Parametric Investigation of Foundation on Layered Soil under Vertical Vibration

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1 IOSR Jornal of Engnrng IOSRJEN ISSN : 5-3 ISSN p: Vol. 4 Iss 7 Jly. 4 V3 PP 3- Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton S. N. Swar P. K. Prahan ** B. P. Mshra *** Assstant profssor Dpartmnt of Cvl Engnrng H-th Insttt of Thnology Bhbanswar INDIA **Profssor & Ha Dpartmnt of Cvl Engnrng V. S. S. Unvrsty of Thnology Brla Sambalpr INDIA-7688 ***Rsarh Sholar Dpartmnt of Cvl Engnrng Natonal Insttt of Thnology Rorkla INDIA Abstrat: - Th papr prsnts th paramtr nvstgaton of fonaton on layr sol nrlan by a rg bas sbt to vrtal vbraton ar fon ot sng on-mnsonal wav propagaton n onbas on th strngth of matral approah. Th stffnss an ampng o-ffnt for a rg masslss rlar fonaton rstng on layr half-spa an homognos half-spa nr vrtal vbraton ar valat sng varos paramtr sh as pth of th layr matral ampng rato Posson s rato. Th stat stffnss prt by th mol for ffrnt pth layr s valat sng thr val of Posson s rato. Th rsonant frqnyamplt an frqny-magnfaton ar also st varyng th nflnng paramtr sh as mass rato Posson s rato. Kywor- Con mol Dynam mpan Crlar fonaton on-mnsonal wav propagaton Fonaton vbraton I. INTRODUCTION Fonaton may b sbt to thr stat loa or ombnaton of stat an ynam loas; th lattr la to moton n th sol an mtal ynam ntraton of th fonaton an th sol. Th sgn of mahn fonaton nvolvs a systmat applaton of th prnpls of sol ngnrng sol ynam an thory of vbraton. Th sor of ynam for ar nmros so to trmnaton of rsonant frqny an rsonant amplt of fonaton has bn sbt to onsrabl ntrst n th rnt yar n rlaton to th sgn of mahn fonaton as wll as th ssm sgn of mportant massv strtr sh as nlar powr plant. Th sty of th ynam rspons of fonatons rstng on sol sbt to varos mo of vbraton s an mportant aspt n th sgn of mahn fonatons an ynam sol-strtr ntraton problm. Th solton of th ynam Bossnsq problm of Lamb94 form th bass for th sty of osllaton of footngs rstng on a half-spa Rssnr936 ; Sng953 ; Rhart97 t al.. Rssnr936 vlop th frst analytal solton for a vrtally loa ylnral sk on last half-spa assmng nform strss strbton nr th footng. Latr xtnng Rssnr s936solton many nvstgators Byroft956 Lysmr 97an Rhar97Wolf994 Lo an Mta987 Prahan48 to nam a fw st ffrnt mos of vbratons wth ffrnt ontat strss strbtons. Gatas98399 prsnt smpl formlas for ynam mpan o-ffnt for both srfa an mb fonatons for varos mos of vbraton. Th on mol was orgnally vlop by Ehlrs 94 to rprsnt a srfa s nr translatonal motons an latr for rotatonal moton Mk an Vltsos 974; Vltsos an Nar 974. Mk an Wolf prsnt a smplf mthoology to valat th ynam rspons of a bas mat on th srfa of a homognos half-spa. Th on mol onpt was xtn to a layr on to ompt th ynam rspons of a footng or a bas mat on a sol layr rstng on a rg rok. Mk an wolf 994 prform ynam analyss of mb footng by alng th sol as a translat on nsta of last half-spa. Wolf an Mk 994 hav fon ot th ynam stffnss offnts of fonatons rstng on or mb n a horontally layr sol sng on frstms. Also Jaya an Prasa st th ynam stffnss of mb fonatons n layr sol sng th sam on frstms. Th maor rawbak of on frstms mtho as rport by Wolf an Mk 994s that th ampng offnt an bom ngatv at lowr frqny whh s physally mpossbl. Prahan t al34hav ompt ynam mpan of rlar fonaton rstng on layr sol sng wav propagaton n ons whh ovroms th rawbak of th abov on frstm mtho. Intrnatonal organaton of Sntf Rsarh 3 P a g

