20.16, 158.7. -188.1. -134.3. -22.31

I

- 1 3

z

1

q

J

-.

.

. . 3

=

.

I

- -


15/22

Although

o

computer program

as

used t o sol - the p m l - m t h e s t i f f n e s s

mat r ix c n b e assembled using the equati ons i n Appendix

I

The or

s t i f f n e s s m a t r ic e s

Ib1Ix

can be determin+d

from

n d o l Al but vero coa-

puted f o r th is problem

u h g

C t o

check Al

CONCLUSIONS

The spring formdacioa

model has r nuder

of advantag.. f o r arod.1

i n g l a t e r a l l o rd i ng including t ho a b i l i t y t o

compute

det loet ions ,

shears, bending momeats and s t r u s o s along the p i l a .

The

corapur t i a ru

a r e s t r a i g h t f o m ar d and

und8rstandable.

Vark

rcautns

t o be done on

impmvlng the soi l - pi la inte ract ion models snd

adding

t o t h o l i b r a r y

of

models mai la bl e to the des igner.

It

i s u s e fu l t o r e a l i z e t h a t when p i l b g a r e hinged o r v e r t ic a l ,

a

number

of variahlu

b u m zero. Io addi t ion, tho Ib ] matrix nay b e

t h e s w i f sever l o r

ll

p i l e s i n t he

fou d tioo

are the same.

f ir-

t he r, s p e t r y of t h e arr an ge ws nt of p i l i n g n a folmdation

m y l l ow

a

d e c re a se i n t h e amount of cosputations.

yrmacem

of loadlag

may

a l so shor ten computa tioos, espec ia l ly i f

a

combination of

gaometrp and

l o ad i ng a l l w t h e f o un da ti on t o

be

analyzcd as

a

plane figuro.

P a r t i a l c o n s t r a i n t o f t h e p i l e t o t h o c a p s accounted f o r through

choice of

d

b e t v e e n and 1.

The add itio n of the finite le ngt h model, C herein, i s

an

impor-

- t a n t

and

valuable Step forvard

alloving

use of shor t p i l ea , var i ab l e

o r

layered

soils

and

var iab le s ec t ion p i l es .

oHram PILEFDN, b J

Dep;.-.tmcnt

of Ci vi l and Environmental Enginrering.

1 4


16/22

PPENDIX

I

- FORHUTAS FOR STIFFNESS

ItWLUOJCE

COEFFICICFTS

Formulas are 8iv .n for s in gle pi loa. St i f fne ss coeff ic ien t

S

S'

i

j i

by reciproci ty nd frurction Bi ar e def ined f or convd8ncm ae followa:

2

B1 bll cos Y - bZ2

2

bj3 in

B2

-

b b 8inY coeY

11 33

B (blS+bZ4) COSY 8 i ~ Ooea

Bb

q sina -

u2

cons

2 2

B5 bll sin y b33

cos y

2 2

B6 bgq cos y

-

bS5 bb6

sin

y

B1 = u3 [b22+i31 c o s 2 d

B u (b22+Bl sh20 )

2 2

B16 b15 s i n a - b24 cos a)cosy

B17

u1 bZ2

-

bZ4 sinY cosa

bg B~ u3 s in a cosu B

18-

sins C O S ~

B19

(b44

-

b

sin y cosy

66

Thus StU

B10 cosa bZ2

S*12 B1 18

13 B1l

Stl4 -u2 Bll

-

Bg-B3

15 B1l 7 '14

.

S'16 I B10 - u2b22 bZ4 sLL~ s im


17/22

2 2

S 44 u2B5+2u2Bl5~iua+B6~~s+bS5+U3 (2u2B13+B8+2B16)

-u

u B +B B -B (u sina+osa)-u (u B +u B +u B +2B3)

45 1 2 5 6 1 8 15 1 3 2 1 1 1 1 3 3 1 1 8

(B -B B 1 u B -B C O ~ Q ~ ~ ~ ( B ~ ~ B ~ + B ~ , )

46 2 2 4 1 16 19

2

U +B sin a++ +2 6 cosWu3(B,+2tyBU+2B14)

55 1 5

6

55

l

15

56

q

B ~ B ~ + B ~ ) - B ~ ~ s ~ u ~ B ~ ~ + u ~shy gBlc~W+b24sino)~2 ~ ~ ~ c o s 0 c b ~ ~ ~ l

2

2 2

2

S'

66

blB4+(~l~Z)bU-2b24(~1~~~~2~i~)~inY+(b44-b66)~b

+b66

-

vhere U1(u1,u2.u3) are the coordinates of the pifa t p in thm f o d r

tion ai is the angle to the direction of batter elockuiss

in

p l w

fron the U1

and

yi

is the angle of batter r om the vertical n

the plane of batter.

.


