Dynamic analysis o piled ounda ions in s a i ied soils by a
BEM-FEM model ∗
L.A. Pad ´on, J.J. Azn´a ez, O.Maeso
Ins i u o Uni e si a io de Sis emas In eligen es y Aplicaciones Num´e icas en Ingenie ´ıa
(SIANI) Uni e sidad de Las Palmas de G an Cana ia
Edi icio Cen al del Pa que Cien ´ı ico y Tecnol´ogico
Campus Uni e si a io de Ta i a, 35017, Las Palmas de G an Cana ia, Spain
{lpad on,jjazna ez,omaeso}@siani.es
4 July 2007
Abs ac
In his pape , a 3D BEM-FEM coupling model is used o s udy he dynamic beha io o piled
ounda ions in elas ic laye ed soils in p esence o a igid bed ock. Piles a e modelled by FEM
as beams acco ding o he Be noulli hypo hesis, and e e y laye o he soil is modelled by BEM
as a con inuum, semi-in ini e, iso opic, homogeneous, linea , iscoelas ic medium. Fi s , he
main poin s o he model a e se ou . Then, se e al esul s o e ical, ho izon al and ocking
impedances o single piles and 2 ×2 pile g oups embedded in a s a um es ing on a igid
bed ock, a e p esen ed. The in luence on he dynamic esponse o s a um dep h, soil s i ness
and piled ounda ion con igu a ion is discussed. Finally, he in luence o he s a ig aphy on
he seismic esponse o a 3 ×3 pile g oup is analyzed, oge he wi h he pile- o-pile kinema ic
in e ac ion and he wa e-sca e ing phenomena.
1 In oduc ion
The s eady-s a e dynamic esponse o piled ounda ions in elas ic soils has been he subjec o much
esea ch, in bo h he kinema ic and he o ced ib a ion analyses (see e.g. [1]). Fo he s udy o piles
and pile g oups embedded in a homogeneous hal -space, se e al equency domain bounda y in eg al
o mula ions in conjunc ion wi h he monodimensional Fini e Elemen Me hod (FEM) ha e been
used by di e en au ho s [2, 3, 4, 5]. Linea analyses o non-homogeneous media o laye ed soils
ha e also been ca ied ou using he same kind o app oach [6, 7, 8, 9]. Subsequen ly, mo e e sa ile
and igo ous linea nume ical models ha e been de eloped using he Bounda y Elemen Me hod
(BEM) o bo h soil and piles [10, 11, 12, 13], bu wi h he disad an age o a high compu a ional
cos .
Al hough a g ea deal o his esea ch has been ocused on he o ced ib a ion p oblem, much
o i has also deal wi h he kinema ic esponse o piled ounda ions. Fo ins ance, pa ame ic
s udies o he seismic esponse o single piles and pile g oups o Rayleigh wa es and o e ically
and obliquely inciden body wa es ha e been epo ed in [14, 15, 16, 17].
∗This is he pee e iewed e sion o he ollowing a icle: L.A. Pad ´on, J.J. Azn´a ez, O.Maeso, Dynamic analysis
o piled ounda ions in s a i ied soils by a BEM-FEM model, Soil Dynamics and Ea hquake Enginee ing 28 (2008)
333-346, which has been published in inal o m a doi:10.1016/j.soildyn.2007.07.005. This wo k is eleased wi h a
C ea i e Commons A ibu ion Non-Comme cial No De i a i es License.
1
On he o he hand, due o he ac ha piles a e commonly used when a oiding shallow soil o
low bea ing capaci y and ans e ing load o deepe soil o ock o high bea ing capaci y is needed,
a pa icula case o in e es in he dynamic analysis o piles is ha o piled ounda ions embedded
in a iscoelas ic s a um es ing on a igid bed ock, o bo h loa ing and hinged piles. Howe e ,
no many pape s ha e been epo ed ocusing on his opic [18, 19], hough some ha e deal wi h i
while s udying he pile-soil-s uc u e in e ac ion p oblem [20, 21, 22, 23].