2 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton. Thrfor a nmbr of smplf approxmat mthos hav bn vlop along wth th xat soltons. Con mol s on of sh approxmat analytal mthos whr n last half-spa s trnat nto a sm-nfnt on an th prnpl of on-mnsonal wav propagaton throgh ths on Bam wth varyng ross-ston s onsr. In ths papr sts th paramtr nvstgaton of fonaton rstng on layr nrlan by rg bas nr vrtal vbraton s fon ot sng wav propagaton n on varyng wly th paramtr lk mass rato Posson s rato pth of th layr matral ampng rato. II. MATHMATICAL FORMULATION To sty th ynam rspons of fonaton rstng on th srfa of a sol layr nrlan by rg bas a rg mass lss fonaton of ras r s sbt to vrtal vbraton shown Fg.a.Th l pth of th layr has th shar mols G Posson s rato ν mass nsty ρ hystrt ampng ξ.th ntraton for P an th orrsponng splamnt U ar assm to b harmon. Th ynam mpan of th masslss fonaton s s xprss by: P K a K [ k a a a] K a Dynam mpan k a = sprng offnt a = ampng offnt a r / s 4Gr =mnsonlss frqny s G shar wav vloty of th sol =Stat stffnss offnt of s on homognos half spa wth matral proprts of th layr. Th ffts of hystrt matral ampng s solat sng an altrnat xprsson to Eq. for ynam mpan P K a K [ k a a a] Usng th qatons of ynam qlbrm th ynam splamnt amplt of th fonaton wth mass m an sbt to a vrtal harmon for Q s xprss as Q K [ k a a a Ba ] 3 Whr = ynam splamnt amplt nr th fonaton rstng on th homognos sol half-spa. Gr Q for amplt an m B b wth b th mass rato. 3 K r Dynam splamnt amplt gvn n Eq. 3 an b xprss n th non-mnsonal form as gvn blow Gr Gr K[ k a a a Ba 4 Q K Magnfaton fator.. th rato of ynam splamnt to th stat splamnt s xprss by Intrnatonal organaton of Sntf Rsarh 4 P a g

3 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton Q / K M [ k a a a Ba ] 5 III. CONE MODEL FOR VERTICAL TRANSLATION Th thory of wav propagaton n a sm-nfnt trnat on s prsnt bas on strngth of matral approah. Fg a shows wav propagaton n ons bnath th sk of ras r rstng on a layr nrlan by a rg bas nr vrtal harmon xtaton P. Lt th splamnt of th trnat smnfnt on b not as wth th val nr th sk fg b molng a sk wth sam loa P on a homognos half spa wth th matral proprts of th layr. Ths splamnt s s to gnrat th splamnt of th layr wth ts val at srfa.ths an also b all as th gnratng fnton. Whn fonaton sbt vrtal vbraton wav s gnrat blow bas of th fonaton an propagatng own war to th sol n th shap of on. Th frst wav gnrat blow th bas fonaton an propagatng ownwar n a on wth apx s all as nnt wav an ts on wll b th sam as that of th half-spa as th wav gnrat bnath th sk os not know f at a spf pth a rg ntrfa s nontr or not. Ths th aspt rato fn by th rato of th hght of on from ts apx to th ras s ma qal for on of th half-spa an frst on of th layr. Sn th nnt wav an sbsqnt rflt wavs propagat n th sam mm n layr th aspt rato of th orrsponng ons wll b sam. From th gomtry knowng th hght of th frst on th hghts of othr ons orrsponng to sbsqnt pwar an ownwar rflt wavs ar fon as shown n Fg. a. Intrnatonal organaton of Sntf Rsarh 5 P a g

4 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton P r b on mol for half a EQUATION OF MOTION ST CONE Th translatonal trnat sm-nfnt on wth th apx hght an ras r s shown for axal storton n Fg. 3 whh s s to mol th vrtal gr of from. Th ara A at pth qals A / A wth A r whr s masr from srfa of sk. Wth notng th approprat wav vloty of omprsson-xtnson wavs latatonal wavs an ρ th mass nsty ρ s qal to orrsponng last mols onstran mols. Also rprsnts th axal splamnt an N th axal for. Raal ffts ar srgar. th qlbrm qaton of an nfnt lmnt strp Fg.3 takng th nrtal loas nto aont N N N A 6 Sbstttng th for-splamnt rlatonshp n Eq 6 N A 7 Apx A P Z N r N+ N Z A= A Z +Z²/Z ² Fg 3 Wav propagaton n sm-nfnt trnat on nr vrtal harmon xtaton Th qaton of moton n tm oman of translatonal on Whh may b wrttn as on-mnsonal wav qaton n 9 Th splamnt amplt of th nnt wav propagatng n a on wth apx n tm oman gvn blow: 8 Intrnatonal organaton of Sntf Rsarh 6 P a g

