BOUNDARY ELEMENT APPROACH TO THE
DYNAMIC STIFFNESS FUNCTIONS OF
CIRCULAR FOUNDATIONS
E.
ALARCON AND J. J. CANO
AND
J. DOMINGUEZ
INTRODUCTION
In he pas wen y yea s nume ous in es iga ions which deal wi h he compu a ion o dynamic
s i ness (also called impedance) ma ices o massless igid ounda ions wi h di e en shapes ha e
been epo ed. Taking ad an age o
he
simpli ica ions de i ed om axial symme y, ci cula and
cylind ical ounda ions ha e been s udied in many cases. Analy ical solu ions o s i ness
unc ions o ci cula ounda ions on a uni o m hal -space we e p esen ed o ho izon al and
ocking mo ions by Vele sos and Wei1 and o e ical and o sional mo ions by Luco and
Wes man.2
The de elopmen o abso bing bounda ies pe mi ed he use o he ini e elemen me hod o
s udy he dynamic esponse o axisymme ic ounda ions on soils ha consis o one o se e al
laye s based on a igid ock.3"5
An al e na i e app oach o he dynamic analysis o igid ounda ions is he use o bounda y
in eg al equa ions which allow o he modelling o many di e en geome ies and soil p o iles.
Bo h he di ec and he indi ec in eg al equa ion o mula ions ha e been used in dynamic
soil-s uc u e in e ac ion p oblems.6"8 Dynamic s i ness unc ions o igid ci cula ounda ions
on uni o m o laye ed soils ha e been compu ed by Apsel9 and Apsel and Luco7 using he indi ec
o mula ion in combina ion wi h G een's unc ion o a ing load in a uni o m o laye ed
hal -
space, de i ed by he same au ho s.9"11 This G een's unc ion is w i en in e ms o in eg al o ms
ha include p oduc s o wo Bessel unc ions. Owing o ha , i has o be e alua ed segmen ally
and i s compu a ion becomes a he in ol ed. An auxilia bounda y whe e he unknown sou ces
a e loca ed has o be de ined.7'9,12
The bounda y elemen me hod, based on he di ec in eg al equa ion o mula ion, has been
used o compu e dynamic s i ness unc ions o ec angula 6 and s ip ounda ions in he equency
domain13,14 as well as in he ime domain.15 In he p esen pape he bounda y elemen me hod is
o mula ed o ime ha monic axisymme ic p oblems using he ull-space poin load undamen al
solu ion. The app oach is applied o he compu a ion o he dynamic s i ness unc ions o igid
ci cula ounda ions on laye ed iscoelas ic soils.
The o mula ion o he BEM o axisymme ic elas os a ic p oblems was p esen ed by
Ke manidis16 and C use e al11 They used he ing load undamen al solu ion. Howe e , a closed
o m exp ession o he ing load does no exis in elas odynamics18 and when he same
o mula ion is ollowed he compu a ions become in ol ed because o he di icul ies in he
e alua ion o he undamen al solu ion a each in eg a ion poin . On he con a y, he poin load
undamen al solu ion is simple and may be in eg a ed nume ically along he azimu hal co-
o dina e wi hou any pa icula di icul y.
As an example o he ange o si es o which he p oposed app oach can be applied, h ee
di e en
si es
a e conside ed; a uni o m hal -space,
a
soil laye on a hal -space, and a soil consis ing
o ou ho izon al laye s and a complian hal -space. The nume ical esul s compu ed by he
p oposed app oach a e compa ed wi h esul s ob ained by di e en p ocedu es.
BOUNDARY ELEMENT FORMULATION FOR ELASTODYNAMIC
AXISYMMETRIC PROBLEMS
Following he idea o Chapel,19 he ull-space poin load undamen al solu ion is used o sol e
ime ha monic axisymme ic p oblemsJ
The
axisymme ic ep esen a ion o he geome y and ield
a iables is main ained. The bounda y o he gene a ing su ace o he body is disc e ized in o line
elemen s and he poin load colloca ed a each node. The undamen al solu ion, in e ms o
cylind ical co-o dina es, is in eg a ed along he bounda y elemen s o he gene a ing su ace and
along he azimu hal co-o dina e.
