Soil-Pile Dynamic Interaction in the Viscous Damping Layered Soils

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1 1 The Open Civil Engineering Journl, 11, 5, 1-18 Soil-Pile Dynmic Inercion in he Viscous Dmping Lyered Soils Open Access Yinhui Wng 1, Kuihu Wng, Zhixing Zh 1, * nd Reno Que 1 Deprmen of Civil Engineering nd Archiecure, Ningo Insiue of Technology, Zhejing Universiy, Ningo, 3151, Chin MOE Key Lorory of Sof Soils nd Geoenvironmenl Engineering, Zhejing Universiy, Hngzhou, Zhejing, 3158, Chin Asrc: Modeling surrounding soil s hree-dimensionl xisymmeric coninuum nd considering is wve effec, soil-pile dynmic longiudinl inercion in viscous lyered soils is sudied. The pile is ssumed o e vericl, elsic nd of uniform secion, nd he soil is lyered nd visco-elsic. Longiudinl virion of pile in viscous dmping lyered soils undergoing rirry lod is heoreiclly invesiged. By ing he Lplce rnsform, he quesion cn e solved in frequency domin. Uilizing wo poenils comined wih impednce rnsfer funcions, nlyicl soluions for oh he impednce funcion nd moiliy he pile hed in frequency domin re yielded. Wih he convoluion heorem nd inverse ourier rnsform, semi-nlyicl soluion of velociy response in ime-domin undergoing hlf-cycle sine pulse force is derived. Bsed on he soluions proposed herein, he effecs of vriey of soil modulus on moiliy curves nd reflecion wve curves re emphiclly discussed. The resuls shows h here is smller pe eween every wo djcen lrger pes on he moiliy curve in lyered soil, nd lrger pe cycle reflecs he locion where he modulus of he soil vries ruply. The conclusions cn provide heoreicl guidnce for non-desrucion es of piles. Keywords: Soil-pile dynmic inercion, Lyered soils, Viscous dmping, Admince curves, Moiliy curves, Reflecion wve curves. 1. INTRODUCTION Dynmic non-desrucion es mehod sed on he pile virion heory is widely used o idenify pile inegriy in civil engineering. Mny soil-pile dynmic inercion models hve een developed o simule he ehvior of longiudinl virion of pile. rom he view poin of differen model of he surrounding soil, hey cn e pu ino wo cegories, h is, Winler model [1-7] nd coninuum model [8-13]. In he former model, soil is modeled y he disriued Voig ody. However, he vlue of prmeers in Winler model, h is, siffness of spring nd dmping of dshpo, cn correle well wih he usul soil esing resul. Wh s more, i is difficul o consider he wve effec of he surrounding soil. The firs ind of coninuum model [8-11] is Plin-Srin model wih ssumpion h soil consiss of independen infiniesimlly hin lyers exending o infiniy horizonlly. In Plin-Srin model, i is ssumed h he grdien of srin nd sress in he vericl direcion is zero nd he wves propge only horizonlly. Such ssumpion does no mch well wih reliy. The second ind of coninuum model [1, 13] es he grdien of sress of he surrounding soil in vericl direcion ino ccoun nd hence cn consider he wve effec of he surrounding soil. Neverheless, wo impor fcors, sy, he rdil displcemen nd he *Address correspondence o his uhor he Deprmen of Civil Engineering nd Archiecure, Ningo Insiue of Technology, Zhejing Universiy, Ningo, 3151, Chin Tel: x: E-mil: zhzx71@16.com xisymmeric wve effec of he surrounding soil, re ll negleced in ove models. urhermore, he foundion soil is lyered nd his lyered nure hs significn impc on he dynmic response of he pile nd hence should