Mechanism Analysis of Dynamic Compaction based on Large Deformation

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1 Th Opn Civil Engining Jounal,,, - Opn Accss Mchanism Analysis of Dynamic Compaction basd on Lag Dfomation Xi Nnggang *, Chn Yun, Y Y and Wang Lu Anhui Univsity of Tchnology, Maanshan, Anhui Povinc, China, Abstact: In th mchanism analysis of foundation consolidation by using dynamic compaction, many kinds of non-lina conditions xist. This pap adopts lag dfomation on th lationship btwn stain and displacmnt. Non-lina govning quation of soil, basd on finit lmnt mthod, is stablishd. Itativ calculation fom is aisd. Finally, non-lina numical analysis is don to a calculation xampl. W not only obtain th changing gulaity of soil displacmnt and stss duing th acting tim of dynamic compaction, but also achiv thi contou maps aft compaction. Kywods: Dynamic compaction, non-lina, lag dfomation.. INTRODUCTION As fo multipl factos and complx of th dynamic compaction(f with Fig. ), many scholas hav studid infocmnt mchanisms of compaction fom th ngining pactic, laboatoy tsts, numical analysis and oth filds. Whn th basic contol quations of dynamic poblms a stablishd, th a th following majo pincipl issus: ) Stss - stain modl; ) Stain - displacmnt modl; ) Th subjcts w considd only soil sklton, o considing using th coupld analysis of soil sklton and po wat (Dynamic consolidation modl is adoptd); ) Calculation mthods of shock loads o dynamic contact stss btwn gound sufac and th bottom hamm whn tamping. Th main diffnc xistnc in vaious studis of thoy and mthods of compaction poblms lis in th handling of som abov pincipl issus. In th lations of stss-stain, dynamic loading and unloading bilina constitutiv modl psntd by Qian Jiahuan [] was widly usd. In th staindisplacmnt lations, most sachs also us th "small stain" assumption. Jiang Png [] in poposd a "lag dfomation" assumption that is, considing gomtic nonlinaity. Dong-Soo Kim [] poposd that fo diffnt vibation souc, th gomtic modls to b built w also diffnt. In [], only th soil sklton was considd to do th compaction. In considing th coupling analysis of soil sklton and po wat, Yang Jun [] povidd th tim solutions on symmtic poblms of fluid dynamic coupling shaft and th impovd mthod of th poblm was givn by Kong Lingwi []. Using th-dimnsional consolidation thoy of poous mdia, Chn Ji [t al. calculatd and analyzd th stuctu of satuatd soil compaction pocss. In th numical simulation pocsss of dynamic consolidation of satuatd soft clay foundation, Ding Zhnzhou [] *Addss cospondnc to this autho at th School of Mchanical Engining, Anhui Univsity of Tchnology, Maanshan, China; Tl: +--(offic); xinnggang@yahoo.com.cn Fig. (). Dynamic compaction action. usd Boit consolidation thoy. H considd th stong nonlina coupling poblm of th soil and po wat and obtaind th dynamic changs of th pocss of soil consolidation. Song Xiuguang [] took fluid dynamic coupling and contact coupling btwn hamm and foundation soil into account and obtaind changs of soil displacmnt and contact dynamic stss und tim piod in th dynamic compaction. Gunaatn M [] usd th numical mthod to simulat dynamic consolidation accoding to th modifid classical Tzaghi static consolidation thoy. H took th contact dynamic stss as th puls stss which was bokn down into a sufficint numb of load stps so as to calculat th chaactistics of po pssu. Zhou XL [] usd Biot's consolidation thoy to study th tansint dynamic spons of foundation und th tiangula puls load and Laplac tansfom to solv th poblm. Thn h obtaind th tim histoy btwn displacmnt and po pssu and th lationships of load foms. Bsids, h [] also studid th dynamic spons of th satuatd soil und th concntatd load and got mo accuat displacmnt, stss and th numical solution of po pssu. In tms of th tansint load simulation gnatd by th dynamic compaction, it usually would b simplifid to a tiangl wav o half-sin basd on xpinc []. Th calculation of dynamic contact btwn th bottom of hamm and th fac of soil is th ky tchniqu poblm of dynamic compaction. In this fild, Scott RA and Pac RW [] put fowad a calculation foum, but it couldn t satisfy th initial condition. Kong LW and Yuan JX [] usd intgal tansfom and tansition -/ Bntham Opn

