Computers and Geotechnics 9 (1990) 133-148 AXIAL RESPONSE ANALYSIS OF pF.l:.q IN VERTICALLY AND HORIZONTALLY NON-HOMOGENEOUS SOILS C.Y. Lee Research Fellow School of Civil and Mining Engineering University of Sydney Australia and H.G. Poulos Professor School of Civil and Mining Engineering University of Sydney Australia ABSTRACT c f T h i s p p , aper presents a modified procedure for the analysis of the axial response ues embedded in multi-layered soils. The results obtained by this procedure are compared with those computed by some previous methods and with a limited number of field test measurements. In the determination of the group settlement interaction between piles embedded in muki-layered soils, an additional simple soil mass stiffness model is dev¢Iol~,d in order to include the horizontal non-homogeneity of the soil due to sod disturbance cause by pile installation. The predictions b y this model agree more ciosciy witll the observed field test group performance than do predictions by the conventional method which assumes lateral homogeneity of the soil. 133 Computers and Geotechnics 0266-352X/90/$03-50 © 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain 134 INTRODUCTION Axial pile and pile group analyses using Mindlin's equations of elasticity have provided a simple and practical means of calculating the settlement of piles and pile groups in the past two decades (e.g. Poulos and Davis, 1980; Butterfield and Banerjee, 1971; Banerjee and Davies, 1977). In general, these analyses lead to adequate solutions in a soil mass with uniform or linearly increasing soil modulus with depth (e.g. Poulos 1979a, 1979b). It has been found that they may not give acceptably accurate solutions for piles embedded in layered soils where the modulus of the adjacent layers differ abruptly (e.g. Poulos, 1979a, Yamashita et al, 1987). In addition, they usually overpredict group interaction effects since they ignore the horizontal non-homogeneity in modulus in each soil layer between piles, due to pile installation (O'Neill et al 1977). The analysis of pile groups in vertically non-homogencous soil can be modelled more accurately by using the infinite layer method (Cheung et al 1988) or the finite element method (Chow 1987, 1989); the latter method can also be used to model horizontally non-homogeneous soil. In this paper, a more general approximation for piles in an arbitrary layered soil profile, involving the value of modulus in all soil layers, is developed. The influence of pile installation on the soil modulus between piles in a group is considered by introducing a simple empirical expression to relate the modulus in the disturbed soil near the pile surface to the modulus in the less disturbed soil mass further away. These two approaches are incorporated into conventional axial pile and pile group analyses based on the boundary element method. The modified analysis generally leads to better agreement with field measurements than do the conventional approaches. Method of Analysis (a) Single Pile The simplified form of boundary element analysis developed by Poulos and Davis (1980) is used in which the pile is represented as an elastic cylinder and the surrounding soil mass as an elastic continuum, as shown in Figure I. 135 The axial displacement of the pile elements may be expressed as follows: (1) {pp} - [AD][FE] {p) ÷ Pb {|} where {Pp} = displacement vector [AD] = summation matrix = pile compression matrix {P} = interaction stress vector Pb = pile base displacement {~} = vector whose elements are unity --dl-- xj Xl t ( i ( ( t I ( i1 t.. ) t ) 1 )~ 4 ) 4 ) t ~pJ t t t ) t b~4 ) 4 Soil Modulus I t t I I t J v$:Constant tpJ ® t t Ittt Pb I bE, j-..] -E,A~ I I t IrJIrrll"~sr~F411r~Jsl..rrJi41Frlrlri.jl.. I. bteract~n shut $ ~ I ~ t i ~ s of ~jtn fn~ surface Sol Pie SQt Llytrs Ck,~ Eb .I of sin] modulus ,,,~t'hdtpth FIG.1 