e considered. Bsed on ove review, he purpose of his pper is o derive n nlyicl soluion for longiudinl virion of pile sujeced o hrmonic longiudinl exciion in lyered soils y ing he xisymmeric wve effec of he surrounding soil ino ccoun. Uilizing he soluion derived herein, he effecs of vrious soil prmeers on he longiudinl virion of pile re discussed.. ORMULARION.1. Pile-Soil Sysem Model The prolem sudied herein is he longiudinl virion of pile emedded in lyered soil wih viscous dmping. The geomeric model is shown in ig. (1. The soil-pile sysem is discreized ino ol of n lyers numered y 1,,, n from he pile oe o pile op. The properies of pile nd soil lyer re ssumed o e homogeneous wihin ech lyer respecively, u my vry from lyer o lyer. In he h soil-pile lyer (1 n, he mss densiy, modulus of compression, hicness, poisson s rio, longiudinl wve velociy, rnsversl wve velociy of he soil re denoed y s, E s, h,, VL nd VS, respecively. The Lme consns re nd μ nd he corresponding viscosiy /11 11 Benhm Open

2 Soil-Pile Dynmic Inercion in he Viscous Dmping Lyered Soils The Open Civil Engineering Journl, 11, Volume 5 11 h n h h 1 n 1 ig. (1. Schemic illusrion of pile-soil sysem. coefficiens re nd μ, respecively. Here E s (1+ μ nd o f ( pile soil lyer z c 1 μ, where is he rnsverse rio of he viscous deformion re. The lengh, mss densiy, elsic modulus, longiudinl wve velociy, cross secionl re nd secionl rdius of he h pile lyer re expressed s H, p,, VP, S nd r, respecively. f ( is n rirry vericl exciing force cing on he pile op nd R (z, is fricion force cing on he surfce of he pile long per uni lengh in he h lyer. The suppor of soil he oe of pile is descried s single Voig ody which consiss of liner spring nd dshpo conneced in prllel. The elsic modulus of he liner spring nd he dmping coefficien of he dshpo re denoed y nd c, respecively. The suppors of surrounding soil he level of pile oe re descried s disriued Voig odies in rdil direcion. The elsic modulus of he liner spring nd he dmping coefficien of he dshpo re expressed s s nd c s, respecively... Assumpions The soil-pile sysem model is developed on he sis of he following ssumpions: (1 The surrounding soil lyers wih liner viscoelsic dmping re ssumed o e isoropic nd homogeneous wihin ech lyer. The soil is infinie in he rdil direcion wih free oundry condiion he surfce of he soil. The soil rdil displcemen he inerfce of he pile shf is regrded o e smll nd hus cn e negleced. ( The exciion is hrmonic. The soil-pile sysem is sujeced o smll deformions nd srins during he virion. Pile nd soil conc perfecly nd herefore oh force equilirium nd displcemen coninuiy re sisfied he inerfce of soil-pile, soil-soil nd pile-pile. H 1 H 1 s H cs H r (3 The pile is elsic nd vericl wih uniform circulr cross secion. (4 The iniil displcemen nd velociy in he soil nd pile re zero..3. Dynmic Equion of Soil The geomeric model of soil-pile in h soil lyer is shown in ig. (. The suppor of he surrounding soil he op nd oom re descried s Voig odies disriued in rdil direcion. The elsic modulus of he liner spring nd he dmping coefficien of he dshpo re denoed y ss, c ss nd sx, c sx, respecively. The virion is xisymmeric. Le u r (r, z,, u z (r, z, o e he rdil nd vericl displcemen, respecively nd hen dynmic equion of soil cn e wrien s follows: 1pile segmen pile segmen 1 pile segmen ig. (. Schemic illusrion for soil lyer. Rdil direcion: G 1 ( + 1 r u r + G Vericl direcion: G 1 u z G ( r + 1 r s where G 1 ( + μ + ( + μ, G ( + μ + ( + μ 1 (u z r u r z. ss c ss soil lyer sx z u r (1 s u z (, r + 1 r r + z nd.4. Dynmic Equion of Pile Assuming he pile o e one-dimensionl coninuum, is dynmic equion cn e expressed s: h c sx