18/22

PPENDIX

I STIFFNESS COEFRCIEfiTS FOR LON P I L E S


19/22

APPENDIX - STIFFNESS KATRIX

FOR

A PILE SEGMENT

where, Tli (C S

- C S q

n (C S - CS )K~

2

T3i 2(CS C S*)KB q

T4i

2(C S CS )KB q

Segment of Pile

vlth

Hembe

T5 (s*

s2)rk

egrees of Fretdoa

T6, 2SS dq

q

l/(sq2-s2) 2

C COSSL

S sia0L


20/22

APPENDIX

I V

- REFERENCES

1

S a d .

W i l l i a m E.

S t a t i c and Dynamic Ana lyst s of P l l e Foundations,

Jo urn al of t he St ru ct ur al Div isio n, ASCE, Vol. 94, No. STS,

Uay.

1968, pp. 1077-1100.

2.

O'Neill, Hiclue1

U,,

discuss ion of Stat ic and Dynamic hualyl i8 of

P l l e Fuun&tlmu,

by

William

E.

Saul , J ou rn a~ f t h e S t ruc tu ra l

Division,

CE,

Vol. 95,

NO.

ST?

Feb., 1969, pp. 289-295.

' .

i 3. ~ b r t h y ,

v.N.s.,

and sht ivast rwa,

s.P.

discusaioa of s t a t i c and

D y n m i c nalpis

o f P i l e

Foua&efoar,

by W i U p p

2 Saul

J o u d

of tho Stmctural Mvia ioo ,

AS=

Vol. 95,

No.

SR, Feb., 1969,

pp. 288-289.

i

-

4

Prakash. Shaadiar,

rrd

Cluadrasokaran,

v.,

d i s c u u i o n o f S t a ti c

and Dynamic Analysis o f PUe ~oun&t ioos , by U i l l k n E. Saul,

Journal o f the S t ruc t u ra l Mvis ion . ASCE. Vol. 95, No.

STIl

&v., 1969, p. 762.

5

S a i l W i l l i a m E., clo su re t o St at ic and Dynamic Analysis of P i l e

Foundations, Jou rnal of the St ru ct ur al Division, ASCE, Vol. 95

No. ST ll , Nov., 1969,

p.

2511.

6

Saul, Willlam

E.,

disc uss ion of Full-Scale Lat era l Load Tes ts of

P il e Groups, by J a i

B. Kim

and Robert J. Brungraber, J o u n d o f

the Gmt ech nka l Div is ion , ASCZ Vol. 10 3, No. CT2 Feb., 1977,

pp. 147-148.

7.

Saul,

W i l l i a m E.

and Wolf, Thomas V., dis cu ss io n of Design of

Hachina Foundations on Piles, by Jogeshvar P. Singh, Neville C.

Ibnovan, and

Adrianus C.

Job sis, Journal of the Ceotechnical

Division, ASCE Vol. 104, No. GTl2, Dec., 1978, pp. 15261530.

8.

.Saul , Uf l l iaa E. and Wolf, ~hontas ., Ap pli ca tio ns For New Research

fo r P i le Supported Hachine Foundations, paper presen ted a t the 1979

nnual

Conventloo,

ACI

H i l v a k e e ,

W i s . ,

March, 1979.

9.

~esi; , Aleksandar S., Design of Pi le ~ou ndat ioos ,

NCARP

Synthesis

42, TRB 1977.

10. Bovles, Joseph E., Found ation An aly si s and Desim, 2d Ed., HcGrar

H i l l ,

N.Y.,

1977.

ll.

Arya, Suresh C., O'Nei l l , Mchael W. and Pfncus, George, Design of

St ru ct ur es and Foundations foe Vi bra tin g Fachinas, 'Gulf Publ. Co.,

Houston, 1979.

?

12

Poulos,

Harry C.,

Group F a ct o rs f o r P i d e f l e e t i o n E a ti rm ti oa ,

Journa l of the Ceo tech nicd Ennineerinx Division,

ASCZ

Vol. 105,

No. GT12, Dec., 1979 , pp. 1489-1509.

13.

Novak, Milos, Ver tica l Vibratio n of Floa ting Pil es , Jou rnal of t h e

Lhgineering Mechanics Division,

ASCE

Vo1. 103, No. EHl Feb. 1977,

pp. 153-168.

14-

O 'burka , Mchar l J. and

Dobry

Picardo, Spririg

and

Dashpot Caeffi-

c i en ts fo r rachine Foundat ious on Pi les ,

paper

presented a t t he 1979

Annual

Convention, ACI. Htlwaukea U i s . , Parch, i979.


21/22

15.

vesif .

A. H

Bending of Beams Resting on Iso tr op ic E la st ic Solid*

Jo ur na l Engi nee rin g KechanLcs Div fsi on, ASCE,

Vol.

87, EHZ Apr.,

1961,

pp 35-53.

16.

Terzaghi,

K.,

Evaluation of Coe ffi cie nt of Subgradm Re. cti om, ~

Ceotechni que, Vol. 5, No. 6 Dec., 1955,

pp

297-326.