ANALYSIS OF SINGLE PILE IN LAYERED SOIL 136 The displacements of the soil adjacent to each pile element may be expressed as follows: {ps} - [~-s]{p} (2) where {Ps} -" [~1 = soil displacement vector matrix of soil influence factors determined from Mindlin's equation (Mindlin 1936; Poulos and Davis, 1980); divided by the soil Young's modulus near the pile surface. When pile-soil interface conditions remain elastic, (ps } - {pp} hence I~S - AD'FEI{pP) - Pb{l} (3) The vertical force equilibrium condition requires: N _~ AiPi = p (4) i-I where Ai = surface area of element i P = of applied load to the pile head N = total number of pile elements. The unknown interaction stress {p} and base displacement Pb, can be evaluated by solving equations (1) and (2). For vertically non-homogeneous soils, Poulos (19"/9a) proposed a simple and practical method in which the homogeneous soil modulus E s is replaced by thc mean values at the influencing and influenced elements, but this method ignores the soil moduli of the other layers. This method does not give particularly accurate results for a pile embedded in layered soils in which the underlying layers are more compressible. Yamashita et al (1987) modified this m ~ h o d by 137 considering the soil modulus at every layer using a o n e - p a r a m e t e r "a" model. This parameter "a" depends on soil and pile properties, but no clear method is suggested for its determination. A similar m u k i - l a y e r e d soil model is developed here (termed the ML model), which considers the effect of the soil modulus at all soil layers, but requires no additional parameter when determining the mean soil modulus at the influencing and influenced elements. This model postulates that, for an element i, the soil modulus Esij due to the influenced of element j is given as follows: Esl j - 0.5(Esa t + Esa j) (6) and N ~. 6k Esk k-1 N ; Esal " for I - i,j (7) Y. 6k k-1 where 6k L [1 . Ixl - xkl Esk]-' - [ = Xl,Xk = Esi,Esk N ------- ~ (8) EslJ total pile length distance from respectively ground surface of elements I and k soil Young's modulus of layer I and k respectively total number of elements. Basically this model assumes that the mean soil modulus depends on the relative soil stiffness and the distance between all the influenced and the influencing elements. (b) Pile Groups For a group of two identical equally loaded piles, only the calculation of the soil displacement at each element requires modification to include the components due to the other pile, and hence equation (2) may be r e - e x p r e s s e d as follows (Poulos and Davis, 1980): 138 (9) where F;-S1 ,Es 2 = Ii,I 2 = soil Young's modulus near the surface of pile.s 1 and 2 respectively, matrices of displacement-influence factors for piles 1 and 2 respectively. This conventional approach assumes that the soil moduli (i.e. Es, and Es2 ) used to determine the matrices 11 and 12 are identical. However it has been found that the soil closer to the pile surface is more disturbed than that further away, due to pile installation (e.g. Cooke ¢t al 1979, Williams, 1979, Francescon, 1983) and hence some horizontal non-homogeneity is induced in the soil mass. Poulos (1988a) has suggested a two-parameter soil model to modify the calculation of group interaction effects. A simpler one parameter horizontal non-homogeneous soil model is proposed here which includes the variation of horizontal soil modulus used to determine the soil displacement influence factors. The soil model is shown diagramatically in Figure 2 and Equation (9) may be modified as follows: {Ps} " [~'~ + -~-d]{P } (10) and Esd E--'~-- LQJ where Es Esd n = average soil Young's modulus within one pile radius from the pile surface, = soil Young's modulus at a distance s greater than one pile radius from pile surface, = soil parameter depending on pile and soil type. E, I FIG.2 HORIZONTALNON-HOMOGENEOUS