3 1 The Open Civil Engineering Journl, 11, Volume 5 Wng e l. S w (z, + R z (z, m w (z, (3 where m nd w (z, re mss of uni lengh nd vericl displcemen, respecively..5. Boundry Condiions nd Iniil Condiions or convenience, locl coordine sysem is doped herein. This mens h he coordine is zero he oom nd h he op in he h soil-pile lyer. (1 Boundry condiions of he soil lyer: Displcemens pproch zero n infinie rdil disnce: u r (r, z, r (4- u z (r, z, r Rdil displcemen is zero he inerfce of soil nd pile: u r, z, (4- A he oom of he h soil-pile lyer, sress oundry condiion is s follows: sx u E z + c u sx z u z s E s z z where sx1 s, c sx1 c s. (4-c A he op of he h soil-pile lyer, sress oundry condiion is s follows: ss u E z + c u ss z s E s z zh ( n + u z z zh ( n 1 (4-d ( ricion force nd displcemen coninuiy condiion he inerfce of he h soil-pile lyer: R (z, r rz, z, u z, z, w (z, (3 Boundry condiions of he h pile segmen: A he op of he h pile segmen: w z f ( S zh (5- (5- (6- where f ( is he force h cs on he op of h pile segmen y +1h pile segmen nd f n ( f ( he level of pile hed. A he oom of he pile: c x w + x S S w w z z (6- where c x1 c, x1. (4 Iniil condiions of he h soil-pile lyer: Iniil condiions of he h soil lyer: u r (r,z,, u r (r,z, u z (r,z,, u z (r,z, Iniil condiions of he h pile segmen: w ( z,, w (7- ( z, (7-3. SOLUTION O THE EQUATIONS 3.1. Virions of he Soil Lyer Two poenil funcions re inroduced o decouple he displcemens u r (r,z, nd u z (r,z, in he Eq. (1 nd Eq. (: u r (r, z, (r,z, s r + s (r,z, rz u z (r, z, (r,z, s z 1 r r r (r,z, s r (9 Le U r (r, z,s U, z (r, z,s, s (r, z,s nd s (r, z,s o e he Lplce rnsform of u r (r, z,, u z (r,z,, s (r, z, nd s (r, z, wih respec o, respecively. Ting he Lplce rnsform of Eq. (1 nd Eq. ( nd comining he iniil condiions of he soil lyer yield: Rdil direcion: G L 1 ( + 1 r U + G L r Longiudinl direcion: (8 z L s s U r (1 G L 1 U z G L ( r + 1 r L s s U z (11 where G 1 L G L L ( + μ + ( + μ s, ( + μ + ( + μ s, 1 (U z r U r z. r + 1 r r + z nd Ting he Lplce rnsform of Eq. (8 nd Eq. (9 yields: U r (r, z,s (r,z,s s r + s (r,z,s rz U z (r, z,s s (r,z,s z (1 1 r r r (r,z,s s r (13

4 Soil-Pile Dynmic Inercion in he Viscous Dmping Lyered Soils The Open Civil Engineering Journl, 11, Volume 5 13 Susiuing Eq. (1 nd Eq. (13 ino Eq. (1 nd Eq. (11 yields: L G r 1 s + ( μ + μ s rz s s s ( s r + s rz (14 ( L G z 1 s μ + μ s s s s z ( s r + 1 r r ( s r + 1 r r (15 And hus Eq. (14 nd Eq. (15 cn e decoupled s follows: s s l s (16 s s v (17 s s where l G L 1, s μ + μs, (1 s s 1 VL + μ nd VS μ re longiudinl wve nd s s sher wve velociy of he h soil lyer, respecively. Solving Eq. (16 nd Eq. (17 wih mehod of seprion of vriles firs nd hen comining Eq. (1 nd Eq. (13, we cn oin he soluions of U r (r, z,s nd U z (r, z,s s follows: U r (r, z,s [A 1 cos( z + B 1 sin( z] [C 1 I 1 ( r D 1 ( r] [A sin( z B cos( z] (18 or convenience, repe Eq. (4- here: u r (r, z, r ( u z (r, z, r Te he Lplce rnsform of Eq. (: U r (r, z,s r (4- U z (r, z,s r Similrly, ing he Lplce rnsform of Eq.