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Dynamic analysis for laterally loaded piles and dynamic p–y curves M. Hesham El Naggar and Kevin J. Bentley Abstract: Pile foundations are often subjected to lateral dynamic loading due to forces on the supported structure. In this study, a simple two-dimensional analysis was developed to accurately model the pile response to dynamic loads. The proposed model incorporates the staticp–y curve approach (wherep is the static soil reaction andy is the pile de- flection) and the plane strain assumptions to represent the soil reactions within the frame of a Winkler model. Thep–y curves are used to relate pile deflections to the nonlinear soil reactions. Wave propagation and energy dissipation are also accounted for along with discontinuity conditions at the pile–soil interface. The inclusion of damping with the static unit transfer curves results in increased soil resistance, thus producing “dynamicp–y curves.” The dynamicp–y curves are a function of the staticp–y curve and velocity of the soil particles at a given depth and frequency of load- ing. The proposed model was used to analyze the pile response to the lateral Statnamic load test, and the predicted re- sponse compared well with the measured response. Closed-form solutions for dynamicp–y curves were established by curve fitting the dynamic soil reactions for a range of soil types and loading frequencies. These solutions can be used to model soil reactions for pile vibration problems in readily available finite element analysis (FEA) and dynamic structural analysis packages. A simple spring and dashpot model was also proposed to be used in equivalent linear analyses of transient pile response. The proposed models were incorporated into an FEA program (ANSYS) which was used to compute the response of a laterally loaded pile. The computed responses compared well with the predictions of the two-dimensional analysis. Key words: dynamic, transient, lateral, piles,p–y curves, inertial interaction. Résumé: Les fondations sur pieux sont souvent soumises à un chargement dynamique latéral dû à des forces appliquées sur la structure portée. Dans cette étude, une analyse unidimensionnelle simple a été développée pour modéliser avec précision la réaction du pieu aux charges dynamiques. Le modèle proposé incorpore l’approche de la courbe statiquep–y et les hypothèses de déformation plane pour représenter les réactions du sol dans le cadre d’un modèle de Winkler. Les courbesp–y sont utilisées pour faire une corrélation entre les déflexions du pieu et les réactions non linéaires du sol. La propagation d’ondes et la dissipation d’énergie sont également prises en compte en même temps que les conditions de discontinuité à l’interface pieu-sol. L’inclusion de l’amortissement avec les courbes de transfert d’unité statique résulte en un accroissement de la résistance du sol, produisant ainsi des « courbesp–y dynamiques ». Les courbesp–y dynamiques sont une fonction de la courbe statiquep–y et de la vélocité des particules de sol à une profondeur et à une fréquence de chargement données. Le modèle proposé a été utilisé pour analyser la réponse du pieu à l’essai de chargement statnamique, et la réponse prédite se comparaît bien à la réponse mesurée. Les solutions exactes pour les courbesp–y dynamiques ont été établies par lissage des courbes des réactions dynamiques du sol pour une plage de types de sol et de fréquences de chargement. Ces solutions peuvent être utilisées pour modéliser les réactions du sol pour les problèmes de vibration de pieux dans les progiciels FEA et d’analyse structurale dynamique couramment disponibles. Un modèle comprenant un simple ressort avec pot amortisseur a aussi été proposé pour être utilisé dans des analyses linéaires équivalentes de la réponse transitoire d’un pieu. Les modèles proposés ont été incorporés dans un programme FEA (ANSYS) qui a été utilisé pour évaluer la réponse d’un pieu chargé latéralement. Les réponses calculées se comparaîent bien avec les prédictions de l’analyse bidimensionnelle. Mots clés: dynamique, transitoire, latéral, pieux, courbesp–y, interaction inertielle. [Traduit par la Rédaction] El Naggar and Bentley 1183 Introduction Pile foundations are often subjected to lateral loading due to forces on the supported structure. The horizontal loads at the pile head can be the governing design constraint for sin- gle piles and pile groups supporting different types of struc- tures in many situations including environmental loading (wind, water, and earthquakes) and machine