2 Th Opn Civil Engining Jounal,, Volum Nnggang t al. matix to study th dynamic contact stss. Th tchniqu also has bn studid analytically by Rosst JM []. Howv, th mthod abov, vn many oth ssays, w mly in spac fild, xplaind th distibuting fatu of dynamic contact stss on th bottom of hamm, thy adoptd a simpl lastic contact modl in tim fild duing th dynamic compaction, and didn t tak th possibl bhavio of hamm into account such as, bouncing contact and spaating. Jiang P and Li RQ [] considd this nonlina contact and suggstd to adopt impuls-dynamic contact modl to calculat th dynamic contact stss, but th contact condition in this modl is static fom.. DIFFERENTIAL GOVERNING EQUATION Fo dynamic compaction, lt activ aas of infocmnt foundation b abstactd to b which is boundd by bounday S, wh X is on of th points. Thn th basic quations of th lag dfomation poblm of soil body a as follows Dynamic balanc quation is [( ik + u i,k ) kj ], j + F i p&& u i = () Rlationship btwn stain and displacmnt is ij = (u i, j + u j,i + u k,i u k, j ) () Rlationship btwn stss and stain is ij = kk ij + G ij () Bounday conditions of displacmnt( S u ) u i = u i () Bounday condition of stss( S p ) ( ik + u i,k ) kj n j = p i () Wh:Each physical quantity is xpssd as a tnso componnt fom. That is, u i is a displacmnt componnt, ij a stss tnso componnt, ij a stain tnso componnt, F i a foc componnt of th volum of soil, th dnsity of soil body. Bsids u i and p i a givn bounday displacmnt and xtnal bods, and G Lam constants, ij a Clinton Sign and n i dnots diction cosin. Engy functional is constuctd as follows t = { [ &u i &u i A( )+ Fu ]d + p ij i i u ds}dt () i i t Hamilton s Pincipl Bounday conditions of displacmnt and lationship btwn stain and displacmnt a mt so that functional displacmnt u i of th xtm valu must satisfy th dynamic quilibium quation, stss bounday conditions and stss-stain lations und th givn conditions of S p displacmnt u i at tim t = t and t = t, that is th tu solution of th poblm. Lt th ngy functional in quation () b vaid into zo = (). GOVERNING EQUATIONS BASED ON FINITE ELEMENT With th finit lmnt mthod and th disct stuctu, th ngy functional fo body of any unit is t = { [ { &u} T [N] T [N]{&u} A( )]dv ij t +{u} T {F} + {u} T {P} }dt Wh, {u},{&u} dnot th displacmnt aay and vlocity aay of lmnt nod spctivly; [N] dnots th shap function matix; {F} is th quivalnt nodal load causd by body foc of th lmnt; {P} is th quivalnt nodal load causd by sufac foc. Lt quation () b vaid into = t t () {{u} T ([m {&& u} + {F} + {P} ) {} T {}d }dt () With {} = [B {u} ( [B is th stain matix fo th unit), quation () is tansfomd into = t t {u} T {[m {&& u} + {F} + {P} [B] T {}d }dt () Wh, [m = [N] T [N]d is th mass matix fo th unit. Lt quation () b vaid into zo and obtain dynamic quilibium quation of th unit body [m {&& u} + [B] T { }d = {F} + {P} () Accoding to th unit assmbly, th dynamic quation of ovall stuctu can b obtaind [M ]{&& u} + [B] T { }d = {F} + {P} (). METHOD FOR SOLVING THE NONLINEAR DYNAMIC EQUATIONS.. Dynamic Equation in Incmntal Fom Accoding to quation (), lations of staindisplacmnt blong to nonlina. So [B is th function of {u} and can b wittn as [B = [B ]+[B L () Wh: [B ] is th matix nty by lina dfomation analysis; [B L is th matix nty by nonlina dfomation,