(4- o (4- d s follows: U r, z,s (4- ( sx + c sx su E s E z (r, z,s U (r, z,s z s z z (4-c ( ss + c ss su E s E z (r, z,s + U (r, z,s z s z zh (4-d [( n + μ n + ( n + μ n s] U zn z + ( + s (ru rn n n rr zhn Susiuing Eq. (18 nd Eq. (19 ino Eq. (4- o (4-d nd solving hem simulneously yields U r (r, z,s nd U z (r, z,s : U r (r, z,s m M m sin( m z m m [ ( m r K ( r 1 m (s m r K (s r] 1 m U z (r, z,s m M m cos( m z m [ K ( r m 1 m m (s m r s K (s r K ( r] m m m m (3 where m is consns o e deermined. m, m, s m, m nd M m cn e oined from he following equions: [C I 1 (s r D (s r] s U z (r, z,s [B 1 cos( z A 1 sin( z] [C 1 I ( r + D 1 K ( r] [A cos( z + B sin( z] [C I (s r + D K (s r]s (19 n( m h (KX + KS m (4 KX KS m m m - s l (5 m s m s (6 s where I ( nd K ( re modified Bessel funcions of order zero of he firs nd second ind, respecively I 1 ( nd ( re modified Bessel funcions of order firs of he firs nd second ind, respecively. A 1, B 1, C 1, D 1, A, B, C, D nd re consns o e deermined y he oundry condiions. nd s sisfy he following relionship: - s l ( s s (1 s n( m KX m (7 M m 1+ ( m KX (8 KS + c s ss ss ( < n E s where KS ( n KX + c s sx sx E s (9

5 14 The Open Civil Engineering Journl, 11, Volume 5 Wng e l. 3.. Soluion for Longiudinl Virions of he Pile Denoe W (z,s s he Lplce rnsform wih respec o ime of w (z, nd e he Lplce rnsform of Eq. (3. Afer using he iniil condiion (7-, we cn ge: d W (z,s + p W dz (z,s R (z,s S where p s VP. Rewrie oundry condiion (5- s : R (z,s r (μ + μ s m M m cos( m z m ( m s m m K 1 ( m r m (3 (31 Then solving Eq. (3 for W nd using oundry condiion Eq.(31 gives: W Asin( pz + B cos( pz + C m cos( m z m (3 C m C m m where C m r (μ + μsm (s m m S( p m m m m ( m r (33 Ting he Lplce rnsform of Eq. (5- gives U z, z,s W (z,s, which cn e furher rewrien s follows: m D m cos( m z m C m cos( m z m + Asin( pz + B cos( pz (34 where D m M m [ m K ( m r m ( m r m (s m r s m K (s m r ] (35 I cn e proved h cos( m z m form n orhogonl se over he inervl [, h ] s follow: h cos( m z m cos( n z n dz (m n h cos( m z m cos( n z n dz (m n (36 Muliplying oh sides of Eq. (34 y h cos( m z m nd hen inegring over he inervl [, h ], we cn ge W : W + 1m m C m cos( m z m + sin( pz A (D m C m 3m C m cos( m z m + cos( pz B (D m C m 3m (37 1m h where m h 3m h h h h sin( pzcos( m z m dz cos( pzcos( m z m dz cos( m z m cos( m z m dz. (38 Denoing (s s he Lplce rnsform of f ( wih respec o nd ing he Lplce rnsform of he oundry condiions Eq. (6- nd Eq. (6- yields: dw dz (s S zh (6- S dw dz ( + c sw (6- x x z Susiuing Eq. (37 ino Eq. (6- o (6- nd solving for A nd B, we cn oin he impednce funcion he h pile segmen hed: Z (s (s W (h,s E S p h {[ where L 1 1m C m m sin( (D m C m m h m + p cos( ph, 3m N 1 m C m m sin( (D m C m m h m p sin( ph, 3m C m 1m [ Z 1 E L p S cos( sin( ] m m m p, (D m C m 3m N C m m [ Z 1 S cos( sin( ] m m m (D m C m 3m + Z 1 S. (39 And hen he velociy dmince funcion he h pile segmen cn e oined s follows: G v (s s Z (s (4 Se i 1, nd T h o e he imginry uni, VP he circulr frequency nd he propging ime in he h pile segmen, respecively. or convenience, some dimensionless prmeers re inroduced s follows: r r h s p v sp VS VP D μ (L 1 N L N 1 h C m cos( m z m (N (D m C m 1m L m +[N sin( pz L cos( pz]} 3m v sp p T r μ D μ T R T x h sx I E x c h sx s E s T