Fo his eason, he aim o his pape is o p esen a me hod o he dynamic analysis o piles
and pile g oups and, making use o i , con ibu e o he opic discussed abo e by in es iga ing,
h ough pa ame ic s udies: a) he in luence o he p esence o a igid bed ock on he dynamic
impedances o piled ounda ions, and b) he in luence o he s a ig aphy on he seismic esponse
o hinged pile g oups. To his end, a BEM-FEM coupling model p e iously p esen ed by he
au ho s [24] is used o compu e ime-ha monic dynamic impedances and he seismic esponse o
piled ounda ions embedded in iscoelas ic zoned-homogeneous laye ed soils. The model has been
alida ed o bo h he kinema ic and he s i ness p oblems wi h se e al esul s aken om he
li e a u e (e.g. [2, 18, 19, 20]), hough hese compa isons a e no p esen ed he e o he sake o
b e i y.
In his app oach, and in he line o a p e ious s a ic model de eloped in [25, 26, 27], he pile-soil
in e ac ion akes place, om he in eg al equa ion poin o iew, h ough in e nal o ces, as i is
assumed ha he soil con inui y is no al e ed by he p esence o he piles. These a e modeled by
FEM as beams acco ding o he Be noulli hypo hesis, and e e y s a um o he soil is modeled by
BEM as a con inuum, semi-in ini e, iso opic, homogeneous, linea , iscoelas ic medium. The model
no only allows he dynamic analysis o piled ounda ions embedded in a hal -space o in a s a um
es ing on a igid bed ock, bu also in mul ilaye ed soils o gene ic s a ig aphy and opog aphy,
including deposi s and inclusions.
Fi s ly, he main poin s o he BEM-FEM coupling model o mula ion a e se ou . Secondly,
se e al esul s o e ical, ho izon al and ocking ime-ha monic dynamic impedances o single piles
and 2 ×2 pile g oups embedded in a s a um es ing on a igid bed ock, a e p esen ed. Di e en
dep hs o he s a um and h ee ounda ion con igu a ions a e s udied, and he e ec s associa ed o
hese pa ame e s a e discussed. Finally, he in luence o he s a ig aphy on he seismic esponse o a
3×3 pile g oup is analyzed. To his end, displacemen ans e unc ions o e ically-inciden plane
ime-ha monic shea wa es and esponse spec a o a pa icula con igu a ion, unde wo di e en
s ong g ound mo ions, a e p esen ed o se e al soil p o iles. Pile- o-pile kinema ic in e ac ion and
wa e-sca e ing phenomena, a e also in es iga ed.
Rega ding he impedances s udy, i is shown ha he e ec o he p esence o he igid bed ock
is i al o he ho izon al impedance o single piles and pile g oups, and also o he ocking beha io
o single piles, while i is almos negligible o he ocking esponse o pile g oups. Rega ding he
e ical impedances o loa ing piled ounda ions, he e ec o he bed ock is no impo an excep
o equencies below 1.5 imes he undamen al na u al equency o he s a um.
As o he pe o med seismic analysis, i can be seen ha he p esence o a so laye a he
op o a s a um yields o apidly dec easing ans e unc ions and, consequen ly, o a weakened
seismic esponse a he pile cap. Howe e , aking in o accoun addi ional laye s wi h shea wa e
eloci y inc easing wi h dep h is o mino impo ance.
