Winkler modulus for axially loaded piles

Transcription

1 Mylonai, G. (2). GeÂotechnique 5, No. 5, 455±46 Winler modulu for axially loaded pile G. MYLONAKIS The problem of elatic oil±pile interaction and it modelling uing the concept of a Winler upport are reviited. It i hown that dividing the vertical oil hear traction and the correponding ettlement along the pile generate depthdependent Winler pring that accurately decribe the pile±oil interaction, contrary to the widepread belief that the Winler repreentation i alway approximate. A impli- ed theoretical model i then derived for analying the repone of an end-bearing cylindrical pile in a homogeneou oil tratum. Explicit olution are obtained for: pile ettlement; depth-dependent Winler modulu; (c) average (depth-independent) Winler modulu to match pile head ettlement. Both in nitely long pile and pile of nite length are examined. The approximate analytical olution compare favourably with nite-element and boundary-element olution. A imple regreion formula for the average Winler modulu i developed. KEYWORDS: dynamic; elaticity; pile; ettlement; tiffne; oil/ tructure interaction. Nou reprenon le probleáme de l'interaction eâlatique entre ol et pile et a repreâentation en maquette par le concept du upport de Winler. Nou montron qu'en diviant de traction de ciaillement de ol verticale et le taement correpondant le long de la pile, on engendre de reort de Winler qui deâpendent de la profondeur, reort qui deâcrivent avec exactitude l'interaction pile-ol, contrairement aá l'opinion qui veut que la repreâentation de Winler oit toujour approximative. Nou deârivon enuite une maquette theâorique impli eâe pour analyer la reâaction d'une pile cylindrique porteue en bout dan une couche de ol homogeáne. Nou obtenon de olution explicite pour : le taement de la pile ; le module de Winler deâpendant de la profondeur ; le module moyen de Winler (ne deâpendant pa de la profondeur) pour preâdire le taement de teãte de pile. Nou eâtudion aá la foi le pile de longueur in nie et le pile de longueur nie. La olution analytique approximative montre une bonne correâlation avec le olution aá eâleâment ni et le meâthode marginale. Nou deâveloppon une imple formule de reâgreion pour le module de Winler moyen. INTRODUCTION A veratile way of modelling oil±pile interaction i through a erie of independent Winler pring ditributed along the pile haft. Although approximate, Winler model are widely accepted in the analyi of both axially and laterally loaded pile ubjected to tatic or dynamic load (Terzaghi, 955; Coyle & Reee, 966; Nova, 974; Poulo & Davi, 98; Randolph & Wroth, 978; Scott, 98; Fleming et al., 992; Gazeta et al., 992). Their popularity tem primarily from their ability to yield prediction that are in atifactory agreement with more rigorou olution incorporate variation of oil propertie with load amplitude (non-linearity) and depth (inhomogeneity) (c) be extended to dynamic load by adding pertinent ditributed damper to the pring bed (d) incorporate group effect through pertinent pile-to-pile interaction model (e) require maller computational effort than rigorou niteelement or boundary-element formulation. The ey problem in the implementation of Winler model lie in the aement of the modulu of the Winler pring. In the cae of axially loaded pile, the pring are de ned a (z) f (z) () W(z) where f (z) denote the vertical oil reaction per unit pile length and W(z) the correponding pile ettlement, at depth (z). Following early application of the model to ettlement analyi of urface footing, (z) i often referred to a the modulu of Manucript received 6 November 2; revied manucript accepted 6 March 2. Dicuion on thi paper cloe 2 November 2, for further detail ee the inide bac cover. Civil Engineering Department, City Univerity of New Yor. { Note that differ from the o-called coef cient of ubgrade reaction (denoted by ), which i de ned a preure over ettlement, and which thereby ha unit of force per length cubed. { The effect of radial diplacement on pile repone i uually minor a, for intance, hown by Matte (969) and Pa & Ji (993). 