5 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton Intrnatonal organaton of Sntf Rsarh 7 P a g t t Convrt Eq. 3n frqny oman an b wrttn as: Th splamnt of th nnt wav at rg bas qal = Th splamnt of th frst rflt pwar wav propagatng n a on wth apx fg a xprss as: 3 Th splamnt of th ownwar wav propagatng n a on wth apx 3 fg a xprss as: 4 Ths aftr th mpngmnt at rg bas th splamnts of pwar an ownwar wavs propagatng n ons wth Fg. a 5 6 Th rsltng splamnt n th layr s obtan by sprposng all th own an p wavs an s xprss n th followng form * 7 8 E F 9 Wth F E An for > E F /

6 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton E F an b all as ho onstant th nvrs of sm of whh gvs th stat stffnss of th layr normal by th stat stffnss of th homognos half-spa wth matral proprts of th layr. b DYNAMIC IMPEDANCE Enforng bonary onton to Eq. yls Also pwar for qal ownwar for t f t P N A 3 Dffrntatng Eq. wth rspt to an sbstttng ts val at n Eq. 6 w gt A P t t A t 4 Th Eqs. 7 ar val for omprssbl sol.. / 3. For nomprssbl sol th onpt of ntrong trapp mass s nfor. Whr K M C 3 wth trapp mass offnt µ; th vals of whh rommn by Wolf ar gvn n M r M P 5 Tabl. Th trapp mass s ntro n orr to math th stffnss offnt of th on mol wth rgoros soltons n as of nomprssbl sol. /3 /. Aftr smplfaton Eq. 8 rs to th form. Tabl Th paramtrs of on mol nr vrtal vbraton Con Paramtrs Paramtr Exprssons nr Vrtal Vbraton Aspt Rato r 4 s Stat stffnss offnt K Normal sprng offnt k a r s a r Normal ampng offnt a s r Dmnsonlss frqny a Approprat wav vloty p r s for / 3 s for/3 / whr p s Intrnatonal organaton of Sntf Rsarh 8 P a g

7 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton P s s a K a a r r Usng Eq. n Eq. 7 th ntraton for splamnt rlatonshp for th layr-rg bas systm rs to 6 s s a a p r r K a K F E Not: If /r s thn th sol bhav lk homognos sol In th xprsson of th ynam mpan K a gvn by Eq. 3 th smmaton of srs ovr s work ot p to a fnt trm as th splamnt amplt of th wavs vansh aftr a fnt nmbr of F mpngmnt. Nmrally s trmnat at a val sh that E E. IV. RESULTS AND DISCUSSIONS A paramtr sty s ont wly varyng th nflnng paramtr sh as mass rato pth of th top layr /r matral ampng rato an Posson s rato. Th rslt ar prsnt n th form of mnsonlss graph whh may prov to b sfl n nrstanng th rspons of fonaton rstng on layr an homognos sol sbt vrtal vbraton.. STATIC STIFFNESS In ths as th stat stffnss of rlar fonaton s st varyng th pth of th layr.. /r rato from.5 to. Th vals of Posson s rato onsr ar..3 an.49. Th normal stat stffnss K L /Gr ar prsnt n Fg. 4. It s obsrv from ths fgr that th Posson s rato affts th stat stffnss of fonaton rstng on a layr ovr rg bas nr vrtal. Also mor th val of Posson s rato mor s th stat stffnss for sa grs of from. Th stat stffnss of th fonaton s fon to b mor whn th pth of th layr s lss Fg. 4. Wth th nras n th pth of th layr th stffnss rass an t approahs to half-spa val at a spf pth pnng on th gr of from. F 7 5 Normals vrtal stat stffnss K V /Gr 5 5 =. =.3 = Dmnsonlss pth /r Fg. 4 Normal stat stffnss of rlar fonaton rstng on a layr ovr rg bas wth varaton of /r for varos vals of. Intrnatonal organaton of Sntf Rsarh 9 P a g