2 Pile-soil in e ac ion model
The beha io o a pile submi ed o ha monically a ying loads, conside ing ze o in e nal damping,
can be desc ibed by he equa ion
¯
K up=Fex +Q qp,(1)
2
whe e ¯
K=K−ω2M, being Kand M he s i ness and mass ma ices o he pile, ωis he ci cula
equency o exci a ion, and up he ec o o nodal ansla ion and o a ion ampli udes along he
pile. Fex includes he o ces a he op F op and he axial o ce a he ip o he pile Fp,qpis he
ec o o ac ions along he pile-soil in e ace, whe eas Qis he ma ix ha ans o ms hese nodal
ac ion componen s o equi alen nodal o ces. By means o Eq. (1), piles a e modelled as e ical
Be noulli beams by h ee-node FEM elemen s on which he e a e de ined 13 deg ees o eedom:
h ee la e al displacemen s on each node, and wo o a ions on each one o he ex eme nodes.
On he o he hand, each s a um o he soil is modelled by BEM as a linea , homogeneous,
iso opic, iscoelas ic, unbounded egion. The bounda y in eg al equa ion o a ime-ha monic
elas odynamic s a e de ined in a domain Ωmwi h bounda y Γmcan be w i en in a condensed and
gene al o m as
ckuk+ZΓm
p∗udΓ = ZΓm
u∗pdΓ + ZΩm
u∗XdΩ,(2)
whe e ckis he local ee e m ma ix a colloca ion poin ‘k’, Xa e he body o ces in he domain
Ωm,uand pa e he displacemen and ac ion ec o s, and u∗and p∗a e he elas odynamic
undamen al solu ion enso s o a ime-ha monic concen a ed load a poin ‘k’. A hys e e ic
damping model is used o he soil h ough a complex alued shea modulus µo he ype µ=
Re[µ](1 + 2iξ), being ξ he damping coe icien . De ails abou BEM o mula ion can be ound
in [28].
Gene ally, body o ces Xa e conside ed o be ze o in mos o elas odynamic p oblems. Ne -
e heless, in his app oach, om he in eg al equa ion poin o iew, he pile-soil in e ac ion akes
place h ough in e nal punc ual o ces placed a he geome ic piles ip and h ough load-lines
placed along he piles axis, as i is assumed ha he soil con inui y is no al e ed by he p esence
o he piles. The load-lines wi hin he soil, he ac ions along he pile-soil in e ace ac ing o e he
pile and wi hin he soil (qpj=−qsj), and he in e nal punc ual o ces Fpja he ip o he piles,
a e ep esen ed in Fig. 1, whe e a ske ch o he model is shown.
Figu e 1: Load-lines ep esen a ion o wo piles embedded in a laye ed soil.
Then, Eq. (2) can be w i en as
ckuk+ZΓm
p∗udΓ = ZΓm
u∗pdΓ +
nm
ll
X
j=1 "ZΓm
pj
u∗qsjdΓpj−δjΥj
kFpj#,(3)
whe e Γm
pjis he pile-soil in e ace along he load-line jwi hin he domain Ωm;nm
ll is he o al
numbe o load-lines in he domain Ωm;δjis equal o one i he load-line jcon ains he ip o a
loa ing pile and ze o o he wise; and Υj
kis a h ee-componen ec o ha ep esen s he con ibu ion
o he axial o ce Fpja he ip o he j h load-line.
3
Once all bounda ies ha e been disc e ized, Eq. (3) can be w i en o each egion o all nodes
in Γmand Γm
pj o ob ain wo ma ix equa ions o he ype
Hssus−Gssps−
nm
ll
X
j=1
Gspjqsj+
nm
ll
X
j=1
δjΥsjFpj= 0,(4)
and
upi+Hpisus−Gpisps−
nm
ll
X
j=1
Gpipjqsj+
nm
ll
X
j=1
δjΥpijFpj= 0,(5)
espec i ely, whe e Hand Ga e coe icien ma ices ob ained by in eg a ion o e he elemen s o he
undamen al solu ion imes he co esponding shape unc ions, usand psa e he ec o s o nodal
displacemen s and ac ions o bounda y elemen s, and upiis he ec o o nodal displacemen s o
he load-line i.