loading on structures such as buildings, bridges, and offshore platforms. Most building and bridge codes use factored static loads to account for the dynamic effects of pile foundations. Al- though very low frequency vibrations may be accurately modeled using factored loads, the introduction of non- linearity, damping, and pile–soil interaction during transient loading may significantly alter the response. The typical fre- quencyranges of interest are 0–10 Hz for earthquakes, Can. Geotech. J.37: 1166–1183 (2000) © 2000 NRC Canada 1166 Received May 5, 1999. Accepted April 28, 2000. Published on the NRC Research Press website on December 5, 2000. M.H. El Naggar and K.J. Bentley. Geotechnical Research Centre, The University of Western Ontario, London, ON N6A 5B9, Canada. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:48:21 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. 0–1 Hz for offshore environmental loading, and 5–200 Hz for machine foundations. The emphasis in the current study is on the seismic loading case. Novak et al. (1978) developed a frequency-dependent pile–soil interaction model; however, it assumes strictly lin- ear or equivalent linear soil properties. Gazetas and Dobry (1984) introduced a simplified linear method to predict fixed-head pile response accounting for both material and ra- diation damping and using available static stiffness (derived from a finite element or any other accepted method). This method is not suitable for the seismic response analysis be- cause of the linearity assumptions. In general, there is much controversy over advanced linear solutions (frequency do- main), as they do not account for permanent deformation or gapping at the pile–soil interface. Nogami et al. (1992) developed a time-domain analysis method for single piles and pile groups by integrating plane strain solutions with a nonlinear zone around each pile using p–y curves, (wherep is the static soil reaction andy is the pile deflection). El Naggar and Novak (1995, 1996) also de- veloped a computationally efficient model for evaluating the lateral response of piles and pile groups based on the Winkler hypothesis, accounting for nonlinearity using a hy- perbolic stress–strain relationship and slippage and gapping at the pile–soil interface. The model also accounts for the propagation of waves away from the pile and energy dissipa- tion through both material and geometric damping. The p–y curves (unit load transfer curves) approach is a widely accepted method for predicting pile response under static loads because of its simplicity and practical accuracy. In the present study, the model proposed by El Naggar and Novak (1996) is modified to utilize existing or developed cy- clic or staticp–y curves to represent the nonlinear behaviour of the soil adjacent to the pile. The model uses unit load transfer curves in the time domain to model nonlinearity and incorporates both material and radiation damping to generate dynamicp–y curves. Model description Pile model The pile is assumed to be vertical and flexible with circu- lar cross section. Noncylindrical piles are represented by cy- lindrical piles with equivalent radius to accommodate any pier–pile configurations. The pile and surrounding soilare subdivided inton segments, with pile nodes corresponding to soil nodes at the same elevation. The standard bending stiffness matrix of beam elements models the structural stiff- ness matrix for each pile element. The pile global stiffness matrix is then assembled from the element stiffness matrices and is condensed to give horizontal translations at each layer and the rotational degree of freedom at the pile head. Soil model: hyperbolic stress–strain relationship The soil is divided into n layers with different soil proper- ties assigned to each layer according to the soil profile con- sidered. Within each layer, the soil medium is divided into two annular regions as shown in Fig. 1. The first region is an inner zone adjacent to the pile and accounts for the soil nonlinearity. The second region is the outer zone that allows for wave propagation away from the pile and provides for the radiation damping in the soil medium. The soil reactions and the pile–soil interface conditions are modeled separately on both sides of the pile to account for slippage, gapping and state of stress as the load direction changes. Inner field element The inner field is modeled with a nonlinear spring to rep- resent the stiffness and a dashpot to simulate material (hysteretic) damping. The stiffness is calculated assuming