8 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton. DYNAMIC IMPEDANCE Rslts for th ynam mpan fntons of a rg rlar sk on th srfa of a sol layr of fnt pth ovr rg bas ar prsnt n Fgs.5 an 6. Fg.5 shows th fft of /r rato on th ynam stffnss offnt k a an ampng offnt a for a sngl val of hystrt matral ampng rato =.5; an Fg. 6 shows th snstvty of k a an a to th varaton of for /r =. Th varaton of stffnss an ampng offnts wth frqny shows a strong pnnt on /r rato Fg. 5. k a an a ar not smooth fntons as on a homognos half-spa bt xhbt nlatons paks an vallys assoat wth th natral frqns of th sol layr. In othr wors th obsrv fltatons ar th otom of rsonan phnomna.. wavs manatng from th osllatng fonaton rflt at th sol layr rg bas ntrfa an rtrn bak to th sor at th srfa. As a rslt th amplt of fonaton moton may sgnfantly nras at spf frqns of vbraton whh as shown sbsqntly ar los to th natral frqns of th post. Wth th nras n /r rato th nlatons bom lss pronon an t approahs th half-spa rv at som spf pth pnng on th mo of vbraton. Th varaton of stffnss an ampng offnts wth frqny for ffrnt hystrt ampng ratos rangng btwn an % ar prsnt n Fg. 6. Smlar typs of nlatons ar obsrv for both stffnss an ampng offnts for varos vals. In gnral k a s not afft by th prsn of matral ampng p to a rtan val of a th natral frqny of th layr pnng on th mo of vbraton byon whh t rass wth nras n. Smlarly obsrvaton of ampng offnts for varos mos of vbraton shows that th fft of s promnant n th lowr frqny an t rass wth nras n frqny an boms nglgbl at hghr frqny. Bt th ampng offnt rvs wth = prly last shows ro ampng p to rtan frqny whh s fon to b vry los to th natral frqny of th layr. 3 =.5 =.5..5 =.5 =.5 /r = 4 6 k V - - /r = a V a Fg. 5 Varaton of mpan fntons wth pth of th layr for a rg rlar fonaton rstng on a layr ovr brok k V =.5 /r = = % 5% % % V.5..5 =.5 /r = = % 5% % % a Fg. 6 Varaton of mpan fntons wth varaton n matral ampng rato for a rg rlar fonaton rstng on a layr ovr a Intrnatonal organaton of Sntf Rsarh P a g

9 Magnfaton fatr Magnfton fatr Dmnsonlss amplt Dmnsonlss amplt Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton 3. FREQUENCY-AMPLITUDE RESPONSE Th frqny vrss amplt rspons rvs for homognos sol ar prsnt n Fgs.7 an 8.Fg.7 prsnts a plot of th rspons of th fonaton for fv ffrnt mass ratos b an =.5. An nras n th amplt an ras n rsonant frqny s obsrv wth nras n mass rato. For sx ffrnt vals of Posson s rato an mass rato b =5 th fonaton rspons s obtan sng on mol an prsnt n Fg.8 It s obsrv that th amplt of vbraton rass an rsonant frqny nrass wth nras n Posson s rato n=. n=. n=. n=.3 n=.4 n= b=5 b= b= b=4 b= Dmnsonlss frqny a Fg.7 frqny-amplt rspons rvs for ffrnt vals of Posson's rato.5.5 Dmnsonlss frqny a 4. FREQUENCY MAGNIFICATION RESPONSE Th frqny vrss magnfaton rspons rvs for homognos sol ar prsnt n Fgs.9 an.fg.9 prsnts a plot of th rspons of th fonaton for fv ffrnt mass ratos b an =.5. An nras n th magnfaton fator an ras n rsonant frqny s obsrv wth nras n mass rato. For sx ffrnt vals of Posson s rato an mass rato b =5 th fonaton rspons s obtan sng on mol an prsnt n Fg.. It s obsrv that th magnfaton fator of vbraton rass p to ν=.3 thn agan nras for hghr val an rsonant frqny nras wth nras n Posson s rato.. Fg.8 frqny-amplt rspons rvs for ffrnt vals of mass rato.4. n=. n=. n=. n=.3 n=.4 n= b=5 b= b= b=4 bo= Dmnsonlss frqny a Fg.9 frqny-magnfaton rspons rvs for ffrnt vals of Posson's rato mnsonlss frqny a Fg. frqny-magnfaton rspons rvs for ffrnt vals of mass rato Intrnatonal organaton of Sntf Rsarh P a g