A coupled sys em o equa ions, o he ype
A{us,ps,qs,Fp,up}T=B,(6)
ep esen ing he laye ed soil-piles sys em, can be ob ained om Eqs. (1), (4) and (5), whe e equilib-
ium and compa ibili y ully-bonded con ac condi ions o e he di e en in e aces o he p oblem
ha e been imposed. Aand Ba e he squa e ma ix o coe icien s and he known ec o espec i ely,
bo h compu ed by ea anging he equa ions and p esc ibing he known bounda y condi ions.
In his wo k, wo di e en kinds o analysis a e ca ied ou , each one o hem wi h a di e en
se o bounda y condi ions, hough, in bo h cases, piles in a g oup a e conside ed o be ixedly
connec ed o a igid massless cap (no in con ac wi h he soil) and ze o- ac ion ee su ace is
assumed.
Fi s ly, dynamic s i ness o piles and pile g oups a e compu ed by p esc ibing o ced ib a ion
a he pile cap. Secondly, he seismic esponse o a piled ounda ion subjec o e ically-inciden
plane ime-ha monic shea wa es is analyzed. To his end, he displacemen and ac ion ields in
he laye ed soil a e conside ed as he supe posi ion o wo: he uni o m iscoelas ic laye ed soil
ields uIand pI, whose analy ic exp ession a e known; and he sca e ed wa e ields uSand pS,
p oduced by he p esence o he piles. Consequen ly, he o al ields a e he sum o hese wo
(u=uI+uS,p=pI+pS), while i is only in he sca e ed wa e ield ha he ac ions along he
pile-soil in e ace and he o ces a he pile ip exis (q=qS,Fp= (Fp)S). In his case, he abo e
equa ions a e w i en o he sca e ed ields in all egions, and hen exp essed in e ms o he o al
and he inciden ields as
Hssus−Gssps−
nm
ll
X
j=1
Gspjqsj+
nm
ll
X
j=1
δjΥsjFpj=Hssus
I−Gssps
I(7)
and
upi+Hpisus−Gpisps−
nm
ll
X
j=1
Gpipjqsj+
nm
ll
X
j=1
δjΥpijFpj=upi
I+Hpisus
I−Gpisps
I(8)
whe e he igh hand e ms a e known.
Only he main poin s o he o mula ion ha e been p esen ed he e. Howe e , mo e de ails can be
ound in Pad ´on e al. [24], whe e he o mula ion, ocused on pile g oups embedded in a hal -space,
has been p esen ed.
4
3 Dynamic s i ness o piled ounda ions in homogeneous s a a
The dynamic s i ness ma ix Kij o a pile ela es he ec o o o ces (and momen s) applied a
he pile op o he esul ing ec o o displacemen s (and o a ions) a he same poin . Fo a g oup
o piles, i is assumed ha he pile heads a e cons ained by a igid pile-cap, and he ounda ion
s i ness is he addi ion o he con ibu ions o each pile. Fig. 2 illus a es he app oached p oblem
o a usual con igu a ion, whe e Land da e used o deno e he leng h and diame e o he piles, s
e e s o he dis ance be ween adjacen piles and Hdeno es he dep h o he s a um.
Figu e 2: 2x2 pile g oup embedded in a s a um es ing on a igid bed ock. P oblem geome y
de ini ion.
The dynamic s i ness e ms o a ime-ha monic exci a ion a e unc ions o equency ω, and
hey a e usually w i en as
Kij =kij + iaocij,(9)
whe e kij and cij a e he equency dependen dynamic s i ness and damping coe icien s, e-
spec i ely, aois he dimensionless equency
ao=ωd
cs
(10)
and csis he soil shea -wa e eloci y.
Figu e 3: 2x2 pile g oup and s a um BEM-FEM disc e iza ion (only a qua e o he geome y).
Fig. 3 shows a ske ch o he disc e iza ions used o ob ain he s i ness o di e en pile g oups
embedded in a iscoelas ic homogeneous s a um es ing on a igid bed ock. As he de eloped
so wa e inco po a es symme y p ope ies, only a qua e o he o al geome y o he p oblem has
o be disc e ized.