plane strain conditions, the inner field is a homogeneous iso- tropic viscoelastic massless medium, the pile is rigid and cir- cular, there is no separation at the pile–soil interface, and displacements are small. Novak and Sheta (1980) obtained the stiffnesskNL under these conditions as [1] k G v v r r r r NL m o o = − − + 8 1 3 4 1 1 2 1 π ( )( ) + − + 2 2 1 2 3 4 1( ) lnv r r r r o 1 o −1 wherero is the pile radius,r1 is the outer radius of the inner zone, andν is Poisson’s ratio of the soil stratum. The ratio r1/ro depends on the extent of nonlinearity, which depends on the level of loading, and on the size of the pile. A para- metric study showed that a ratio of 1.1–2.0 yielded good agreement between the stiffness of a composite medium (in- ner zone and outer zone) and that of a homogeneous me- dium (no inner zone) under small strain (linear) conditions. Gm is the modified shear modulus of the soil and is approxi- mated, according to the strain level, by a hyperbolic law as [2] G Gm = − + max 1 1 η η where Gmax is the maximum shear modulus (small strain modulus) of the soil according to laboratory or field tests. In the absence of actual measurements, maximum shear modu- lus for any soil layer can be calculated in this model by (Hardin and Black 1968) © 2000 NRC Canada El Naggar and Bentley 1167 Fig. 1. Element representation of the proposed model. m1, m2 represent the mass of the inner field lumped at two nodes; half (m1) at the node adjacent to the pile, and the other half (m2) at the node adjacent to the outer field. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:48:27 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. [3] G e e max )= − + 3230(2.97 o 0.5 2 1 σ wheree is the void ratio, andσo is the mean principal effec- tive stress (in kN/m2) at the soil layer. The parameterη = P/Pu is the ratio of the horizontal soil reaction in the soil spring,P, to the ultimate resistance of the soil element,Pu. The ultimate resistance of the soil element is calculated us- ing standard relations given by the American Petroleum In- stitute (1991). For clay, the ultimate resistance is given as a force per unit length of soil by [4] Pu = 3cud + γXd + JcuX [5] Pu = 9cud where Pu is the minimum of the resistances calculated by eqs. [4] and [5],cu is the undrained shear strength,d is the diameter of the pile,γ is the effective unit weight of the soil, X is the depth below the surface, andJ is an empirical coef- ficient dependent on the shear strength. A value ofJ = 0.5 was used for soft clays (Matlock 1970) andJ = 1.5 for stiff clays (Bhushan et al. 1979). The corresponding criteria for the ultimate lateral resis- tance of sands at shallow depthsPu1 or at large depthsPu2 are as follows (American Petroleum Institute 1991): [6] P A X K X u cos 1 0= − + − γ φ β β φ α β β φ tan sin tan( ) tan tan( ) × + +( tan tan ) tan (tan sind X K Xβ α β φ β0 − − tan )α K da [7] P A Xd K Ku a2 8 0 41= − +γ β φ β[ (tan ) tan tan ] whereA is an empirical adjustment factor dependent on the depth from the soil surface,K0 is the earth pressure coeffi- cient at rest,φ is the effective friction angle of the sand,β = φ /2 + 45°,α = φ /2, andKa is the Rankine minimum active earth pressure coefficient defined asKa = tan 2(45 – φ /2). In the derivation of eq. [1], the inner field was assumed to be massless (Novak and Sheta 1980). Therefore, the mass of the inner field is lumped equally at two nodes on each side of the pile: node 1 adjacent to the pile, and node 2 adjacent to the outer field, as shown in Fig. 1. Far-field element The outer field is modeled with a linear spring in parallel with a dashpot to represent the linear stiffness and damping (mainly radiation damping). The outer zone allows for the propagation of waves to infinity. Novak et al. (1978) devel- oped explicit solutions for the soil reactions expressed in terms of complex stiffness,K, of a unit length of a cylinder embedded in a linear viscoelastic medium given by [8] K = Gmax[Su1(ao, ν, D) + iSu2(ao, ν, D)] whereao = ωr1/Vs is the dimensionless frequency,ω is the frequency of loading,Vs is the shear wave velocity of the soil layer, i is the imaginary unit (= −1 ), andD is the ma- terial damping constant of the soil layer. Figure 2 shows the general variations ofSu1 andSu2 with Poisson’s ratio and ma- terial damping. Rewriting eq. [8], the complex stiffness,K, can be represented by a spring coefficient,kL, and a damping coefficient,cL, as [9] K = kL + iaocL Figure 2 shows that for the dimensionless frequency range between 0.05 and 1.5,Su1 maintains a constant value andSu2 increases linearly with an increase inao. The majority of dy- namic loading on foundations falls within this frequency range, including destructive earthquake loading. Therefore, for the purpose of the time-domain analysis, the spring and dashpot constants,Su1 and Su2, respectively, are considered frequency independent and depend only on Poisson’s ratio and are given as follows: [10] kL = GmaxSu1(ν) [11] c G r V S a vL 1 s u2 o 0.5= = 2 max ( , ) © 2000 NRC Canada 1168 Can. Geotech. J. Vol. 37, 2000 Fig. 2. Envelope of variations of horizontal stiffness and damping stiffness parameters betweenν = 0.25 andν = 0.4 (after Novak et al. 1978). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:48:29 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. http://nrc.literatumonline.com.nrc.literatumonline.com/action/showImage?doi=10.1139/t00-058&iName=master.img-000.png&w=358&h=196 Pile–soil interface The pile–soil interface is modeled separately on each side of the pile, thus allowing gapping and slippage to occur on each side independently. The soil and pile nodes in each layer are connected using a no-tension spring, that is, the pile and soil will remain connected and will have equal dis- placement for compressive stresses. The spring is discon- nected if tensile stress is detected in the soil spring to allow a gap to develop. This separation or gapping results in per- manent displacement ofthe soil node dependent on the mag- nitude of the load. The development of such gaps is often observed in experiments, during offshore loading, and after earthquake excitation in clays. These gaps eventually fill in again over time until the next episode of lateral dynamic loading. The pile–soil interface for sands does not allow for gap formation, but instead the sand caves in, resulting in the virtual backfilling of sand particles around the pile during repeated dynamic loading. When the pile is unloaded, the sand on the tension side of the pile follows the pile with zero stiffness instead of remaining permanently displaced as in the clay model. In the unloading phase, the stiffness of the inner field spring is assumed to be linear in both the clay and sand models. Soil model: p–y curve approach The soil reaction to transient loading comprises stiffness and damping. The stiffness is established using thep–y curve approach and the damping is established from analyti- cal solutions that account for wave propagation. A similar approach was suggested by Nogami et al. (1992) usingp–y curves. Based on model tests,p–y curves relate pile deflections to the corresponding soil reaction at any depth (element) below the ground surface. Thep–y curve represents the total soil reaction to the pile motion (i.e., the reactions of the inner and outer zones combined). The total stiffness,kpy, derived from the p–y curve is equivalent to the true stiffness (real part of the complex stiffness) of the soil medium. Thus, re- ferring to the hyperbolic law model, the combined inner zone stiffness (kNL) and outer zone stiffness (kL) can be re- placed by a unified equivalent stiffness zone (kpy) as shown in Fig. 3a. Hence, to ensure that the true stiffness is the same for the two soil models, the flexibility of the two mod- els is equated, i.e., [12] 1 1 1 k k kpy = + L NL The stiffness of the nonlinear strength is then calculated as [13] k k k k k py py NL L L = − The constant of the linear elastic spring,kL, is established from the plane strain solution (i.e., eq. [10]). The static soil stiffness,kpy represents the relationship between the static soil reaction,p, and the pile deflection,y, for a givenp–y curve at a specific load level. Thep–y curves are established using empirical equations (Matlock 1970; Reese and Welch 1975; Reese et al. 1975) or curve fit to measured strain data using an accepted method such as the modified Ramberg- Osgood model (Desai and Wu 1976). In the present study, internally generated staticp–y curves are established based on commonly used empirical correlations for a range of soil types. Damping The damping (imaginary part of the complex stiffness) is incorporated into both thep–y approach and the hyperbolic model to allow for energy dissipation throughout the soil. The nonlinearity in the vicinity of the pile, however, drasti- cally reduces the geometric damping in the inner field. Therefore, both material and geometric (radiation) damping are modeled in the outer field. A dashpot is connected in parallel to the far-field spring, and its constant is derived from eq. [11]. If the material damping in the