The basic BEM equa ion o ze o body o ces can be w i en in Ca esian co-o dina es as usual:
CV+ Tud = U dT (1)
whe e u and s and o he displacemen s and ac ion ec o s, espec i ely; U and T a e,
espec i ely, he ma ices ep esen ing displacemen s and ac ions p oduced by he poin load
applied a poin / and C is he independen coe icien ma ix such ha Ckl =
(1/2)<5W
when he
su ace is smoo h a he poin i.
The ela ion be ween Ca esian and cylind ical co-o dina es may be w i en o ec o s u and
a a ce ain poin y as
u = Qu
(2)
=Q
cylind ical (Figu e 1) and he ans o ma ion
Q
COS&
sind
0
sind 0
cos 0 0
0 1
(3)
By
subs i u ion o equa ion
(2)
in o equa ion
(1)
and p emul iplying
by
Q, he ollowing equa ion
is ob ained:
QiT C
Q'
u^
+ J
(Q*'-T
TQ)ucd (Q*'-TUQ) cd (4)
whe e QJ is he ans o ma ion ma ix o he colloca ion poin
i.
Equa ion
(4)
can now be w i en
as
C^+
j Tc
ucd
uc c d (5)
which is he same as equa ion (1) w i en in cylind ical co-o dina es. [
The 9 co-o dina e o he colloca ion poin
j is
ixed
du ing he in eg a ion p ocess and can be se
equal o any desi ed alue; o ins ance $' = 0. In such a case
T i.T
Q UTQ
ncos0+
7 2sin$
!T21cos9 + 22sin9
31cosd+ 32sin0
nsind+ 12cos3 Tx
21sin»+ 22cos» T
31sin9+ 32cos9 T
3
23
33
(6)
whe e he e ms Tkl a e he well-known Ca esian co-o dina e componen s o he ac ions in he /
di ec ion a he in eg a ion poin , due o a uni poin load in he k di ec ion a
he
poin
i.
The same
equa ion can be w i en o Uc.
x
i
x
2
Figu e 1. Desc ip ion o co-o dina e sys ems
When a ci cula ounda ion unde e ical o o sional mo ion is conside ed, he ield a iables
a e axisymme ic. Assuming ha he bounda y is smoo h a i, equa ion (5) becomes
1
2
u
i
p
3
4
+ 2 P
^cS+ ^sd
o
31cd
+
32s0
o
T2ls$+T22c&
0
7-13
0
^33
d&
up
"3
"z
d
2 P
l/nc&+ /12s3
0
U3! cd + U32 sd
0
U21sa+L/22c9
0
l>13
0
/33
dO
',
'3
2
d (7)
whe e T* is he bounda y o he gene a ing su ace (Figu e 2), s and c s and o sin and cos,
espec i ely and a ze o has been placed o he skewsymme ic e ms, whe e he in eg als a ound
he azimu hal axis a e null. I should be men ioned ha he o sional and e ical adial mo ions
a e uncoupled in equa ion (7) and ha he ke nels o be in eg a ed along d a e known unc ions.