SOILMODELIN PILE GROUPINTERACTIONANALYSIS Disturbed Soil due to Pile Installation Less Disturbed Soil Mass S n O.S 0 I I I 6 ~ 0.7 {b)N o n - H o m o ~ (Gibson) Sod I I (a) H o m o ~ n=O h L/~,SI I I "~ 8 10 ~L i ~ ~~ ~ 15 20 _~-,.... ,. "::'..:. Conventional ~pproach ~,/[,.ml, 2 Z, 6 6 10 20 Pile Spacing/Diameter Is/d) FIG3 EFFECTOFn VALUESONINTERACTIONFACTOR .o= 0.25 u_ ,Y, =o 0.25 - L o 0.5 co co 140 Figure 3 demonstrates the effect of the value of n in the so -called "horizontally non-homogeneous" (HNH) soil model on the computed interaction factor c~. For a homogeneous soil (n = 0) the values of interaction factor c~ are equivalent to those computed by the conventional approach (Poulos and Davis, 1980). The value of interaction factor decreases as the value of n increases. It appears to decrease more significantly with pile spacing for higher n values, than the conventional approach (n = 0). The values of interaction factor a in a non-homogeneous ("Gibson") soil also vary similarly with n and pile spacing, except that the interaction factor values from the conventional approach lie below those computed by the horizontal non-homogeneous (HNH) soil model for n = 0. This horizontal non-homogeneous (HNH) soil model may also be used to analyse any general configuration of piles in a group. Using this model in conjunction with the multi-layered soil model, it is believed that a more realistic simulation of pile group behaviour may be made. This combined approach will be referred to as the ML/HNH model. Evaluation of the Modified Approaches Single piles in layered soil The present approach using the multi-layered (ML) soil model has been used to analyse three idealised cases (Poulos, 1979a). The results are compared with solutions obtained from other approaches, as shown in Figure 4. The resuks computed by all the approaches appear to agree closely with those obtained by the finite element approach for Case 1 and Case 3. However, for Case 2, in which the soil modulus decreases with depth, the solutions from the present approach and the finite element approach only differ by about 5%, whereas the difference between the other approaches and the finite element method exceeds 20%. Engeling and Reese (1974) performed a compression loading test on a drilled pile shaft of length 42 ft (12.8 m) and diameter 30 inches (0.76 m), embedded in soft to hard clay west of Bryan in Texas, USA. The soil shear strength decreased from the ground surface to about 30 ft (9.1 m) depth and increased beyond that depth. In the calculations performed by the authors, the soil modulus has been assumed to be 750 times the shear strength obtained from the triaxial tests (Aschenbrenner et al 1984). Figure 5 compares the measured results with those predicted by the present approach, the conventional approach (Poulos 1979a) and the Yamashita et al (1987) approach. The comparisons ] l 2 Ep L h ~--~s= 1000 ~ ~ - = 25 ~ ~ = YamiJl'e el d le=O.SI Prmnt ~ d l e w ~ 19";9al F..qubllmlUnifcm5ai Y~i et d WB?I a=0.SI ~Mnt A ~ Sd l 0.2 • • 19(finite element) O.Z. 0.6 0.8 I i i I" I I• I I I •1 I I* 1.0 ito l/H//H////////// 2 ~ vs = 0.3 FIG./, COMPARISONSOLUTION COMPUTED BY VARIOUS APPROACHES Case 3 Case 2 Ihu~. e'nd Equ~ U~ la=0.SI Ap~oach e et d CBf/I ~wmd 9== PHe settlement Case 1 Case 3 Soil Young's Modulus Distribution with Depth 1 ",'/,'/H///////////,','//,'/////////H/H/[//~,~////H//H/////,*//H °'3LI II P ~ 12 / " 15 12 - 6 " / ~f } // / I 0 0.5 Predicfed Measured PiLe Head S e t f l e m e n t Present Approach • • ----Moasu~,, c[~Ii.g a.d Reese.197/,) "-----[onventiona[ Approach !p~o~.l.gal IPoulos 19"/9a) ..... -Y~ashita et al 1987 Present Approach Conventional Approach (Poulos. 1979a) Yalashita et al (1987J (a=0.51 .