6 Soil-Pile Dynmic Inercion in he Viscous Dmping Lyered Soils The Open Civil Engineering Journl, 11, Volume 5 15 KX (R x + ii x p R s ss h E s I s c ss h E s T h n( [KX + KS ] [ KX KS ] (41 KS (R s + ii s p s ( s 1+ id μ p s h VS M m 1+ ( m KX ( s + i[( D + D μ ]p R h 1 S I c h 1 ST 1 Z R + ii p 1 r (1+ id p M (s μ m m m m ( m r C m m p m D m M m [ m ( m r m (s m r s m K (s m r m K ( m r ] Z 1 h S Z 1 1m cos( m cos[( p + m m ] p + m + cos( m cos[( p m + m ] p m m sin[( p + m m ]+ sin( m p + m + sin[( p m + m ] sin( m p m 3m sin[( m m ]+ sin( m m +1 L 1 1m C m m sin( m m + p cos( p (D m C m 3m C L m 1m [Z 1 cos( m m sin( m ] p (D m C m 3m N 1 m C m m sin( m m p sin( p (D m C m 3m C N m m [Z 1 cos( m m sin( m ] + Z 1 (D m C m 3m DN N [ 1m C m cos( m m + sin( p ] (D m C m 3m. L [ m C m cos( m m + cos( p ] (D m C m 3m where nd m cn e deermined uilizing he following equions: n( m KX (4 m where equion (41 is complex rnscendenl equion nd cn e solved y numericl mehod. Susiuing s i ino Eq. (39 o (4, he displcemen impednce funcion nd velociy dmince he hed of h pile segmen cn e expressed s follows: K d S h H v ( K d (41 1 p S VP H vl (4 where K d nd H vl re dimensionless displcemen impednce funcion nd velociy dmince funcion, respecively: K d Z (L N L N 1 1 (43 DN H vl 1 Z ip (44 n in Eq. (41 nd Eq. (4 nd hen he Se displcemen impednce funcion nd velociy dmince he pile hed cn e oined. When he exciion cing on he pile hed is hlf-sine pulse such s f ( Q mx sin(, (,T, where T T denoes he impulse widh, hen he semi-nlyicl velociy response of he pile hed cn e expressed s: g n ( Q mx IT (H vn T T (1+ eit Q mx p S VP ( (45 g n ( 1 T H (1+ v ln T ei T e i d (46 n h where T c denoes he propgion ime of elsic i1 VP longiudinl wve propging from he pile hed o pile ip., T T nd T T c T c denoe he dimensionless c ime, dimensionless impulse widh nd dimensionless circulr frequency, respecively. 4. PARAMETRIC STUDY AND DISCUSSION In he following, he effecs of vriey of soil modulus on he velociy dmince curves nd reflecion wve curves g n

7 16 The Open Civil Engineering Journl, 11, Volume 5 Wng e l. re discussed, which re he heoreicl sis of mechnicl impednce nlysis mehod nd reflecion wve mehod. or convenience, he suscrips mring he soil lyer re elimined if he prmeer vlues re sme in ech lyer in he following figures. Numer sequence of he soil lyer is shown in ig. (1. Prmeers using in he nlysis re shown s follows: V ij VS i / VS j VP 35 r.5 p 5 c μ sx E s 1 c sx 1 4 ss E s ( < n 1 c sx ( < n 1 4 T 1.5ms. The cse of wo soil lyers: Influence of modulus of upper soil lyer on he velociy dmince curve nd reflecion wve curve he pile op ig. (3. shows h he vriion of modulus of he upper soil lyer hs significn effec on he velociy dmince curve. Compred wih he cse h he soil is homogeneous, he velociy dmince curve of he pile op hs phenomenon h here is smller pe eween every wo djcen lrger pes. When he upper soil lyer is siffer hn he lower soil lyer, i cn e seen h he mpliude of he velociy dmince increse firs s he frequency increses, u hen decrese s he frequency furher increses fer he mpliude is eyond he mximum. When he upper soil lyer is sofer hn he lower soil lyer, i cn e seen h he mpliude of he velociy dmince decrese s he frequency increses nd he mpliude increse s he frequency furher increses fer he mpliude is eyond he minimum. or wo cses, he lengh eween every wo djcen lrger pes (lrge pe