When a ci cula ounda ion is unde ho izon al mo ion along Xx o ocking a ound X2, he
ield a iables a e o he o m (Figu e 3)
u
u
3
ulpcos9 ;
"id sin » ;
lpcosd
3 ld sin $ (8)
u2 = ulz cos 3 ; z = Xz cos $
whe e ulp, ul , ul , lp,
x</>
and j2 a e independen o 9. Ins ead o equa ion (7) he ollowing
P
(a)
(b)
Figu e 2. Gene a ing su ace o an axisymme ic domain: (a) gene a] p oblem; (b) su ace ounda ion on a hal -space
X 2
X 1
X 3
X 2
Ho izon al
X 1
X 1
Rocking
Figu e 3. Ho izon al and ocking displacemen
equa ion is ob ained:
1^
2
u
IP
0
u
i
+ 2 P
2
nc2d + 12s&cd ns2o- 12sdcd
o
0
2
31c29 + 32s9c9 31s2d- 32sdcd
,
3
c»
0
33cd
dd
"IP
«1S
"l*
d
2 P
2
l/n^S+l/jaSacd (/nS29-l/12sdc9 l/i
3
0 0 0
2
l/31c2d+l/32s9c0 l/31s29-U32sdcd
U
33
c&
c9
dd
h.
h*
*i.
d
(9)
Equa ion (9) can
be
used
o
ep esen
he
adial and e ical displacemen s
bu no he
azimu hal displacemen o which an iden i y would be ob ained. The ampli ude o a unc ion such
as ud
=
ux
a
sin 9 canno be compu ed wi h a colloca ion poin whe e he unc ion is ze o. Because
o his,
a
di e en colloca ion poin whe e he shape unc ion —sin 3' has
a
uni ampli ude
(»'
7 /2) is selec ed o he azimu hal equa ion:
T
o
,«.T
Q'-'TQ
T°2lc$-T°22sB
^ca+ ^sa
?,cs
?,s9+ ?2c»
T°3is$+T°32c$
n
23
7^3 (10)
whe e he supe index '0' indica es ha he colloca ion poin is a
8'
ep esen s he h ee componen s
is
7 /2.
The equa ion ha

1
2
id
i
«u
«i.
+
2
P
*
J
(7ilC2s+ 12
s9c9)dS
72
( j^s+ jjsScSjdd
-«/2
2 P
*
I
(7-3,
c29+T32s9c9)d9
c29+l/I2s9c9)d9
72 (C/?1c2»+{/o2s8c8)d8
-«/2
["(U"
c29+l/32s9c9)d9
J>
s23- 12s&cS)d»
«/2
( ?jsa»- ?288o»)d»
-«/2
Jo
s29-7"32s9c9)d9
s29-l/,2s9c9)d9
72 0 „2
(l/?,s29-l/?2s9c9)d9
-«/2
T
I *""
s2»-l/32$3c9)d3
c9)d3
:/2 ( ?3c»)ds
-«/2
(733cS)d»
u
ip
«i»
u
Xx
d
^
c9)d9
n (l/?3c»)dS
-*/2
(l/33c»)d»
IP
'ia
'i.
d
(11)
In gene al, non-axisymme ic bounda y condi ions can be analyzed using a plane model by
ep esen ing he
ield
a iables by a Fou ie se ies along he azimu hal co-o dina e. The se ies
is
o
he o m
00
U
P
(u*np
cos n§ + ulp sin nS)
« = o
00
uB
= £ (-149
sin
n9 +
u^cos
nO) (12)
n = 0
ao
U (u*zcos
nO
+
wJJ
sin nO)
Fl = 0
e ms
and a bounda y equa ion may be w i en o he ampli ude o he symme ic and he
an isymme ic pa s o each mode. The equa ions o he ampli ude o he e ms o he o m sin n3
a e ob ained using a poin a 0' in as he colloca ion poin excep
elemen ha con ains he colloca ion poin . In he nex sec ion, an in eg a ion p ocedu e o
cons an elemen s con ained in planes x3 = cons an is p esen ed. This kind o elemen is he only
one needed o he analysis o ci cula ounda ions on uni o m o laye ed soils.
INTEGRATION OVER THE BOUNDARY ELEMENTS
Cons an bounda y elemen s wi h one node pe elemen a e conside ed (Figu e
4).
The bounda ies
o he ci cula ounda ion p oblems analyzed a e pe pendicula o he X3-axis and he line
bounda y co-o dina e T* coincides wi h he adial cylind ical co-o dina e p. Equa ions (7) and
(11) can be w i en a e disc e iza ion as
4
N
«£
+
j-i
p
Pi
[JT.d»]d,>.