~ / /',v 1.0 E - - - 890 FIGS PREDICTIONS OF COMPRESSION PILE IN CLAY SOIL C3 ~9 E lJ.5 Applied Load, KN 142 indicate that the predictions of head settlement and load distribution by the present approach agree well with the measured values, whereas the other two approaches seem to underestimate the settlement by more than 15%. The three approaches have also been employed to predict the behaviour of an offshore steel tube pile driven into marine sediments at Plancoet in France (Puech, 1982). The soil profile consisted of three distinct layers, as shown in Figure 6, and the pile was 13 m long, 0.27 m in diameter with a 6.3 mm wall thickness. The pile was loaded in tension in three different stages. For the theoretical calculation, the soil modulus was assumed to be 15 times the static cone resistance (Poulos, 1988b). As shown in Figure 6, the present approach predicts the measured settlements more accurately than the other two approaches, although all three approaches underpredict the load distributions. The present approach predicts a more gradual transfer of load with depth than the other two approaches. ApptiedLoad (kN) 0 100 Sandy Silt 200 T'100 200 '/I 100 200 I j v-/.0% /a c'=0 . O*=/.2 ° "~" r~ 0S == Loose Sand w-/*S% #'=/.3" c~ ca Q. 1.0 Silty Clay v-~-5% , %=57 c'=20KPa . vp=29 e'=26"30' ~ s t Conve~tkmat ,4.~oKh 52 I TesP S3A I Test S3B IPoulUI ~l?9,~J Ymm~hita et of 11~$7) :e=O.5l Present Approach Leg=rid: ----- Measured Eonvmltional 0 Approach (Poulos. 19?9ai - - * - - YamtshJto et el 1987 - - - - - - Present Approlch 1.0 1.0 PredicPed Measured (Pite Setttement) FIG.6 PREDICTIONS OF OFFSHORE TENSION PILES IN MARINE SOIL 1.0 143 Pile groups in layered soil Cooke et al 0980) performed field tests on steel tube piles of 168 nun diameter with 6.4 nun wall thickness and approximately 5 m long embedded in London clay, and measured the settlement interaction factors. The vertical soil modulus was assumed to be represented as a "Gibson" soil profile with Es(z) - 35z MPa where z is the soil depth. As shown in Figure 7, the conventional theoretical approach assuming lateral homogeneity of the soil overestimates the interaction factor values significantly. For the horizontally non-homogeneous soil model, various values of the parameter n have been tried to obtain a fit with the measured values. It appears that n = 0.5 gives the best agreement with the measured values, and this value has been used in the predictions of another series of pile group tests. O.SO .~"~ ~ Measured - - - - Cenventic~ Approach(PouIo$, 1979a) . . . . Present Approa¢h(n=03) ~';~ ~'~--~- Present Approach(n=0.51 g o.z5 N o 0 2 I, 6 8 10 12 s/d FIG.7 MEASUREDANO COMPUTEDINTERACTIONFACTORS O'Neill et al (1981) have reported results of axial loading tests on full-scale pile groups and single piles in clay. The piles were 10.75 inches (0.273 m) diameter steel tubes with a 0.365 inches (9.3 ram) wall thickness and 43 ft (13.1 m long). The tests were carried out at a site at the University of Houston, Houston, Texas, USA, and the geotechnical data at the site is summarised in Figure 8. In this case, a remoulded near-pile soil modulus of 25 times the static cone resistance qc (Poulos, 1988a) was used for the theoretical predictions of the behaviour of the pile groups and single piles using the following approaches: (a) conventional approach (Poulos and Davis, 1980, with n = 0); (b) approach using HNH model (with n = 0.5); (c) present approach (MIdHNH model, with n = 0.5). 144 Average Cone Undrained Shear Water Content OCR SPT Resistance Strength IkN/mz) Blows/0.3m (kN/m z) I'/.I 20 ~0 0 ,000 10000 250 500 20 z,0 80 0 2 z, t I Stratigraphy 0 -I 0 V. stiff 9ray i tan clay Still day, sand seams L~ S-~ ..~ Stiff-V. stiff rod i gray clay i~XJ -----1Q-I~.~ Stiff-V.stiff gray & I~XI tan sandy day. " ~ . ~ vltb sand pockets ' t° ~LL , .:l-o i, o " ,4-0 . ~ X ;O'JC "FOX 0 I ~,'o~ "° ~.IOY---.x 1s-111111o ~ , red ~ gray t,,i~HT /11111~t. ~ith day. ~lt /11111~ sand layers 2 0 & ~ V. stiff red & gray "Triaxial FIG.8 SUMHARY OF :~ ~'~ ~=Nat. W/C ;':-°Ons°l I j ~iriaxialJ I . Consol. I GEOTECHNICALDATA AT TEST SITE i ~ I 9 l x I• .