inner zone is to be considered, a parallel dashpot with a constantcNL, coeffi- cient of material damping in the inner zone, to be suitably chosen may be added as shown in Fig. 3b. The addition of the damping resistance to static resistance represented by the static unit load transfer (thep–y curve) tends to increase the total resistance as shown in Fig. 4. Static p–y curve generation for clay The general procedure for computingp–y curves in clays both above and below the groundwater table and correspond- ing parameters are recommended by Matlock (1970) and Bhushan et al. (1979), respectively. Thep–y relationship was based on the following equation: [14] p P y y n u 0.5= 50 wherep is the soil resistance,y is the deflection correspond- ing to p, Pu is the ultimate soil resistance from eqs. [4] and [5], n is a constant relating soil resistance to pier–pile de- flection, andy50 is the corrected deflection at one-half the ultimate soil reaction determined from laboratory tests. The tangent stiffness constant,kpy, of any soil element at time step t + ∆t is given by the slope of the tangent to thep–y curve at the specific load level as shown in Fig. 4. This slope is established from the soil deflections at time stepst andt – ∆t and the corresponding soil reactions calculated from eq. [14], i.e., © 2000 NRC Canada El Naggar and Bentley 1169 Fig. 3. Soil model: (a) composite medium andp–y curve, and (b) inclusion of damping. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:48:31 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. [15] k p p y y py t t t t t t t t ( )+ − − = − −∆ ∆ ∆ Therefore, eqs. [10] and [15] can be substituted into eq. [13] to obtain the nonlinear stiffness representing the in- ner field element in the analysis. Thus, the linear and nonlin- ear qualities of the unit load transfer curves have been logically incorporated into the outer and inner zones, respec- tively. Static p–y curve generation for sand Several methods have been used to experimentally obtain p–y curves for sandy soils. Abendroth and Greimann (1990) performed 11 scaled pile tests and used a modified Ramberg-Osgood model to approximate the nonlinear soil resistance and displacement behaviour for loose and dense sand. The most commonly used criteria for development of p–y curves for sand were proposed by Reese et al. (1974) but tend to give very conservative results. Bhushan et al. (1981) and Bhushan and Askari (1984) used a different pro- cedure based on full-scale load test results to obtain nonlin- ear p–y curves for saturated and unsaturated sand. A step- by-step procedure for developingp–y curves in sands (Bhushan and Haley 1980; Bhushan et al. 1981) was used to estimate the static unit load transfer curves for different sands below and above the water table. The procedure used to generatep–y curves for sand differs from that suggested for clays. The secant modulus approach is used to approxi- mate soil reactions at specified lateral displacements. The soil resistance in the staticp–y curve model can be calcu- lated using the following equation: [16] p = (k)(x)(y)(F1)(F2) wherek is a constant that depends on the lateral deflectiony (i.e., k decreases asy increases) and relates the secant modu- lus of soil for a given value ofy to depth (Es = kx); x is the depth at which thep–y curve is being generated; and F1 and F2 are density and groundwater (saturated or unsaturated) factors, respectively, and can be determined from Meyer and Reese (1979). The main factors affectingk are the relative density of the sand (loose or dense) and the level of lateral displacement. The secant modulus decreases with increasing displacement and thus the nonlinearity of the sand can be modeled accurately. This analysis assumes a linear increase of the soil modulus with an increase in depth (but varies nonlinearly with displacement at each depth) which is typi- cal for many sands. Equation [16] was used to establish thep–y curve at a given depth. The tangent stiffnesskpy (needed in the time- domain analysis), which represents the tangent to thep–y curve at the specific load level, was then calculated using eq. [15] based on calculated soil reactions from the corre- sponding pile displacements for two consecutive time steps (using eq. [16]). Degradation of soil stiffness Transient loading, especially cyclic loading, may result in a buildup of pore-water pressures and (or) a change of the soil structure that causes the shear strain amplitudes of the soil to increase with an increasing number of cycles (Idriss et al. 1978). Idriss et