USAM*™ (13)
Massless ounda ion
1
In eg a ion
poin
Colloca ion
• •
Colloca ion poin
Figu e 4. Bounda y elemen disc e iza ion
whe e N is he numbe o bounda y elemen s, pi indica es he segmen ha o ms he elemen
j9
and Tc and Uc s and o he ma ices in equa ions
(7)
o
(11).
The double in eg als ex end o e one
hal o he ci cula c owns ep esen ed by he line elemen s (Figu e
4).
The colloca ion poin o
equa ion (13) can be ei he a 9l =0 o a &
n12
and o each e m he in eg a ion domain is
ei he he elemen o which he colloca ion poin belongs (i=y) o a di e en one
(i
#7). The la e
case is analysed i s . The in eg als a e done nume ically using a Gaussian quad a u e o mula
a e a special co-o dina e ans o ma ion. The in eg a ion domain is shown in Figu es 5(a) and
5(b),
he la e being o he case when he colloca ion poin is a
S1
=
—
i /2.
In o de o apply he Gaussian quad a u e he domain is ans o med in o a squa e in he
dimensionless co-o dina es
(—
1
^ £
<;
1,
— 1
^
*/
< 1). The adius p is ans o med in o
by
9 1
2 l(Rx+R2) +
i(R2
1
dp = ^(R2-Rx)d i
*i)] (14)
+
A linea ans o ma ion o he same kind was es ed o he co-o dina e 0; howe e , he
accu acy o he compu ed alues o he in eg als was poo
e en
when a la ge numbe o in eg a ion
poin s was used. The accu acy was imp o ed, and sa is ac o y esul s we e ob ained, by using a
quad a ic ans o ma ion o mula ha inc eases he numbe o in eg a ion poin s in he icini y o
he X
j-axis
o he X2-axis in he cases o Figu es 5(a) o 5(b), espec i ely.20 The ollowing
(a)
X 1 (c)
* 5
(b)
Figu e 5. In eg a ion domains and coo dina es when he colloca ion poin is ou side he in eg a ion domain: (a) 9' = 0,
(b) $' = - 7 /2
pa abolic ans o ma ion has been applied;
n
a =^<i + 02
n (15)
dd=-(l + £)d£
when he colloca ion poin is a $' = 0 (Figu e 5(a)) and
4
n (16)
d9=-(l+£)d£
when he colloca ion poin is a 01
7c/2
(Figu e
5(b)).
In his way one hal o he o al numbe o
in eg a ion poin s a e loca ed in he qua e o he domain close o he colloca ion poin .
The numbe o quad a u e poin s is wo o he adial co-o dina e and wen y o he azimu hal
co-o dina e. Thus, he in eg als in equa ion (13) become
H*u
P
pi
2 20
dSJdp-
X Z(7T/4)[(/?1+/?2)
+ nm(R 2
= 1 n=l
Rl)1Tc( n.'lm)lR2-RllV+U">*Wm
2
20
G'j
= | p
Ucd»
dp- £
ZinM i^ +
R )
Pj L JS
J
m~lm*l
+ ^/?2-^i)]Uc(^,0[«2-^i][l+^]> ll> m
(17)
whe e Tc(^, jj and UJ&,, m) ep esen he alues o he ma ices in equa ions (7) and (11)
compu ed a he (n,
m)
in eg a ion poin and wn, wm a e he weigh s o he Gaussian quad a u e
o mula.
When he in eg a ion is ca ied ou o e he elemen which con ains he colloca ion poin ,
a
singula i y exis s in he undamen al solu ion and he in eg a ion scheme is di e en . Fi s o all,
he pa con aining he singula i y is sepa a ed om he es o he undamen al solu ion o be
in eg a ed. This
is
done by sub ac ing he s a ic undamen al solu ion om he dynamic
one.