5

In wha ollows, e ical, ho izon al and ocking impedances o di e en piled ounda ion con ig-
u a ions a e p esen ed. In he case o single piles, he s i ness and damping unc ions a e no malized
by he espec i e s a ic s i ness. As o pile g oups, he e ical and ho izon al impedance unc ions
a e di ided by he espec i e single pile s a ic s i ness (ks
zzoand ks
xxo) imes he numbe (N) o
piles in he g oup. Finally, he ocking impedances a e no malized wi h espec o he sum o he
p oduc s o he espec i e single pile e ical s a ic s i ness (ks
zzo) imes he squa e o he dis ance
o he o a ion axis (xi). All esul s a e plo ed e sus he dimensionless equency de ined by
Eq. (10), and se e al a ios H/L ha e been conside ed.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
4
5
6
7
ao
czz/(kzzo
)
H=L
H=1.5L
H=2L
H=3L
H=5L
H=10L
hal −space
Figu e 4: Ve ical impedances o a single pile embedded in a homogeneous s a um.
In he i s place, e ical impedances o a single pile embedded in a homogeneous s a um o
dep h H es ing on a igid bed ock a e shown in Fig. 4 o a ios be ween s a um dep h and pile
leng h o H/L = 1 (hinged pile), 1.5, 2, 3, 5 and 10, oge he wi h he esponse o he hal -space.
I is assumed ha he s a um p ope ies a e: in e nal damping coe icien βs= 0.05 and Poisson
a io νs= 0.4. The a io be ween densi ies is ρs/ρp= 0.7, he aspec a io o he pile is L/d = 15,
and he pile/soil modulus a io is Ep/Es= 103.
As can be seen om he igu es, he p esence o a igid bed ock below a loa ing pile has a s ong
in luence on he impedances in he equency band om he s a ic alue o app oxima ely 1.5 imes
he dimensionless undamen al na u al equency o he s a um in comp ession-ex ension mode.
Abo e his equency, s i ness and damping coe icien a e coinciden wi h hose o a loa ing pile in
a hal -space, which e eals ha he main damping mechanism a in e media e and high equencies
is he ene gy dissipa ion h ough su ace wa es o bo h he hal -space and he s a um. Below he
i s na u al equency, he damping coe icien s a e a lowe han he ones co esponding o he
loa ing pile in a hal -space, because he e canno be su ace wa es in a s a um a low equencies
and, hus, he ene gy is con ined in i . As expec ed, he s a ic alue o he e ical s i ness o a
hinged pile is much highe han he co esponding o a loa ing pile. Mo e p ecisely, i is 5.1 imes
highe han he co esponding o a loa ing pile embedded in a s a um o dep h H= 1.5L. On he
o he hand, he undamen al na u al equency o he sys em ela ed o he comp ession-ex ension
mode is clea ly highligh ed in each case. No e ha he in luence o he p esence o he igid bed ock
is s ill no iceable o a s a um dep h i e imes he pile leng h, while i is ha dly signi ican when
he s a um dep h is en imes he pile leng h.
Ho izon al and ocking impedances o a single pile embedded in a homogeneous s a um o dep h
H es ing on a igid bed ock a e shown in Figs. 5 and 6 espec i ely. The cha ac e is ics o piles and
soils o hese and u he cases a e he ones de ined abo e and, o he sake o cla i y, only esul s
o a ios be ween s a um dep h and pile leng h o H/L = 1 (hinged pile), 1.5 and 2, oge he wi h
he esponse o he hal -space, a e p esen ed. The ange o in e es in each s i ness igu e has been
enla ged, and he na u al equencies o which e e y peak is associa ed ha e been labeled o he
6
ho izon al cases.