al. (1978)reported that the shear stress amplitude decreased with an increasing number of cycles for harmonically loaded clay and saturated sand specimens un- der strain-controlled, undrained conditions. These studies suggest that repeated cyclic loading results in the degrada- tion of the soil stiffness. For cohesive soils, the value of the shear modulus afterN cycles,GN, can be related to its value in the first cycle,Gmax, by [17] GN = δ Gmax where the degradation index,δ, is given byδ = N –t, and t is the degradation parameter defined by Idriss et al. (1978). This is incorporated into the proposed model by updating the nonlinear stiffness,kNL, by an appropriate factor in each loading cycle. Time-domain analysis and equations of motion The time-domain analysis was used to include all aspects of nonlinearity and examine the transient response logically and realistically. The governing equation of motion is given by © 2000 NRC Canada 1170 Can. Geotech. J. Vol. 37, 2000 Fig. 4. Determination of stiffness (kpy) from an internally generated staticp–y curve to produce a dynamicp–y curve (including damping). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:48:35 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. [18] [ ]{ &&} [ ]{ &} [ ]{ } { ( )}M u C u K u F t+ + = where [M], [C], and [K] are the global mass, damping, and stiffness matrices, respectively; and{&&}, { &}, { }u u u, and F(t) are acceleration, velocity, displacement, and external load vec- tors, respectively. Referring to Fig. 1, the equations of mo- tion at node 1 (adjacent to the inner field) and node 2 (adjacent to the outer field) are [19] m u c u u k u u F1 1 1 2 1 2 1&& (& & ) ( )+ − + − =NL NL [20] m u c u u k u u F2 2 1 2 1 2 2&& (& & ) ( )− − − − =NL NL whereu1 andu2 are displacements of nodes 1 and 2, respec- tively; F1 is the force in the nonlinear spring including the confining pressure; andF2 is the soil resistance at node 2. The equation of motion for the outer field is written as [21] cu k u F&&2 2 2+ = −L Assuming compatibility and equilibrium at the interface between the inner and outer zones leads to the following equation, which is valid for both sides of the pile: [22] F1 0 = Am B c c k k Bc k Bc k Am B c c 1 2 + + + − − − − + + + ( ) ( L NL NL NL NL NL NL NL L NL L) + k × + − − u u F F i i 1 2 1 1 2 1 where F i1 1− and F i2 1− are the sums of inertia forces and soil reactions at nodes 1 and 2, respectively; andA and B are constants of numerical integration for inertia and damping, respectively. The linear acceleration assumption was used and the Newmarkβ method was implemented for direct time integra- tion of the equations of motion. The modified Newton- Raphson iteration scheme was used to solve the nonlinear equilibrium equations. © 2000 NRC Canada El Naggar and Bentley 1171 Fig. 5. Soil modulus variation for profiles considered in the analysis.d, pile diameter;L, pile depth;z, depth below the surface. Fig. 6. Calculated dynamic soil reactions at 1.0 m depth (for a prescribed harmonic displacement at the pile head with amplitude 0.03d = 0.015 m,L/d = 30). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:48:45 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. Verification of the analytical model Verification of clay model Different soil profiles were considered in the analysis. Figure 5 shows the typical pile–soil system and the soil pro- files considered including linear and parabolic soil profiles. The p–y model was first verified against the hyperbolic model (El Naggar and Novak 1996). Figures 6 and 7 com- pare the dynamic soil reaction and pile head response for both the hyperbolic andp–y curve models for a single rein- forced concrete pile in soft clay. A 0.5 m diameter, 15 m long pile was used with an elastic modulus (Ep ) equal to 35 GPa. A parabolic soil profile with the ratioEp /Es = 1000 at the pile base was assumed. The undrained shear strength of the clay was assumed to be 25 kPa. Figure 6 shows the cal- culated dynamic soil reactions for a prescribed harmonic displacement of a single amplitude equal to 0.03d (0.015 m) at a frequency of 2 Hz at the pile head. Figure 6 shows that the soil reactions obtained from the two models are very similar and approach stability after five cycles. The pattern shown in Fig. 6 is also similar to that obtained by Nogami et al. (1992) showing an increasing gap and stability after ap- proximately five cycles. Figure 7 shows the displacement– time history of the pile head installed in the same soil pro- file. A harmonic load with single amplitude equal to 10 kN was applied