The
di e ence
is
non-singula a any poin o he in eg a ion domain and can be in eg a ed by means
o
he same kind o nume ical quad a u e o mula, combined wi h he quad a ic ans o ma ion o
he co-o dina e $, as be o e.
The coe icien s o equa ion (13) o which i==/ can be w i en as
whe e H " and G|J a e he s a ic coun e pa s o H*" and G", espec i ely, and
H&'=
p{
Tc«Tc(s a ic)jddjdp=
pj
[Tc,dinda|dp
(18)
2
20
X ( /4)l(Ri
+
R2) + im(R2
~
*i)]TCld aK„ jm)
=
1 n=l
[«2-«l3[l+^]wnW
G&=
P<
[Uc-Uc(s a ic)]d3Vdp
= pi
Uc,djnd9Vdp
Pi
IJB
2
20
l.(n/4 mRl+-R2) + im(R2-Rln
=
1 11=1
Uc,din(^,^)[/?2-K1][l+^]wnw (19)
whe e
TCidin(<^„,
m)
and
UK<6in(£H,
jm)
s and o he alues a
£„,
jm
o ma ices o he same o m o
hose
in
equa ions (7) and (11) bu wi h he di e ence be ween he dynamic and he s a ic
undamen al solu ion ins ead o he dynamic undamen al solu ion
(Tkldin
=
Tkl
—
! , (s a ic) and
*Aj.din
=
Uki —
(A/(s a ic) ins ead
o
Tkl and Ukl, espec i ely). Figu e
6
shows he in eg a ion
domains and he colloca ion poin s o his
case.
The p ocess is comple ed wi h he in eg a ion o
he s a ic undamen al solu ion. The coe icien s
Tu
(s a ic) and
Ukl
(s a ic)
a e known in e ms o
he dis ance
and i s de i a i es (Figu e 6):
2
. D2
=
J{p2 +
R2
-
2 Rpcos 3)
R
R
+ R2
2
/-,
=
(/?
—
pcos$)/
2
=
(psin9)/ (20)
Figu e 13. Ci cula ounda ion on a mul ilaye ed soil on a hal -space. Bounda y elemen disc e iza ion
CONCLUSIONS
Dynamic s i ness unc ions o igid ci cula ounda ions on a uni o m o laye ed iscoelas ic soil
ha e been compu ed by a bounda y elemen app oach ha makes use o he comple e space poin
load undamen al solu ion. This solu ion is simple and easy o in eg a e o e he bounda y
elemen s. The p ice paid o using such a simple undamen al solu ion is ha he soil ee su ace
and he laye in e aces ha e o be disc e ized. Howe e , he disc e iza ion o
he
soil su ace and
he laye in e aces can be educed o a limi ed zone so ounding he ounda ion and is
au oma ically done by he code.
App oaches based on G een's unc ions o a laye ed hal -space7,9-11 ha e he ad an age o
equi ing only he disc e iza ion o he soil ounda ion in e ace bu ha e he disad an age o
dealing wi h a undamen al solu ion ha
is
no known in closed o m bu in e ms o in eg al o ms
ha include p oduc s o wo Bessel unc ions and ha e o be e alua ed segmen ally by elabo a ed
nume ical in eg a ion p ocedu es. The p oposed app oach
is
easie o implemen han hose o he

1.0
k hh
0.5
0.0
1.0
hh
0.5
0.0
V
1
^—J p^4
i.d
k
0.5
0.0
| • j
>s>^ |
"«».•
^
.2 .4 .6 .8 1 1.2 1.4 .2 .4 .6 .81.1.21.4
1 .0
k
0.5 •
0.0
1 .5
c
1.0
0.5
.2 .4 .6 .8 1. 1.2 1.4
Luco
B.E.M
Figu e 14. S i ness coe icien s o ci cula ounda ion on a mul ilaye ed soil on a hal space
app oaches. The amoun o compu e ime equi ed o sol e
a
pa icula p oblem is smalle o
one case o he o he depending on he numbe o laye s in he soil p o ile.