3 Mchanism Analysis of Dynamic Compaction basd on Lag Dfomation Th Opn Civil Engining Jounal,, Volum which is lativ to th nod displacmnt {u} of th unit body. Equation () can b wittn as [{u} ] = [m {&& u} + [B] T { }d ({F} + {P} ) = () Lt {u} b divd on both sids in quation () and w can obtain d[{u} ] = d[b T { }d + [B T d{ }d () As {} = [D]({} { })+ { } () Wh: [D] is an lastic matix; { },{ } a th initial stain and initial stss spctivly. Hnc, d{} = [D]d{} = [D][B d{u} () In quation (), [B ] is ilativ to th nod displacmnt {u}. Thfo, d[b = d[b L () Typ quations () and () into quation () and w can gt d[{u} ] = d[b L ] T { }d +[k d{u} () Wh: T [k = [B [D][B d = [k () Typ quation ()into ()and w can obtain [k = [B ] T [D][B ]d () [k L = {[B ] T [D][B L +[B L ] T [D][B L +[B L ] T [D][B ]}d () Lt d[b L ] T { }d = [k d{u} () Wh: [k is th gomtic stiffnss matix. Typ quations fom () to ()into ()and w can gt d[{u} ] = {[k +[k }d{u} () Assum that th systm und xtnal loads, lmnt nod displacmnt at tim t is {u(t)}. Thn th fist-od fo () at {u(t)} + {u} is appoximatly [{u(t)} + {u} ] = [{u(t)} ] d +( {u} ) {u} ={u(t )} d{u} = Typ () and () into () and w can obtain [{u(t)} + {u} ] = [m {&& u(t)} }d ({F(t)} + {P(t)} ) + [B T { () +{[k +[k }{u} = () Accoding to () and w tak initial stain and initial stss as zo, thn w can gt [B] T {}d = ([k ){u(t)} () Typ () into (), and accoding to th unit assmbly and considing damp, w can obtain th ovall dynamic quilibium quation in th incmntal fom. [{u(t)} + {u}] = [M ]{&& u(t)} +[C]{ &u(t)} () +[ K ]{u(t)} ({F(t)} + {P(t)}) +[K]{u} = Wh: [K] = {[k +[k }, [ K ] = {[k }, [C] is th damping matix... Itativ Solution Accoding to th tim stp t, dynamic impact tim is dividd into sval tim piods. ) Fo any tim t, lt {u(t)} small, { &u(t)} small and {&& u(t)} small which a obtaind by solving th small dfomation dynamic quation of th systm b th initial itativ valus {u(t)}, { &u(t)} and {&& u(t)} of lag dfomation dynamic quation. Small dfomation dynamic quation of th systm is [M ]{&& u} +[C ]{ &u} +[K ]{u} = {F} + {P} () Wh: [K ] = [k ; [C ] = [M ]+ [K ]. ) Accoding to th nonlina lationships btwn th stain and displacmnt of lag dfomation poblm, w typ {u(t)} into () and gt [B L fo vy lmnt. Thn on th basis of (), () and (), [k, [k L, [k a obtaind fo vy lmnt and [ K ] and [K] a got though assmbly. ) Typ {u(t)}, { &u(t)}, {&& u(t)},[ K ] and [K] into () and gt {u} ) Using lina acclation mthod, w gt th valus of th nxt itation {u(t)}, { &u(t)} and {&& u(t)} at tim t. ) Rpat stps fom ) to ), th itativ fomat is

4 Th Opn Civil Engining Jounal,, Volum Nnggang t al. { &u(t)} n+ = {u(t)} n+ t {u(t t)} t {&u(t t)} {&& u(t t)} t {&& u(t)} n+ = {u(t)} n+ {u(t t)} t t { &u(t t)} t {&& u(t t)} Wh: t is th tim stp; {u(t t)}, { &u(t t)} and {&& u(t t)} a, spctivly, th displacmnt, vlocity and acclation at t t fo th nonlina poblm of ().. STRAIN MATRIX OF AXISYMMETRIC PROBLEM BASED ON THE GEOMETRIC NONLINEARITY Fo axisymmtic poblms, w us th cylindical coodinats (,, z). Lt z b th axis of symmty and all th stss, stain and displacmnt hav nothing to do with but th functions of and z. Th a only two displacmnt componnts at any point, that is, th adial displacmnt along to and axial displacmnt along th diction of z. Bcaus of th symmty, th cicumfntial displacmnt of th diction is zo. Tak a unit cll and lmnt nods a i, j, m,l, wh th displacmnt aay of lmnt nods is {} = { u i w i u j w j u m w m L} T () Unit displacmnt of any point can b xpssd as {} = u w = [N]{} () Wh: [N] = N i N j N m L. N i N j N m L Th nonlina lationship btwn stain and displacmnt is {} = z z = u u w z (u z + w ) u u + w u + u u u z z + w z u u z + w w w z w z () Lt { } = = u u w z (u z + w ) N i u + N j i u + N m j u +L m N i u i + N j u j + N m u m +L N i z w i + N j z w j + N m z w m +L [(N i z u i + N j z u j + N m z u m +L)+ ( N i w i + N j w j + N m N i N i = N i z N i z N i N j N j N j z N j z N j N m N m N m z w m +L)] L u i w i L u j w j N m L z u m N w m m L M = [L][N]{} () Wh: [L] = z. z Thfo, th stain matix of small displacmnt is [B ] = [L][N] () u u + w w u { L } = u u u z z + w w z z u u Lt z + w w z u u = u z u w u u u w z z z w u w w z w z