455 ubgrade reaction and i meaured in unit of force per length quared. y It i well nown that (z) i not a property of the oil alone, but depend on the characteritic of both pile and oil, and varie with depth even in a homogeneou layer. Under the aumption of negligible Poion effect in the pile, { introducing the function (z) reduce the continuum problem to the one-dimenional equation (Scott, 98) d 2 W(z) dz 2 (z)w(z) ˆ (2) which i amenable to analytical treatment, often leading to cloed-form olution (Nova, 974; Scott, 98; Guo, 2). Note that uing the exact ratio (z) ˆ f (z)=w(z) in equation (2) reproduce the pile repone accurately, contrary to the widepread view that the Winler repreentation i alway approximate. Current method for determining (z) can be clai ed into three main group: experimental method calibration with rigorou numerical olution (c) impli ed theoretical model. In the experimental method (group a), (z) i obtained directly from equation () by meauring the longitudinal train along an axially loaded pile and computing analytically the correponding ditribution of vertical oil reaction and pile ettlement (Coyle & Reee, 966). With the method of group b, average value along the pile can be determined by matching a ey repone parameter (e.g. pile head ettlement) with reult from Winler model, a for intance done by Thoma (98) and Sanchez-Salinero (982) uing nite-element formulation. Group c conit of approximate analytical method, notably the plane-train model (Nova, 974; Randolph & Wroth, 978), which introduce impli cation to derive imple theoretical etimate of. Notwithtanding the igni cance of the above method in engineering reearch and practice, they all can be criticied for certain drawbac. For intance, experimentally determined value (expreed through the well-nown `p± y' or `t±z' curve) have been developed primarily for inelatic condition, and do not properly account for the low-train tiffne of the oil (Reee & Wang, 996). On the other hand, calibration with

2 456 MYLONAKIS rigorou numerical olution in group b may encounter numerical dif cultie in certain parameter range, a, for example, in the cae of long compreible pile (El-Sharnouby & Nova, 99). Alo, thee approache are often limited by the analytical and computational complexitie aociated with the underlying numerical procedure, which can mae them unappealing to geotechnical engineer. Finally, the plane-train theorie of group c involve empirical parameter that need to be calibrated with other method (Randolph & Wroth, 978), and do not account for important factor uch a the continuity of the medium in the vertical direction and the tiffne mimatch between pile and oil. With reference to the method in group c, it eem that a imple rational model capable of providing improved etimate of (z) to be ued in engineering application would be deirable. In the framewor of linear elaticity, an approximate analytical olution i preented in thi paper for an axially loaded pile in a homogeneou oil tratum. While maintaining analytical implicity, the propoed olution ha ditinct advantage over the exiting model in group c. Speci cally (c) it account for the continuity of the medium in both the horizontal and vertical direction it account for pile±oil tiffne ratio, pile length to diameter ratio, and compreibility of the oil material it doe not involve empirical contant. Apart from it intrinic theoretical interet, the propoed approximate olution may be ued to provide a more rational bai for aeing and improving other related rz z r ˆ (3) where ó z i the vertical normal tre and ô rz i the aociated hear tre. Fundamental to the approximate analyi preented i the aumption that the normal tre, ó z, and hear tree, ô rz,in the vertically loaded medium are controlled excluively by the vertical diplacement, u z ; the in uence of radial diplacement, u r, on thee two tree i conidered to be negligibly mall. Baed on thi phyically motivated impli cation, the tre± diplacement relation for ó z and ô rz are repectively ó z ' z ô rz ' G where M i a pertinent contant to be dicued later on. Equation (4) and (5) were apparently rt employed by Nogami & Nova (976) for analying the