To illus a e he capabili ies
o
he p oposed app oach, dynamic s i ness unc ions
o a
uni o m hal -space and wo laye ed soils ha e been ob ained. The esul s ob ained o he
hal -
space ha e been alida ed by compa ison wi h analy ical solu ions; hose o he laye ed soils by
compa ison wi h he esul s
o
an app oach based on nume ical solu ion
o
se s
o
in eg al
equa ions and, in he case o a single soil laye on
a
hal -space, also by compa ison wi h o he
bounda y elemen esul s. The compa isons indica e
a
good deg ee o accu acy.
ACKNO WLEDGEM ENTS
The wo k desc ibed he ein is pa ially suppo ed by a g an p o ided by he local go e nmen o
Andalusia, Spain. The au ho s exp ess hei since e app ecia ion o his suppo .
APPENDIX
In eg als o he s a ic pa o he undamen al solu ion o e he elemen whe e he poin load is
loca ed, a e gi en in his Appendix.
Te ms co esponding o e ical o o sional mo ion o he ounda ion (equa ion (7)):
J
Pi
Jo
2/4 +^3-cos0+—cosa-^3-(l+cos29) pdOdp
2/4 7,cos9d8+ /.cos&d& + /?2 J,cos&d9
VI-
R j /3(l+cos2d)d3
2
0
(24)
whe e A = (l/16); p(l
-
)
and
B
=
3
-
4 .
G,2,2c,s,= 2|
I
l/22c>$lpd3dp = 2,4j
J
!-_+£_ JpdSdp
2/4
B
/jCOsSdd + K /3sin2Odd (25)
G^C.«=2J , /33c.,,pdddp=2/l
[
[*p<&dp = 2AB
1,6$ (26)
P i
J
O
J
Pi
G
il
_
i%i
_ iu _ in __ iu _ in _ (
12c,s — °13c,s — u23c,s — u21c,« — u31c,s — u32c,* — u
(27)
«?!*..«-2
J
j
13c>4 pd3dp
= 2^J
^(pcosd-/?)Jpdadp
2AB
|
/3cos9d9-.R |/2d9 (28)
#5iU=2J 31c,$ pdddp = 2,4
J
-^(p-Rcosd)"|pd»dp
2^B|
I
Add-/? YcosOdS
,•
]
£7*ii
_
H*"
—
*7*"
—
77*"
—
77*"
—£7*"
_
77*"
13 llcs — /722c,s — "33c,s
— A
12c,s
~
"21c,s
—
"23c,$ — "32c,s
Te ms co esponding o ho izon al o ocking mo ion (equa ion (11)):
H-
G?M
= 2A B /1cos2»d9+ /4cos2»dd
0
JO
+
R2
J2cos23-R J3(l+cos29)cos3d9
G 2Ci l
= 2i4 B Ixsm2&d$ + R2 /2sin2dd»-/?
|
/3sin29cos9dd
L
Jo Jo Jo
G
22c,
s
—
2-4
j^ l
j"
cos2
a da +
7*
73sin2acosada
G*2!ic,s
= 2i4
73
71sin2ada+ 74sin2ada —7?
73
sin2 a cosa da
G3''3c,s
= 2/IB 7lCos»dd
u13c,sl
—
u23c,s
—
u31c,s
—
°32c.s
—
u
?3c.. = 2i4£ /3cos2&d9-K /2cos»d9
0
JO
/!&,„ =
2,4
J? /3 sin2 Odd
J
o
H i^=-2AB
^cosOdO-K J2cos20d3
/*«
#*"c.s = 2/4B/? 72sin29d9
77*ii
77*" 77*" 77*" 77*" A
11
lics
—
/322c,si
—
/733c,s
"**
" 12c.s
—
n
21c,s —
u