In his case, he esonance e ec s a e associa ed o bo h he shea and comp ession-ex ension
modes. The in luence o he p esence o he igid bed ock is signi ican app oxima ely un il he
second na u al equency ela ed o he shea mode. In his ange, he impedances luc ua e abou
he hal -space solu ion o bo h he s i ness and he damping coe icien s. As o he la e , hey
a e small a low equencies as discussed abo e. Fo highe equencies, he impedance beha io is
simila o he one co esponding o a pile embedded in a hal -space.
Ve ical and ho izon al impedances o a 2×2 pile g oup embedded in a homogeneous s a um o
dep h H es ing on a igid bed ock a e shown in Figs. 7 and 8, espec i ely. Ra ios be ween pile
sepa a ion and diame e o s/d = 2, 5 and 10 a e p esen ed. Fo scale easons, he peaks associa ed
o he s a um na u al equencies do no appea e y clea ly in he igu es, bu hei magni udes
a e p opo ional o hose o he single pile case, and e en inc ease wi h he a io s/d. Besides, he
e ical impedances a in e media e and high equencies a e equi alen o hose o a loa ing pile
g oup embedded in a hal -space. Howe e , he g oup e ec is p edominan o e he in luence o he
p esence o he igid bed ock. On he o he hand, o he ho izon al case, he in luence o he igid
bed ock is ela i ely much mo e impo an , e en hough he g oup e ec is s ill p edominan . The
impedances luc ua e abou he hal -space solu ion, and he ampli ude o his luc ua ion inc eases
wi h he a io s/d.
Table 1 p esen s he compa ison be ween he alues o dimensionless na u al equencies ob ained
wi h he p esen ed coupling model and hose analy ically compu ed o an undamped s a um. Only
he alues ob ained o he ho izon al impedance o a pile g oup wi h a a io s/d = 10 a e p esen ed.
As can be seen, he p oposed me hod p edic s adequa ely he ac ual undamen al equencies.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
ao
cxx/(kxxo
)
H=L
H=1.5L
H=2L
hal −space
Figu e 5: Ho izon al impedances o a single pile embedded in a homogeneous s a um.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
ao
cφφ/(kφφo
)
H=L
H=1.5L
H=2L
hal −space
Figu e 6: Rocking impedances o a single pile embedded in a homogeneous s a um.
7
S a um na u al equencies in
shea mode comp ession-
ex ension mode
as
o(1)as
o(2)ap
o(1)H/L
Undamped s a um 0.11 0.31 0.26 1
Pile g oup-soil sys em 0.12 0.33 0.25
Undamped s a um 0.07 0.21 0.17 1.5
Pile g oup-soil sys em 0.09 0.23 0.17
Undamped s a um 0.05 0.16 0.13 2
Pile g oup-soil sys em 0.07 0.16 0.12
Table 1: Compa ison be ween na u al equencies o an undamped s a um and o a 2 ×2 pile
g oup-s a um sys em. s/d = 10.
Finally, ocking impedances o a 2×2 pile g oup embedded in a homogeneous s a um on a igid
bed ock a e shown in Fig. 9. Only he a io s/d =10, o which he in luence o he igid bed ock
is s onge , has been displayed in o de o illus a e he ac ha he ocking impedances o pile
g oups a e li le in luenced by he p esence o he bed ock.
8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
10
ao
cG
zz/(N ks
zzo
)
H=L
H=1.5L
H=2L
hal −space
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
0
2
4
6
8
ao
kG
zz/(N ks
zzo
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
10
ao
cG
zz/(N ks
zzo
)
H=L
H=1.5L
H=2L
hal −space
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
0
2
4
6
8
ao
kG
zz/(N ks
zzo
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
10
ao
cG
zz/(N ks
zzo
)
H=L
H=1.5L
H=2L
hal −space
s
d= 2
s
d= 5
s
d= 10
Figu e 7: Ve ical impedances o a 2x2 pile g oup embedded in a homogeneous s a um. s/d=2, 5
and 10 ( om op o bo om).