10 Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton V. CONCLUSION In ontrast to rgoros mthos whh arss th vry omplat wav pattrn onsstng of boy wavs an gnral srfa wavs workng n wav nmbr oman th propos mtho bas on wav propagaton n ons onsrs only on typ of boy wav pnng on th mo of vbraton.. latatonal wav for th vrtal gr of from. Th stonal proprty of th ons nrass n th rton of wav propagaton ownwars as wll as pwars. Bas on th paramtr sts th followng onlsons an b rawn. a. Mor th val of Posson s rato mor s th stat stffnss. b. Wth nras n th pth of th layr th stat stffnss rass.. Wth nras n Posson s rato th rsonant frqny rass bt ynam stffnss o-ffnt rmans nhang for homognos sol.. Th rsonant amplt rass an rsonant frqny nrass wth nras n Posson s rato.. Wth nras n mass rato th rsonant frqny rass an rsonant amplt nrass. f. Wth nras n th mass rato magnfaton fator nrass an rsonant frqny ras. Rslt of paramtr sty prsnt n th form of mnsonlss graph prov a lar nrstanng of th vrtal ynam rspons of th fonaton rstng on sol layr nrlan by rg bas. REFERENCES [] Chn S. an Sh J.6. Smplf Mol for vrtal vbratons of srfa fonatons. Jornal of Gothnal an Gonvronmntal Engnrng Vol.3 No [] Gatas G Analyss of mahn fonaton vbratons: stat of th art. J. Sol Dynams an Earthqak Engrg. -4. [3] Gatas G. 99. Formla an harts for mpans of srfa an mb fonatons. J. Goth. Engrg. ASCE [4] Lang V.C.974 Dynam rspons of strtrs n layr sols n R74-. Dpartmnt of Cvl Engnrng Massahstts Insttt of Thnology Cambrg MA. [5] Lamb H. 94. On th propagaton of trmors ovr th srfa of an last sol. Phlosophal Transatons of th Royal Soty of Lonon A3-4. [6] Lo J. E. an Mta A Rspons of a rlar fonaton on a nform half-spa to last wavs Earthqak Engng an Strt Dyn [7] Lysmr J Wass G.97 Shar wavs n plan nfnt strtrs. Jornal of Engnrng Mhans ASCE ;98:85 5. [8] Lysmr J975 t al. Effnt fnt lmnt analyss of ssm sol strtr ntraton n Rport: EERC Earthqak Engnrng Rsarh Cn- tr Unvrsty of Calforna Brkly C.A. [9] Lysmr J Khlmyr R.L.969 Fnt ynam mol for nfnt ma. Jornal of Engnrng Mhans Dvson ASCE ;954: [] Mk J.W Wolf J.P.99 Con mols for sol layr on rg rok. J Goth Engng Dv ASCE ;85: [] Mk J.W Wolf J.P.994 Con mols for an mb fonaton. J Goth Engng Dv ASCE ;:6 8. [] Mk J.W. an Wolf J.P. 99. Con mols for homognos sol. J. Goth. Engrg. Dv. ASCE [3] Prahan P. K. Baya D. K. an Ghosh D. P. 4. Dynam rspons of fonatons rstng on layr sol by on mol Jornal of Sol Dynams an Earthqak Engnrng Vol [4] Prahan p. k. Manal A. Baya D. K. Ghosh D. P. 8 Dynam Rspons of Mahn Fonaton On Layr Sol: Con Mol Vrss Exrmntal Jornal of Goth Go Engnrng Vol [5] Qnlan P.M953 Th Elast Thory of Sol Dynam ASTM Spal Thnal pblaton No 56 Symposon on Sol Dynam3-34. [6] Rssnr E 936 Statonary an axally symmtral vbratons of a Homognos Elast Half-Spa Cas by a Vbratng MassIng Arhv Ban VII [7] Rssnr. E an Sago H.F 944 For torsonal Osllaton of an Elast Half Spa J of APPL.Phys Vol [8] Rhart F. E. Jr. Hall J. R. Jr. an Woos R. D. 97. Vbratons of sols an fonatons. Prnt- Hall In. Englwoo Clffs Nw Jrsy. [9] Sng.T.Y 953 Vbraton n Sm-nfnt Sol to Pro Srfa Loang ASTM Spal Thnal Pblaton no 58 Symposon on Sol Dynam Jly [] Warbrton GB. For vbraton of a boy on an last stratm. J Appl Mh Trans ASME 957;55 8 Intrnatonal organaton of Sntf Rsarh P a g