correponding dynamic problem. In that wor, however, the radial diplacement of the medium wa aumed to be zero. In the preent tudy the aumption would be le retrictive: u r ha negligible in uence on ó z and ô rz, but i not zero. The importance of thi modi cation i dicued later on. From equation (3), (4) and (5), the equation of vertical equilibrium of the oil medium i u 2 r ˆ (6a) PROBLEM DEFINITION AND MODEL DEVELOPMENT The ytem conidered in thi tudy i depicted in Fig. : a olid cylindrical pile embedded in a homogeneou oil layer over a rigid bae, ubjected to an axial head load, P. Both oil and pile are aumed to be homogeneou, iotropic and linearly elatic. The pile i decribed by it length, L, diameter, d, and Young' modulu, E p, and the oil by it Young' modulu, E, and Poion' ratio, í. Stree and diplacement are aumed to be uniformly ditributed within the pile cro-ection. Perfect contact (i.e. no gap or lippage) i conidered at the pile±oil interface. With reference to the cylindrical coordinate ytem of Fig., the vertical equilibrium of the oil medium in the axiymmetric tate of deformation i written a L d P z d θ Fig.. Sytem conidered r σ θ σ z τ rz σ r u θ = u z u t where ç i a dimenionle parameter given by ç 2 ˆ M (7) G Note that if the variation with depth of the vertical normal tre ó z i neglected z =@z ˆ ), equation (6) impli e z which i the governing equation of the plane-train model. The olution to thi equation i u z ˆ c ln r c 2 (8) which clearly diverge with increaing radial ditance from the pile. To overcome the problem, Randolph & Wroth (978) and Baguelin & Fran (979) conider an empirically determined `magical radiu' around the pile beyond which oil diplacement i aumed to be zero. A will be hown below, the olution to equation (6a) i free of thi problem. Introducing eparation of variable and accounting for the boundary condition of zero normal traction at the oil urface and bounded diplacement at large radial ditance from the origin, equation (6a) yield the olution u z (r, z) ˆ BK (áçr) co áz (9) where K ( ) denote the modi ed Beel function of zero order and rt ind, and á i a poitive variable. B i a contant to be determined from the boundary condition. Becaue of the approximate nature of the analyi employed, the equilibrium of force in the horizontal direction i not ati ed in thi approach, nor i the boundary condition of vanihing hear tree at the oil urface. Neverthele, a demontrated in tudie of everal related probem (Tajimi, 969; Nogami & Nova, 976; Veleto & Younan, 994), thee violation have typically only a minor in uence on the olution. Thi will be demontrated further in thi paper through comparion with reult from pertinent numerical tudie.

3 In nitely long pile For an in nitely long pile, the diplacement and hear tree in the medium are obtained by integrating equation (5) and (9) over the poitive variable á: u z (r, z) ô rz (r, z) ˆ G ç BK (áçr) co áz dá () ábk (áçr) co áz dá () With reference to the pile, the differential equation of vertical equilibrium 2 2 ˆ ð dô rz (d=2, z) F(z) (2) where F(z) repreent body force ditributed along the pile axi. For the problem at hand, F(z) i determined by reolving the force at the pile head into equivalent body force through the Coine tranformation 2P F(z) ˆ co áz dá (3) ð From equation ()±(3), and conidering perfect bonding at the pile±oil interface [i.e. W(z) ˆ u z (d=2, z)], an explicit olution i obtained for the pile ettlement W: W(z) ˆ 2P ð á 2 K (çá d=2) co áz K (çá d=2) ð dçg dá á K (çá d=2) (4) End-bearing pile For a pile of nite length, one hould conider the condition of vanihing oil diplacement at the bae of the oil layer. Impoing thi requirement on equation (9) yield á ˆ á m ˆ ð (2m ), m ˆ,... (5) 2L which correpond to the olution of the eigenvalue problem co(ál) ˆ. In addition, in the ame pirit a with the in nitely long pile, the pile-head force, P, can be expanded in Coine erie a F(z) ˆ X mˆ 2P L co á mz (6) The olution to equation (2) i obtained by replacing the integral in equation () and () with correponding in nite um involving á m : W(z) ˆ 2P L X mˆ á 2 m K (çá m d=2) co á m z K (çá m d=2) ð d çg K (çá m