5 Mchanism Analysis of Dynamic Compaction basd on Lag Dfomation Th Opn Civil Engining Jounal,, Volum = [][G][N]{} () Wh: [G] = z u w u ; [] = u w z z u u w w z z z Thfo, th stain matix causd by th nonlina dfomation is [B L = [][G][N] (). ACTUAL EXAMPLE ANALYSIS Calculating Explanation Tak th highway ngining aound JiNan city fo xampl, th soil is loss with th dnsity of.kn/m, th mass of hamm is kn and its coss sctional aa is.m, th hight at which th hamm is doppd is m. Th loading modulus is kpa and th unloading modulus is kpa, th Poisson ation μ is., tim stp t = ms. Th vtical hight of dispsing aa of finit lmnt is H=.m and hoizontal adius is.m. Th bottom bounday condition of soil is u = w = ; th sid bounday condition is w = and contact foc of noncompaction aa of th sufac is zo. Bsids, initial conditions a u t= =, w t= =, &u t= =, &w t= =, u&& t= = and &&w t= =. Th fist attack on th compaction is calculatd, wh th maximum input contact stss is kpa and contact tim is ms.tim-intval cuvs a simplifid into th load typs, that is, isoscls tiangl load, half-sin load with damping and nomal distibution cuv of load (s Fig. (). Tiangl, sin and nomal is calld fo shot blow).. E+. E+. E+. E+. E+. E+. E+. E+. E+. E+. E+ Load( Pa) Fig. (). Tim-intval cuv of load. Tiangl Si n No mal t ( ms) Rsults and Analyss Calculation sults on No. nod (th contact point of Hamm cnt with th gound sufac) a compad and analyzd. Stss and tim-intval cuvs fom Fig. () to Fig. () flct mo consistntly that soil has shown a significant bound at aound ms in th compaction pocss. Rbound flctd in th displacmnt cuv of Fig. () coms lat about ms than that of th stss and timintval cuv, wh bounds of Tiangl, sin and nomal occu at ms, ms and ms spctivly. _ (Pa) Fig. (). Tim-intval cuv of fo No. nod. Fig. (). Tim-intval cuv of fo No. nod. _ z (Pa) σ (Pa). E+ -.E+ Tiangl t ( - s) -.E+ -.E+ -.E+ -.E+ -.E+ -.E+ -.E+ -.E+ Si n No mal Fig. (). Tim-intval cuv of z fo No. nod.

6 Th Opn Civil Engining Jounal,, Volum Nnggang t al. Tiangl Si n No mal uz(cm) Fig. (). Tim-intval cuv of u z fo No. nod max. (a) tiangl (b) sin (c) nomal Fig. (). Contou maps of fo foundation aft compaction (MPa). t( - s) max. max max max max (a) tiangl (b) sin (c) nomal Fig. (). Contou maps of fo foundation aft compaction (MPa).

7 Mchanism Analysis of Dynamic Compaction basd on Lag Dfomation Th Opn Civil Engining Jounal,, Volum Viwd fom th pak of th tim-intval cuv, th paks of sin and nomal a much clos, wh th diffnc is lss than %. Compad thm with th tiangula pak, th diffnc is mo than %. Compaing th dg of bound und ths th cass, w can s that th bound dg of Sin is lativly lag whil th bound dg of Tiangl is vy clos to that of nomal. Rbound valus of Tiangl, sin and nomal account fo.%,.% and.% of th pak spctivly. Accoding to th stss and tim-intval cuv and th displacmnt and tim-intval cuv, w can obtain that, in th compaction, half sin load with damping is mo pcis whn dg of bound, bound tim, vntually amg pit valu and th stss-displacmnt changing pocss a takn into account fo th foundation soil. Contou maps of, and z fo foundation aft compaction a shown in Figs. (, and ). Fom Figs. ( and ), gulaitis of distibution of and fo th th cass a consistnt, wh th valu of th nomal cicumstanc is th maximum. Fom Fig. (), w can s that distibution uls a basically consistnt, wh th sult of nomal distibution cuv of load is th maximum and that of sin load is th imum. Fom Fig. (), w can s that max max max (a) tiangl (b) sin (c) nomal Fig. (). Contou maps of z fo foundation aft compaction (MPa) max max max (a) Tiangl (b) sin (c) nomal Fig. (). Contou maps of u z fo foundation aft compaction (m).

8 Th Opn Civil Engining Jounal,, Volum Nnggang t al. distibution uls a basically consistnt, wh th sult of nomal distibution cuv of load is th maximum and that of tiangl load is th imum.. CONCLUSIONS In th analysis of dynamic compaction poblms, as th gomty non-linaity hav bn considd, th mchanics analysis is fin, and th modl cosponds to th actual fatu of dynamic compaction. Though th analysis and computation, w can gt th changing gulaity of th soil displacmnt and dynamic stss on sufac duing th acting tim of dynamic compaction. Th ffctiv way of analyzing and simulating dynamic compaction mchanism is aisd. Th last on should b lucidatd is that hypothsis is not considd in dynamic contact condition. ACKNOWLEDGMENTS This pap is suppotd by th Nw Cntuy Excllnt Talnts in Univsity (Gant No.), Stat Ky Laboatoy of Hydology-Wat Rsoucs and Hydaulic Engining (Gant No. ) and th Natual Scinc Foundation of Anhui Povinc of China (Gant No.M). REFERENCES [] J. H. Qian and F. S. Shuai, Application of BEM to dynamic consolidation, Sci. Sin., Vol., pp. -,. [] P. Jiang, R. Q. Li and D. F. Kong, Numical analysis of lag dfomation impact and collision poptis duing dynamic compaction, China J. Gotch. Eng., Vol., pp. -,. [] D. S. Kim and J. S. L, Popagation and attnuation chaactistics of vaious gound vibations, Soil Dyn. Eathquak Eng., Vol., pp. -,. [] Y. C. Li, J. Chn and S. H. Zhou, Numical simulation fo a foundation infocmnt pocdu using dynamic consolidation tchniqu, Rock. Soil. Mch., Vol., pp. -,. [] J. Yang and S. M. Wu, Tansint solutions of axisymmtic dynamic poblm fo inhomognous two phas mdia, Acta. Mch. Sin., Vol., pp. -,. [] L. W. Kong, An impovmnt of tansint solutions of axisymmtic dynamics poblm fo two phas mdia, Acta. Mch. Sin., Vol., pp. -,. [] J. Chn, Y. C. Li and S. H. Zhou, Numical simulation fo foundation infocmnt pocdu using dynamic consolidation tchniqu, J. Shanghai. Tidao. Uni., Vol., pp. -,. [] Z. Z. Ding and Y. R. Zhn, Numical simulation of satuatd soft clay gound impovmnt by DCM, China J. Und. Spa. Eng., Vol., pp. -,. [] X. G. Song, Y. Y. Li and J. Han, Multi-coupling analysis of dynamic consolidation mthod fo infocing foundations, Rock. Soil. Mch., Vol., pp. -,. [] M. Gunaatn, M. Rangansth, S. Thilakasii, tal, Study of po pssus inducd in laboatoy dynamic consolidation, Comput. Gotch., Vol., pp. -,. [] X. L. Zhou, J. H. Wang and J. F, Lu Tansint dynamic spons of poolastic mdium subjctd to impulsiv loading, Comput. Gotch., Vol., pp. -,. [] X. L. Zhou, J. H. Wang and J. F. Lu, Tansint foundation solution of satuatd soil to impulsiv concntatd loading, Soil Dyn. Eathquak Eng., Vol., pp. -,. [] J. L. Pan and A. R. Slby, Simulation of dynamic compaction of loos ganula soils, Adv. Eng. SoftWa., Vol., pp. -,. [] R. A. Scott and R. W. Pac, Soil compaction by impact, Go. Tchniqu, Vol., pp. -,. [] L. W. Kong and J. X. Yuan, Study on sufac contact stss distibution poptis fo multi-layd foundation duing dynamic consolidation, Acta Mch. Sin., Vol., pp. -,. [] J. M. Rosst, Impact of wight falling onto th gound, J. Gotch. Eng., Vol., pp. -,. [] P. Jiang and R. Q. Li, Numical analysis of lag dfomation impact and collision poptis duing dynamic compaction, China J. Gotch. Eng., Vol., pp. -,. Rcivd: August, Rvisd: Novmb, Accptd: Dcmb, Nnggang t al.; Licns Bntham Opn. This is an opn accss aticl licnsd und th tms of th Cativ Commons Attibution Non-Commcial Licns ( by-nc/./) which pmits unstictd, non-commcial us, distibution and poduction in any mdium, povidd th wok is poply citd.

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