-~ S * I• x z 1 A I - - - Measured x Conventional Approach (n=0) (Poulos and Davis, 1980) Approach using HNH HodeI (n=0.5) • Present Approach In=0 5) I I I Ix I I I I I 300 600 900 1000 1500 (a) Pile Head Stiffness HN/m ,=.9 .:- 5 cL. L" I I I &. I I zl I 2 (b) Settlement Ratio Rs 3 FIG9 PREDICTIONSOF PILE HEAp STIFFNESS AND SETTLEMENT RATIO O.S AI x I ! l I I Ix I (b) S Pile Group il 0.875 Load AverageLoad x .i (a) 9 Pile Group II 0.875 Load AverageLoad x • [email protected] IA l I x 1.25 1.2S • A .... x Heasured Conventional Approach (n=O) (Poulos and Davis, 1980) Approach using HNH Model In:O.S) Present Approach (n=O.S) Legend: FIG10 PREDICTIONSOF PILE HEAD LOAD DISTRIBUTIONS ! 0.S ? ! 1.0 [-Z 0S 10 Z_ 0.S L Edge Pile tl r 7 r)- 9 Pi(e Group / 11 11 11 Pile o~,~i!~or ner Pile o} /'~ 12" /,Y ~/o Centre Pile.I" ,J/Corner !;I° -- 11 S Pile Group ~ C e n t r e Pile ,Yr . . . . 1.1 FIGJI PREOICTIONS OF IOA0 01STRIBUTIONS ALONG Ptl£ lENGTH o Heasured -----Conventional Approach In:01 (Poulos and Davis. 1980) -- ~ Approach using HNH Hodel In:0S) ------ Present Approach In=0S) L_egend: o/~ / '?e-~tL~'7- Predrill Pile Group F o,~°~J ~7 Predri[I Depth Load Average Pile Head Load t46 The predicted and measured values of the pile head stiffness and group settlement ratio Rs are shown in Figure 9. The conventional approach (with n = 0) appears to underestimate group stiffness and overpredict the pile settlement, the difference increasing as the number of piles in the group increases. However, the values predicted by the modified conventional approach and the present approach agree much more closely with the measurements. Figure 10 also demonstrates that the group pile head load distributions predicted by the HNH approach (with n = 0.5) and the present approach (ML/HNH) are in better agreement with the measurements than those predicted by the conventional approach. Despite the fact that the HNH and ML/HNH approaches seem to predict the measured pile head response similarly well, the main difference in the predicted performance from these approaches is illustrated in Figure 11 where the load distribution along the pile is plotted. It can be seen that the load distributions predicted by the present ML/HNH approach agree more closely with the measurements than do the predictions by the other two approaches. CONCLUSIONS The conventional approach, using Mindlin's equations for the analysis of the settlement of a single pile in a layered soil profile, is generally adequate except when significant differences in soil modulus exist between adjacent soil layers or if a soil layer is underlain by a much more compressible layer. In order to overcome this limitation, a more general soil profile approximation model (the ML model) has been developed, involving the value of soil modulus at all layers in the soil profile. Comparisons with some field measurements for piles embedded in a layered soil demonstrate that this modified approach leads to more realistic predictions of pile head response and load distribution than does the conventional approach. An alternative simplified pile group analysis has been developed, and involves a computational model (the horizontal non-homogeneous or HNH model) which relates the remoulded near-pile soil modulus to the value for the less disturbed soil mass further away, via the normalised pile spacing and an exponent parameter n. The value of parameter n may depend on the pile and soil type, but a value of n = 0.5 appears to fit limited available data. This HNH model 147 will reduce the overprediction of group interaction effects commonly experienced when using the conventional approach. Comparisons of the predictions by this modified approach with some field measurements of pile groups