d=2) á m (7) where á m i given by equation (5). The above equation can be obtained directly from equation (4) by replacing the integral with an in nite um, and the factor ð outide the integral with L, to account for the difference in the forcing function F(z). Determination of coef cient ç Mention ha been already made of the Nogami & Nova (976) dynamic olution baed on the aumption of vanihing radial diplacement, u r. The tatic part of that olution can be deduced from equation (7) by aigning factor ç the value ç 2 2( í) ˆ (8) 2í which expree the ratio of the contrained modulu to the WINKLER MODULUS FOR AXIALLY LOADED PILES 457 hear modulu of the oil material. A problem ariing from the ue of thi equation i that the olution will exhibit a high enitivity to Poion' ratio (recall that the contrained modulu tend to in nity a í approache 5), a behaviour that ha not been oberved in rigorou numerical olution of uch problem (ee, for intance, Butter eld & Banerjee, 97; Selvadurai & Rajapae 985). A an alternative, one may aume that the two horizontal normal tree, ó r and ó è, in the vertically loaded medium are zero, which i analogou to the aumption ued by Veleto & Younan (994) for the laterally-loaded problem. In uch a cae, equation (8) hould be replaced by ç 2 ˆ 2( í) (9) Perhap a better choice for the problem at hand i to conider ó r ˆ and å è ˆ, which capture better (though only approximately) the condition of zero tangential diplacement in the domain. With the latter aumption, ç 2 ˆ 2 (2) í Reult obtained from equation (8)±(2) are compared graphically in Fig. 2. It can be een that the prediction of equation (9) and (2) remain cloe over the entire range of í value, wherea equation (8) exhibit a ingular behaviour a í approache 5 and ceae to be acceptable. Except where peci cally otherwie indicated, the olution preented herein are baed on equation (2). MODEL VALIDATION Figure 3 compare reult for the tiffne of end-bearing pile computed with the propoed approximate model and with four nite-element and boundary-element olution by Poulo & Davi (98), Blaney et al. (975), Sanchez-Salinero (982), and El-Sharnouby & Nova (99). It can be een that with mall E p =E ratio (Fig. 3) the numerical reult are enitive to the dicretiation of the pile. For intance, when a mall number of element i ued, an increae in tiffne with increaing pile length i oberved in ome of the olution for L=d. 5Ðan obviouly erroneou trend for end-bearing pile. El-Sharnouby & Nova (99) report that a dene dicretiation (of the order of 5 pile element) i generally needed to remove thi anomaly. In contrat, the preent olution exhibit a table behaviour and agree well with the mot rigorou reult by El- Sharnouby & Nova. Similar good agreement i oberved with large E p =E ratio in Fig. 3. With reference to a long hollow pile in homogeneou halfpace, Table compare reult for pile tiffne obtained with the propoed model and with a rigorou elato-tatic olution by Pa & Ji (993). Although the two olution are not trictly comparable (hollow againt olid pile), the agreement between the prediction i very atifactory, with the average difference 4 3 η 2 Uncontrained medium: 2 equation (9) 73 2 Contrained medium: equation (8) Partially contrained medium: equation (2) Poion' ratio, ν Fig. 2. Senitivity of compreibility coef cient, ç, to Poion' ratio

4 458 MYLONAKIS 6 K (çá d=2) co áz dá (z) ˆ ðd çg Poulo á[k (çá d=2) ð dçg ( element) á K (çá d=2)] Blaney K (çá d=2) co áz dá á 2 [K (çá d=2) ð dçg (2) (2 element) 5 El-Sharnouby á K (çá d=2)] (5 element) For pile of nite length, the correponding olution i X Salinero K (çá m d=2) co á m z (>2 element) (z) ˆ ð dçg mˆ á m [K (çá m d=2) ð dçg 4 K (çá m d=2)] á m Normalied pile tiffne Pd/2W()E A p Uncontrained medium: equation (7) and (9) Partially contrained medium: equation (7) and (2) Blaney Poulo El-Sharnouby Salinero Dimenionle pile length, L/d Fig. 3. Normalied tiffne of end-bearing pile in a homogeneou oil tratum over rigid bedroc; comparion of the propoed approximate model with reult from four numerical olution. Soft pile, E p =E ; tiff pile, E p =E. Modi ed from El-Sharnouby & Nova (99); í :5 X mˆ K (çá m d=2) co á m z á 2 m [K (çá m d=2) ð dçg K (çá m d=2)] á m (22) With reference to an ini nitely long pile, the variation of (z) with depth i preented in Fig. 4 a a function of pile±oil tiffne ratio, E p =E. A decreaing trend with depth i oberved in all curve. For point located between about 3 and 2 pile diameter from the urface, (z) varie between approximately one and two time G, which i in agreement with value reported in the literature (Thoma, 98; Sanchez- Salinero, 982; Fleming et al., 992). With mall E p =E ratio, (z) tend to increae cloe to the urface but decreae more rapidly with depth. The ingularity oberved at z ˆ i analogou to that encountered in elatic analye of urface Depth, z/d E p /E = being about 5%. The minor effect of the pile Poion' ratio on the olution i evident. 5 EVALUATION OF WINKLER MODULUS For an in nitely long pile, the Winler modulu, (z), i obtained by dividing the ide hear traction, f [ˆ ð dô rz (d=2)] (equation ()), by the correponding pile ettlement, W (equation (4)). Accordingly, Normalied Winler modulu, /G Fig. 4. Variation with depth of Winler modulu for an in nitely long pile in homogeneou halfpace; í :5 Table. Normalied tiffne of a long hollow pile of wall thicne t and Poion' ratio í p in a homogeneou halfpace. Comparion of the propoed approximate model with a rigorou elato-tatic olution by Pa & Ji (993); í :25, t=d :5 í p log(g p =G ) E p =E { Normalied pile tiffne: Pa & Ji (993) Pd 2W()E A p Propoed model Difference :% { E p ˆ Ep [ ( 2t=d) 2 ] denote the Young' modulu of an `equivalent' olid pile having the ame axial rigidity a the hollow pile.

5 footing, and ha been reported in the literature (Pa & Ji, 993). The effect of pile length on (z) for end-bearing pile i examined in Fig. 5. It can be een that in a hort pile (z) i alway larger than in a more lender pile having the ame E p =E ratio. For intance, with L=d ˆ 5, (z) can exceed the value 2G over the entire pile length, which i more than twice that of the correponding in nitely long pile. The decreaing trend with depth i analogou to that oberved in Fig. 4. AVERAGE (DEPTH-INDEPENDENT) WINKLER MODULUS A common approximation in Winler analye i that the ratio (z)=g i contant along the pile length. While thi introduce ome error in the olution, it uually impli e the analyi by allowing equation (2) to be olved in cloed form within a homogeneou oil layer. Correponding average Winler moduli can be derived by matching a ey repone parameter (e.g. pile head ettlement) with reult from Winler formulation. For intance, auming =G to be contant within a homogeneou layer over rigid roc, the olution to equation (2) i (Scott, 98) W(z) ˆ P (coh ëz tanh ël inh ëz) (23) ë where ë i a parameter (unit of =length) given by ë ˆ (24) Enforcing the ettlement at the pile head in equation (7) and (23) to be equal, the following implicit olution for i obtained: tanh L L ˆ 2 L 2 X mˆ á 2 m K (çá m d=2) K (çá m d=2) ð dçg (25) K (çá m d=2) á m which can be eaily olved iteratively once the value of the right-hand ide ha been determined. For an in nitely long pile, etting tanh(hl) ˆ in equation (23) and uing equation (4) lead to the explicit olution WINKLER MODULUS FOR AXIALLY LOADED PILES ð K (çá d=2) dá ˆ E p A >< p 4 á 2 K (çá d=2) ð dçg >= >: á K (çá d=2) >; (26) Reult obtained from the above expreion are plotted in Fig. 6. The following point are worthy of note. For the pile length of the mot practical interet (ay 5, L=d, 5), varie between about 2 7 G and 8 G, and tend to decreae with increaing E p =E and L=d. In the limiting cae of E p =E! and E p =E!, it can be hown from equation (25) and (26) that tend to zero and in nity repectively. For lenderne ratio le than about 5, i practically independent of pile±oil tiffne ratio. In addition, the effect of Poion' ratio on the olution wa found to be of econdary importance, a hown in Fig. 7. In Fig. 8, reult from the model are compared graphically againt four empirical expreion from the literature. Thee expreion repreent average Winler moduli obtained by curve tting baed on the numerical olution by Banerjee, Blaney, and Poulo (ee lit of reference). In general, the