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PILE FOUND TION N LYSIS
illiam E
aul
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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.
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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 ,
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a hod
b
Displacrrrotr
Fig 1
~ o ~ d a t i o n
ardr and Dfsplacawntr
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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
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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
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C
Elevation
v i t h
espect
xis of Pile
Ng P i l e Cap with P i l e
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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
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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
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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
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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
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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
.
.-
-*.
. - .-.-..---
.
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Fig Example rob lem
1 2
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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
- -
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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
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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
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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.
.
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PPENDIX
I STIFFNESS COEFRCIEfiTS FOR LON P I L E S
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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
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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.
8/17/2019 PILE Foundation Analysis_ASCE William Saul-1980_searchable
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.
8/17/2019 PILE Foundation Analysis_ASCE William Saul-1980_searchable
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,
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