9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T (s)
accele a ion (g)
F ee ield
Hal −space
1 s a um
1 so s a um
2 s a a
3 s a a
4 s a a
Figu e 16: 5 pe cen -damped accele a ion esponse spec a o El Cen o (1940) ea hquake speci-
ied a he ee su ace.
pa ame e s such as densi ies, damping coe icien s and Poisson a ios, conclusions om his s udy
may no be gene alized. On he o he hand, only he ho izon al esponse has been ea ed he e bu ,
depending on he p oblem, he ocking beha io could also be an impo an pa ame e .
5 Conclusions
In his pape , a BEM-FEM coupling model p e iously p esen ed by he au ho s has been used o
compu e dynamic impedances and kinema ic esponse o piled ounda ions embedded in iscoelas ic
homogeneous s a a es ing on a igid bed ock. This way, he wo i s s eps o he subs uc u ing
me hod a e pe o med.
Piles ha e been modelled by FEM as beams acco ding o he Be noulli hypo hesis, and s a a
ha e been modelled by BEM as con inuum, semi-in ini e, iso opic, homogeneous, linea , iscoelas-
ic media. The model no only allows he dynamic analysis o piled ounda ions embedded in
a hal -space o in a s a um es ing on a igid bed ock, bu also in mul ilaye ed soils o gene ic
s a ig aphy, including deposi s and inclusions. Any kind o geome y can be modelled by bound-
a y disc e iza ions, hus, mo e complex s a ig aphies and opog aphies han hose epo ed in his
pape can be analyzed.
Fi s ly, se e al esul s o e ical, ho izon al and ocking impedances o single piles and h ee
di e en con igu a ions o 2 ×2 pile g oups embedded in a s a um ha e been p esen ed. Se e al
dep hs o he s a um ha e also been s udied. F om he analysis o hese esul s, he ollowing
conclusions can be d awn:
•The pile-soil sys em p esen s peaks associa ed o he s a um na u al equencies ha he
nume ical model is able o p edic adequa ely.
•The e ical impedance unc ions o a piled ounda ion embedded in s a a o he analyzed
dep hs a e equi alen o hose embedded in a hal -space a equencies abo e 1.5 imes he
undamen al na u al equency o he s a um in he comp ession-ex ension mode. Besides,
he ounda ion e ical beha io is only in luenced by his na u al equency.
•The in luence o he s a um dep h o e he ho izon al impedance unc ions o a piled ounda-
ion is signi ican o e a b oade band o equencies. Thus, se e al peaks associa ed o bo h
he shea and he comp ession-ex ension modes appea in he ounda ion esponse.
16

0 2 4 6 8 10
−0.8
−0.4
0
0.4
0.8
(s)
accele a ion (g)
F ee ield
0 2 4 6 8 10
−0.8
−0.4
0
0.4
0.8
(s)
accele a ion (g)
1 s a um
0 2 4 6 8 10
−0.8
−0.4
0
0.4
0.8
(s)
accele a ion (g)
1 so s a um
0 2 4 6 8 10
−0.8
−0.4
0
0.4
0.8
(s)
accele a ion (g)
2 s a a
0 2 4 6 8 10
−0.8
−0.4
0
0.4
0.8
(s)
accele a ion (g)
3 s a a
0 2 4 6 8 10
−0.8
−0.4
0
0.4
0.8
(s)
accele a ion (g)
4 s a a
pga=0.778 g pga=0.769 g
pga=0.529 g pga=0.532 g
pga=0.567 g pga=0.566 g
Figu e 17: Response accele a ion his o ies o El Cen o (1979) ea hquake speci ied a he ee
su ace.
•This in luence is e en mo e e iden in he damping unc ions, which a low equencies a e
much smalle han he co esponding o he hal -space. On he con a y, hei alues a e
simila a highe equencies, which e eals ha he main damping mechanism a in e media e
and high equencies is he ene gy dissipa ion h ough su ace wa es o bo h he hal -space
and he s a a.