prediction of the model lie cloe to the average of the empirical value. Neverthele, it i evident that thee formulae do not atify the limiting behaviour of the olution for mall and large value of E p =E and L=d, a dicued above. An improved regreion formula i preented below. Winler modulu, /G 3 2 L/d = Pile oil relative tiffne, E p /E Fig. 6. Depth-independent (`average') Winler modulu for endbearing pile in a homogeneou oil layer over rigid bedroc; í :5 5 2 L/d = 5 Depth, z/d Winler modulu, /G /2 ν = /3 / Normalied Winler modulu, /G Fig. 5. Variation with depth of Winler modulu for end-bearing pile in a homogeneou oil layer over rigid bedroc; E p =E, í : Pile oil relative tiffne, E p /E Fig. 7. Effect of oil Poion' ratio on average Winler modulu for an in nitely long pile in homogeneou halfpace

6 46 MYLONAKIS Sanchez-Salinero (982): 7 5(E p /E ) G and 8 G, and tend to decreae with increaing E p =E and L=d. In the limiting cae of very tiff (E p =E! ) and very oft pile (E p =E!), tend to zero and in nity repectively. On the other hand, remain nite with in nitely long pile, while it tend to in nity a L=d!. tend to increae with increaing oil Poion' ratio, but thi increae i of econdary importance. Normalied Winler modulu, δ Thoma (98): 2 55(E p /E ) 33 Thi paper (Eqn 27) (Eqn 25) Modified Randolph & Wroth (978): 2π/ln(δL/d) O'Roure & Dobry (978): + 5 5(L/d) (L/d) 3 (E p /E ) 4 5 Pile oil relative tiffne, E p /E Fig. 8. Depth-independent Winler modulu for end-bearing pile in a homogeneou oil layer over rigid bedroc. Comparion of reult from the propoed model with four empirical expreion from the literature: L=d 5; í :5 NON-LINEAR REGRESSION ANALYSES For practical application, it appear deirable to have a impli ed formula to etimate. Uing non-linear regreion analye baed on the Levenberg±Marquardt method (Bevington & Robinon, 992), the following equation wa derived: " # ' :3 E =4 p 7 L : 6 (27) G E d which correpond to the depth-independent modulu in equation (25) and (26). It i evident that thi equation ati e the limiting behaviour of the olution, i.e. ˆ at E p =E!;! at E p =E! or L=d! ; ˆ finite a L=d!. The correlation coef cient between the value in Fig. 6 and thoe computed with equation (27) i 973, and the tandard error i 4. CONCLUSIONS An approximate analytical model wa developed for the oil reaction along an axially loaded pile in a homogeneou elatic oil tratum. The model wa hown to be reaonably accurate and exhibit table behaviour over a wide range of parameter without developing the numerical intabilitie with long pile oberved in numerical olution. The main concluion of the tudy are: (z) i ingular at the pile head, and tend to decreae with depth. With long pile and at depth between 3 and 2 pile diameter, (z) varie between approximately one and two time the oil hear modulu, G. With end-bearing pile, (z) increae with decreaing pile length, and may exceed the value of 2 G over the entire pile length. Depth-independent (`average') Winler moduli can be obtained by matching prediction of pile head ettlement with reult from Winler formulation. In the parameter range of mot practical interet (ay, E p =E, and 5, L=d, 5), range between approximately NOTATION A p pile cro-ectional area B integration contant d pile diameter G p, E p pile hear modulu, Young' modulu G, E oil hear modulu, Young' modulu f (z) vertical oil reaction per unit pile length F body force along pile axi, (z) Winler modulu of ubgrade reaction coef cient of ubgrade reaction L pile length (= oil thicne) M oil compreion modulu P pile head load r radial coordinate t wall thicne of hollow pile u q, u r, u z oil diplacement component W(z) pile diplacement (ettlement) z vertical coordinate á, á m poitive variable ç compreibility coef cient ë Winler parameter í oil Poion' ratio í p pile Poion' ratio ô rz oil hear tre ó r, ó q, ó z oil normal tree REFERENCES Baguelin, F. & Fran, R. (979). Theoretical tudie of pile uing the nite element method. In Numerical method in offhore piling, pp. 83±9. London: Intitution of Civil Engineer. Bevington, P. R. and Robinon, D. K. 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