in layered soils have shown generally good agreement between predicted and measured group performance, although some inaccuracy remains in the predicted load distribution characteristics along the pile. This inaccuracy can be overcome by incorporating the more general soil profile approximation model into the analysis, thus leading to more realistic predictions of both the pile head performance and the load distribution characteristics within a pile group. ACKNOWI.EDOEMENT The work described in this paper forms part of a research project into the Mechanics of Calcareous Sediments, supported by the Australian Research Council. REFERENCES 1. Aschenbrenner, T.B. and Oslen, R.E. (1984). Prediction of Settlement of Single Piles in Clay. Anal. and Design of Pile Foundns., ASCE, pp. 41-58. 2. Banerjee, P.K. and Davies, T.G. (1977). Analysis of Pile Groups Embedded in Gibson Soil. Prec. 9th ICSMFE, Tokyo, Vol. I, pp. 381-386. 3. Butterfield, R. and Banerjee, P.K. (1971). Elastic Analysis of Compressible Piles and Pile Groups. Geotechnique, Vol. 21, No. I, pp. 43-60. 4. Cheung, Y.K., Tham, L.G. and Guo, D.J. (1988). Analysis of Pile Group by Infinite Layer Method. Geotechnique, 38(30) pp. 41.5-431. 5. Chow, Y.K. (1987). Axial and Lateral Response of Pile Groups Embedded in Nonhomogeneous Soils. Int. Journal for Numerical and Analytical Methods in Geomechanics, 11(6), pp. 621-638. 6. Chow, Y.K. (1989). Axially Loaded Piles and Pile Groups Embedded in a Cross-Anisotropic Soil. Geotechnique, 39(2), pp. 203-211. 7. Cooke, R.W., Price, G. and Tart, K. (1979). Jacked Piles in London Clay: A Study of Load Transfer and Settlement under Working Conditions. Geotechnique 29, No. 4, pp. 461-468. 8. Cooke, R.W., Price, G. and Tart, K. (1980). Jacked Piles in London Clay: Interaction and Croup Behaviour under Working Conditions. Geotechnique 30, No. 2, pp. 97-136. 9. Engeling, D.E. and Reese, L.C. (1974). Behaviour of Three Instrumented Drilled Shafts under Short Term Axial Loading. Research Report 176-3, University of Texas, Austin, Texas. l16p. 148 10. Francescon, M. (1983). Model Pile Tests in Clay: uDiSmPvlacements due to Installation and Axial Loading. ersity of Cambridge. Stresses and PhD Thesis, 11. Mindlin, R.D. (1936). Force at a Point in the Interior of a Semi-lnfinite Solid Physics, Vol. 7, pp. 195-202. 12. O'Neill, M.W., Ghazzaly, O.I. and Ha, H.B. (1977). Analysis of Three-Dimensional Pile Groups with Non-linear Soil Response and Pile-Soil-Pile Interaction. Proc. 9th Annual OTC, Houston, Paper OTC 2838, pp. 245-256. 13. O'Neill, M.W., Hawkins, R.A. and Mahar, L.J. (1981). Field Study of Pile Group Action. Rep. No. FHWA/Rd-81/002, Fed. Highway Admin. 14. Poulos, H,G. (1979a). Settlement of Single Piles in Non-Homogeneous Soil. Jnl. Geol. Eng. Divn., ASCE, Vol. 105, No. GTS, pp. 627-641. 15. Poulos, H.G. (1979b). Group Factors for Pile-Deflection Estimation. Jnl. Geot. Eng. Divn., ASCE, Vol. 105, No. GT12, pp. 1489-1.509. 16. Poulos, H.G. (1988a). Modified Calculation of Pile Group Settlement Interaction. Jnl. Geot. Eng., ASCE, Vol. 114, No. 6, pp. 697-106. 17. Poulos, H.G. (1988b). Marine Geotechnics. London, Unwin Hyman. 18. Poulos, H.G. and Davis, E.H. (1980). Pile Foundation Analysis and Design. John Wiley and Sons, New York. 19. Puech, A. (1982). Basic Data for the Design of Tension Piles in Silty Soils. Proc. 3rd Int. Conf. Behaviour of Offshore Structures, Vol. 1, pp. 141-157. 20 Williams, D.J. (1979). The Behaviour of Model Piles in Dense Sand Under Vertical and Horizontal Loading, PhD Thesis, University of Cambridge. 21. Yamashita, K., Tomono, M. and Kakurai, M. (1987). A Method for Estimating Immediate Settlement of Piles and Pile Groups. Soil and Foundns., Vol. 27, No. I, pp. 61-76. Received 25 October 1989; revisedversion received7 July 1990; accepted 10 July 1990

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