cycle is lmos equl, which reflecs he locion where he soil impednce vries ruply wih he relionship h VP / d mx 1.18m. The lengh eween one lrger pe nd is corresponding djcen smller pe (smller pe cycle reflecs he pile lengh wih he relionship H VP / d 19.64m. ig. (3. shows h wve curve he division surfce concves nd is ou of phse wih he inpu pulse when he upper soil lyer is siffer hn he lower soil lyer. The wve curve he division surfce convexes nd is in phse wih he inpu pulse when he upper soil lyer is sofer hn he lower soil lyer. ig. (4. nd ig. (4. show h he mpliude of he velociy dmince decreses s he modulus of he upper soil lyer increses nd oscillion is we. The dynmic siffness he low frequency rnge increses nd he mpliude of he inpu impulse nd he reflecion mpliude he pile ip decreses. The mpliude of he refleced wve mpliude he division surfce increses. I is due o more energy dissipion in he shllow lyer s he modulus of he upper soil lyer increses. The cse of hree soil lyers: Influence of modulus of surrounding soil on he velociy dmince curve nd reflecion wve curve he pile op ig. (3. Influence of Vriions of modulus of upper soil lyer on velociy dmince curve nd reflecion wve curve d mx 56 d h h 1 1m V1.5 V11. V1 /VP V1.5 V11 V1 h h 1 1m /VP h h 1 1m V1. V11.5 V11 /VP V1 V11.5 V11 h h 1 1m /VP ig. (4. Influence of vrying degree of modulus of upper soil lyer on velociy dmince curve nd reflecion wve curve.

8 Soil-Pile Dynmic Inercion in he Viscous Dmping Lyered Soils The Open Civil Engineering Journl, 11, Volume 5 17 ig. (5. nd ig. (5. show h he velociy dmince curve is similr o he cse of wo soil lyers when here is siffer or sofer inerlyer in he surrounding soil. Lrge pe cycle reflecs he locion of he inerlyer wih he relionship h3 VP / dmx 6.5m. A he firs pe, he mpliude of he cse wih siffer inerlyer is lrger hn h of he cse wih homogeneous soil nd he mpliude of he cse wih sofer inerlyer is smller hn h of he cse wih homogeneous soil. Consequenly, he properies of he upper soil lyer hve significn effec on he mpliude of he firs pe. The wve curve he division surfce is ou of phse wih he inpu pulse for siffer inerlyer cse nd in phse wih he inpu pulse for sofer inerlyer cse. The mpliude of he reflecion wve he pile ip decrese s he modulus of he siffer or sofer inerlyer increses d mx d mx1 17 d mx 13 d mx3 18 h m Hm 8m V311. h V m /VP m 1m h m Hm V311 V1.5 /VP4 168 d mx V311 V1.5. V11 /VP4 V1 h 1 1m h m V1.5 V11 V1 V311 /VP4 h 1 1m h m ig. (6. Influence of locion of sof inerlyer on velociy dmince curve nd reflecion wve curve h m Hm h. 4m V311 h V1.5 /VP ig. (5. Influence of hrd or sof inerlyer on velociy dmince curve nd reflecion wve curve. ig. (6. nd ig. (6. show h he occurring ime of he refleced wves of he inerlyer is rerded s he uried deph of he sofer lyer ecomes deeper. The lrge pe cycle of he velociy dmince curve reflecs he uried deph of he inerlyer wih he relionship h VP / d / d / d 6.39 m/ 8.46 m/1.18m. 3 mx1 mx mx3 ig. (7. nd ig. (7. show h he mpliude of he velociy dmince increses u he reflecion mpliude of he inerlyer ends o e weer s he hicness of he sofer inerlyer increses. The reflecion mpliude he pile ip increse. I is due o less energy dissipion in he inerlyer s is hicness increses.. h m h 4m h Hm V311 V1.5 /VP ig. (7. Influence of lengh of sof inerlyer on velociy dmince curve nd reflecion wve curve.