•The in luence o he p esence o he igid bed ock is s ill no iceable o a s a um dep h i e
imes he pile leng h.
•The g oup e ec is p edominan o e he in luence o he p esence o he igid bed ock, es-
pecially in e ical and ocking impedances, in which he peaks associa ed o he na u al
equencies o he s a um a e o small magni ude. Howe e , he e ec s o he p esence o
he bed ock on he ho izon al beha io a e s onge and become appa en along a b oade
equency band. In all cases, he in luence o he igid bed ock inc eases wi h he a io s/d.
•The ocking beha io is he less in luenced by he p esence o a igid bed ock.
Also, addi ional expe imen s (no shown he e o he sake o b e i y) ha ha e been ca ied
ou in o de o in es iga e he ole o he a io Ep/Es, show ha he equency band in which he
ounda ion esponse is in luenced by he igid bed ock b oadens wi h he inc ease o he soil s i ness.
Also, a la ge numbe o na u al equencies o he s a um in bo h he shea and he comp ession-
ex ension modes become e iden o he ho izon al and ocking cases as he soil s i ness inc eases.
A he same ime, he g oup e ec becomes mo e e iden as he a io Ep/Es u ns highe .
17
Figu e 18: 5 pe cen -damped accele a ion esponse spec a o El Cen o (1979) ea hquake speci-
ied a he ee su ace.
Secondly, he seismic esponse o a squa e 3 ×3 hinged pile g oup embedded in di e en soil
p o iles and unde e ically-inciden plane ime-ha monic shea wa es and wo di e en s ong
g ound mo ions, ha e been s udied. Gene al conclusions canno be d awn wi hou mo e and deepe
analysis in he subjec bu , as o he p esen ed case, i has been shown ha :
• he p esence o a so laye a op he soil yields o much mo e apidly dec easing displacemen
ans e unc ions han hose co esponding o ounda ions in he e e ence homogeneous soil.
In ac , wo di e en ends a e clea ly de ined, depending on whe he he so laye is con-
side ed o no .
• he addi ion o u he in e media e shea wa e eloci y laye s in be ween is o mino impo -
ance.
• he inciden ield a he ee su ace is pe u bed by he p esence o he piled ounda ion,
being he magni ude o his pe u ba ion la ge when a so laye exis s a op. The maximum
alue o his pe u ba ion is o simila o de in bo h he di ec ion o mo ions induced by he
S-wa es and i s pe pendicula .
•pile- o-pile in e ac ion unde seismic exci a ion is almos nonexis en .
•piled ounda ions embedded in he non-homogeneous soil p o iles s udied he e il e ou a
g ea pa o he ha monic componen s o he seismic inpu , in such a way ha he esul ing
accele a ion esponse spec a a e signi ican ly lowe han hose co esponding o he ee ield.
• he simpli ica ion o he soil p o ile o jus one s a um o o a hal -space would lead o
o e - o unde es ima ing, depending on he chosen p ope ies, he exci a ion a he base o a
supe s uc u e in a subs uc u ing analysis.
• hese conclusions a e alid o ea hquake mo ions speci ied a bo h he ee su ace o a he
bed ock.
Acknowledgemen s
The au ho s would like o hank he e iewe s o hei aluable commen s, ha ha e con ibu ed
o imp o e he pape . This wo k was suppo ed by he Minis y o Educa ion and Science o
Spain h ough esea ch p ojec BIA2004-03955-C02-02 and co- inanced by he Eu opean Fund o
18
Figu e 19: 5 pe cen -damped accele a ion esponse spec a o El Cen o (1940) ea hquake, spec-
i ied a he bed ock.
Regional De elopmen . L.A. Pad ´on is ecipien o he FPU esea ch ellowship AP-2004-4858 om
he Minis y o Educa ion and Science o Spain. The au ho s would like o hank o his suppo .
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