17. Robinson, K.E., Horfzone;ll Subgrade Re ac ti ons Estimated f r o m

Lateral

Loading

ests on

Timber Pi le s, Behavior of Deep Founda-

t ions , STH

Sr

670, Raymond Lundgren,

Ed.,

1979,

pp. 520 536.

8. Novak, W o s , Dynamft Stf f fneas nd Damping of Pi le s, Canu firo

C e o t e c M c a l Jou- Vol.

11

No.

4,

Elov.

1974,

19. Podos,

H.G., and TIIvts E X

The Set tle men t Behavior of S ing le

Axi all y Loaded Incom press ible P i l es and Pie rs, Ccotechaique, Vol.

18 , 1968, pp. 351-371.

20.

Randolph, H F and Wroth,

C.P., A

Simple Approach

t o

P i l e

D w i g u

and Evaluation of Pi le Tests, Behavior of P il e Foundations,

ASTM STP 670, RayraondLundgren,

Ed.,

1979,

pp

484-499.


22/22

E r r a t a D i s c us s i o n s R e l a t i n g t o P a pe r

"S ta t i c a nd Dynamic Ana lys i s o f P i l e F ounda t ions"

by Will iam

E .

S a u l

Pu bl ish ed i n th e S t r u c t u ra l Jo ur na l , ASCE, May 1968, pp . 1077=1100

1.

E r r a t a :

pp. 1080-81-82 1 Change s u b s c r i p t s i n e x p r e s s i o n f o r b t o i n

Eqs.

3 ,

1 3

1 9 .

11

2. Change s u b s c r i p t s i n e x p r e s s i o n f o r b t o 1 i n

Eqs. 4 , 14 , 20 .

22

3. Change s u b s c r i p t s i n e x p r e s s i o n f o r b = b t o

2 i n Eqs. .9, 1 7 , & 2 3 . 1 5 5 1

4 . Change s u b s c r i p t s i n e x p r e s s i o n f o r b =b t o

1 i n Eqs. 10 , 18 , 24. 24 42

5.

C hanqe e xp r e s s io n i n Eqs . 19 20 t o

1 2 B i k ) ;

( i . e . , c h a n g e - t o +.)

p.

1084 Note u n o t shown i n F ig . 5 ,

it

i s

t h e v e r t i c a l c o m p o n e n t .

3

p. 1086

1

Change Eq. 47 t o

Qi = m r

2 a'

i )

2.

Chanqe Eq. 49 t o -[S] {A}= r n [ ~ ] { h }

3. Change Eq. 50 t o [

-

X ~ [ I ] ] { A ) = 0

n

p . 1 0 9 1 ( c a u t i o n ) I t h a s b een s a i d and n o t y e t v e r i f i e d t h a t t h e

c h a r a c t e r i s t i c v e c t o r s i n T ab le 4 ( g ) a r e i n c o r r e c t .

p. 1095

F o l l ow i n g t h e Eq. f o r

S h 5

change 8

46

t o S i 6 .

2. D i s c u s s i o n s o f t h e a bo v e p a p e r w er e p u b l i s h e d a s f o l l o w s :

1

O N e i l l ,

M . W . , S t r u c t u r a l Jo u r na l , ASCE, Fe b. 1969 , pp. 289-295.

2. V.

N . S.

Murthy and

S.

P. S h r i v a s t a v a , S t r u c t u r a l J o u r n a l

ASCE,

Feb. 1969, p. 288.

3. Shamsher Pr akas h and

V.

C h an dr as ek ar an S t r u c t u r a l J o u r n a l ,

ASCE, Nov. 19 69 , p.

7 6 2

4 . C l o s u r e , s t r u c t u r e s j o u r n a l ASCE, N o v . 1 9 6 9 , p . 2 5 1 1 .

3. D i s c u s s i o n s o r o t h e r p a p e r s wh ic h a dd t o t h e a bo v e.

1. Jo ur n a l o f t he Ge o te c nn ic a l D i v i s ion , ASCE, Feb 1977 , pp. 147-148.

E xt en d m odels t o i n c lu d e s e m i - i n f i n i t e p i l e i n

s c i l

w i t h

l i n e a r l y i n c r e a s i n g modulus o f s u bg r ad e r e a c t i o n .

2. J o u r n a l o f t h e G e o t e c h n i c a l D i v i s i o n ASCE Dec 1978 , pp. 1526-1530.

3.

"A pp l ic a t i on s f o r New Research f o r P i l e Suppor ted Machine Founda-

t i o n s " by W i l l ia m E Saul Thomas W . W ol f, p r e s e n t e d a t A C I Annual

Conference , Milwaukee ,

W I ,

March 1979.

4

" P i l e

F ounda t ion Ana lys i s "

by Willian

E. S a u l ,

Preprint 80-102,

PILE Foundation Analysis_ASCE William Saul-1980_searchable

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