9 18 The Open Civil Engineering Journl, 11, Volume 5 Wng e l. 5. CONCLUSIONS (1 Modeling surrounding soil s hree-dimensionl xisymmeric coninuum nd considering is wve effec, soil-pile dynmic longiudinl inercion in viscous lyered soils is sudied. The nlyicl soluion of moiliy in he frequency domin nd semi-nlyicl soluion of velociy response in he ime domin undergoing hlf-cycle sine pulse force hve een derived. Bsed on he soluions herein, he effec of siffer or sofer inerlyer on he moiliy curves nd reflecion wve curves of inegred pile is sudied ( Compred wih he cse of homogeneous soil, here is smller pe eween every wo djcen lrger pes on he moiliy curve in lyered soil. The difference eween wo djcen lrger pes cn e used o loce where rup vriey of soil modulus hppens. A he division surfce, he reflecion wve curve is ou of phse wih he inpu pulse if soil modulus increses nd in phse wih he inpu pulse if soil modulus decreses. REERENCES [1] K. H. Vn, P. Middendorp, nd B. P Vn, An nlysis of dissipive wve propgion in pile, Inl seminr on he pplicion of Sress-Wve Theory onpiles/socholm, 198. [] D. W. Chng nd S. H. Yeh, Time-domin wve equion nlyses of single piles uilizing nsformed rdiion dmping, Soils nd oundions, JGS, vol. 39, no., pp , [3] W. Teng, W. Kuihu, nd X. Knghe, Sudy on virion properies of piles in lyered soils, Chin Civil Engineering Journl, vol. 35, no. 1, pp , 1. (in Chinese. [4] L. Dongji, Dynmic xil response of muli-defecive piles in nonhomogeneous soil, Chinese Journl of Geoechnicl Engineering, vol., no. 4, pp ,. (in Chinese. [5] W. Kuihu, Virion of inhomogeneous viscous-elsic pile emedded in lyered soils wih generl Voig model, Journl of Zhejing Universiy (Engineering Science, vol. 36, no. 5, pp , 595,. (in Chinese. [6] W. Kuihu nd Y. Hongwei, Virion of inhomogeneous pile emedded in lyered soils wih generl Voig model, Ac Mechnic Solid Sinic, vol. 4, no. 3, pp , 3. (in Chinese. [7]. Shijin, C. Yunmin, nd L. Mingzhen, Anlysis nd pplicion in engineering on vericl virion of viscoelsiciy piles in lyered soil, Chin Journl of Highwy nd Trnspor, vol. 17, no., pp , 4. (in Chinese. [8] T. Nogmi nd M. Nov, Soil-pile inercion in vericl virion, Erhque Engineering nd Srucurl Dynmics, vol. 4, pp , [9] M. Nov nd. Aoul-Ell, Impednce funcions of piles in lyered medi, Journl of he Engineering Mechnics Division, ASCE, vol. 14, pp , [1] M. Nov, T. Nogmi, nd. Aoul-Ell, Dynmic soil recions for plne srin cse, Journl of he Engineering Mechnics Division, ASCE, vol.14 (EM4, pp , [11] G. Milino nd R. K. N. D. Rjpse, Dynmic response of pile in muli-lyered soil o rnsien orsionl nd xil loding, Geoechnique, vol. 49, no. 1, pp , [1] H. Chngin, W. Kuihu, nd X. Knghe, Time domin nlysis of vericl dynmic response of pile considering he effec of pile-soil inercion, Chinese Journl of Compuionl Mechnics, vol. 1, no. 4, pp , 4. (in Chinese. [13] H. Chngin, W. Kuihu, nd X. Knghe, Time Domin Axil Response of Dynmiclly Loded Pile in Viscous Dmping Soil Lyer, Journl of Virion Engineering, vol. 17, no.1, pp. 7-77, 4. (in Chinese. Received: Sepemer 1, 1 Revised: Novemer 1, 1 Acceped: Jnury 3, 11 Wng e l. Licensee Benhm Open. This is n open ccess ricle licensed under he erms of he Creive Commons Ariuion Non-Commercil License (hp://creivecommons.org/licenses/ y-nc/3./ which permis unresriced, non-commercil use, disriuion nd reproducion in ny medium, provided he wor is properly cied.