at the pile head. The hyperbolic andp–y curve models show very similar responses at the pile head and both stabilize after approximately five cycles. The dynamic soil reactions are, in general, larger than the static reactions because of the contribution from damping. Employing the same definition used for staticp–y curves, dynamicp–y curves can be established to relate pile deflec- tions to the corresponding dynamic soil reaction at any depth below the ground surface. The proposed dynamicp–y curves are frequency dependent. These dynamicp–y curves can be used in other static analyses that are based on thep–y curve approach to approximately account for the dynamic effects on the soil reactions to transient loading. Figures 8 and 9 show dynamicp–y curves established at two different clay depths for a prescribed harmonic displace- ment at the pile head with single amplitude equal to 0.05d, for a frequency range from 0 to 10 Hz. The shear modulus of the soil was assumed to increase parabolically along the pile length. A concrete pile 12.5 m in length and 0.5 m in di- ameter was considered in the analysis. The elastic modulus of the pile material was assumed to be 35 GPa and the ratio Ep /Es = 1000 (at the pile base). Both thep–y curve and hy- perbolic models were used to analyze the pile response. The dynamic soil reaction (normalized by the ultimate pile ca- pacity, Pult) obtained from thep–y curve model compared well with that obtained from the hyperbolic relationship model, especially for lower frequencies, as shown in Figs. 8 and 9, which also show that the soil reaction increased as the frequency increased. This increase was more evident in the results obtained from thep–y curve model. Verification of sand model The p–y curve and hyperbolic models were used to ana- lyze the response of piles installed in sand. The sand was as- sumed to be unsaturated and a linear soil modulus profile was adopted. The pile used in the previous case was consid- ered. Figure 10 shows the calculated dynamic soil reactions at 1 m depth for a prescribed harmonic displacement with an amplitude equal to 0.038d at the pile head with a frequency of 2 Hz. The two models feature very similar dynamic soil reactions. It should be noted that the soil reactions at both sides of the pile are traced independently. The upper part of the curve in Fig. 10 represents the reactions for the soil ele- ment adjacent to the right face of the pile, when it is loaded from the left. The lower part represents the reactions of the soil element adjacent to the left face of the pile as it is loaded from the right. Both elements offer zero resistance to the pile movement when tensile stresses are detected in the nonlinear soil spring during unloading of the soil element on either side. However, the soil nodes remain attached to the pile node at the same level, allowingthe sand to “cave in” and fill the gap. Observations from field and laboratory pile testing confirmed that, unlike clays, sands usually do not experience gapping during harmonic loading. Thus both analyses model the physical behaviour of the soil realisti- cally and logically. The pile head displacement–time histories obtained from the p–y curve and hyperbolic models for a pile installed in a sand with linearly varying elastic modulus due to an applied © 2000 NRC Canada 1172 Can. Geotech. J. Vol. 37, 2000 Fig. 7. Pile head response under applied harmonic load with single amplitude equal to 10 kN (L/d = 30, Ep /Es(L) = 1000, linear profile). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:48:50 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. harmonic load with a single amplitude of 15 kN are shown in Fig. 11, which shows good agreement between the results from the p–y curve model and the hyperbolic model. Figures 12 and 13 show dynamicp–y curves established at two different depths for a prescribed harmonic displacement equal to 0.05d at the pile head for a steel pile driven in sand © 2000 NRC Canada El Naggar and Bentley 1173 Fig. 9. Calculated dynamicp–y curves at 3.0 m depth (for a prescribed harmonic displacement at the pile head with an amplitude of 0.05d) using (a) hyperbolic model, and (b) p–y curve model. Fig. 8. Calculated dynamicp–y curves at 1.5 m depth (for a prescribed harmonic displacement at the pile head with amplitude of 0.05d) using (a) hyperbolic model, and (b) p–y curve model.ρp , pile mass density;ρs , soil mass density;Pult , lateral ultimate load of the pile. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:09 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. for a frequency range from 0 to 10 Hz. The results from both the p–y curve and hyperbolic models displayed the same trend, as shown in Figs. 12 and 13. Validation of dynamic model with lateral Statnamic tests To verify that thep–y curve model can accurately predict dynamic response, it was employed to analyze a lateral Statnamic load test, and the computed response was com- pared with measured values. The test site was located north of the New River at the Kiwi maneuvers area of Camp Johnson in Jacksonville, North Carolina. The soil profile is shown in Fig. 14 and con- sists of medium dense sand extending to the water table, un- derlain by a very weak, gray, silty clay. There was a layer of © 2000 NRC Canada 1174 Can. Geotech. J. Vol. 37, 2000 Fig. 10. Calculated dynamic soil reactions at 1.0 m depth (for a prescribed harmonic displacement at the pile head with an amplitude of 0.038d, L/d = 25, Ep /Es(L) = 1000). Fig. 11. Pile head response to applied harmonic load with single amplitude of 15 kN of the pile lateral capacity (L/d = 25, Ep /Es(L) = 1000, linear profile). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:29 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. gray sand at a depth of 7 m and a calcified sand stratum be- low the gray sand. The pile tested at this site was a cast-in- place reinforced concrete shaft with a steel casing having an outer diameter of 0.61 m and a casing wall thickness of 13 mm. More details on the soil and pile properties and the loading procedure can be found in El Naggar (1998). Statnamic testing was conducted on the pile 2 weeks after lateral static testing was performed. Statnamic loading tests were performed by M. Janes and P. Bermingham, both of Berminghammer Foundation Equipment, Hamilton, Ontario. The computed lateral response of the pile head is com- pared with the measured response in Fig. 15 for two separate © 2000 NRC Canada El Naggar and Bentley 1175 Fig. 12. Calculated dynamicp–y curves at 3.0 m depth (for a prescribed harmonic displacement at the pile head with an amplitude of 0.05d) using (a) hyperbolic model, and (b) p–y curve model. Fig. 13. Calculated dynamicp–y curves at 4.0 m depth (for a prescribed harmonic displacement at the pile head with an amplitude of 0.05d) using (a) hyperbolic model, and (b) p–y curve model. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:43 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. tests with peak load amplitudes of 350 and 470 kN. The agreement between the measured and computed values was excellent, especially for the first load test. The initial dis- placement was slightly adjusted for the computer-generated model to accommodate initial gapping that occurred due to the previous static test performed on the pile. The staticp–y curve for the top soil layer was reduced significantly to model the loss of resistance due to the permanent gap devel- oped near the surface. Dynamic p–y curve generation The dynamicp–y curves presented in Figs. 8–10, 12, and 13 show that a typical family of curves exists related to depth, much like the staticp–y curve relationships. Thus, dy- namic p–y curves could be established at any depth and be representative of the soil resistance at this specific depth. In this study, they were obtained at a depth of 1.5 m which was found to illustrate the characteristics of the dynamicp–y curves. More dynamic p–y curves were generated using pre- scribed harmonic displacements applied at the pile head which allowed for the development of plastic deformation in the soil along the top quarter of the pile length. Steel pipe piles were considered in the analysis. It was assumed that sand had a linear soil profile and the clay had a parabolic profile to match the soil profile employed to derive the static p–y curves used in the analysis. The soil shear wave velocity profiles and the pile properties are given in Fig. 16. The tests were divided into two separate cases involving clays (case I) and sandy soils (case II). Table 1 summarizes the character- istics of each case and relevant pile and soil parameters. The dynamicp–y curves were generated over a frequency range © 2000 NRC Canada 1176 Can. Geotech. J. Vol. 37, 2000 Fig. 14. Soil profile and Statnamic pile test setup at Camp John- son, Jacksonville. SPT, standard penetration test. Test case no. Soil type cu (kPa) Dr (%) φ (°) ν d (m) L/d Ep /Es Gmax (kPa) Vs (m/s) Case I (clays) C1 Soft clay <50 0.45 0.25 40 10 000 6.6×106 70 C2 Medium clay 80 0.45 0.25 40 4 500 1.6×107 150 C3 Stiff clay >100 0.45 0.25 40 1 600 8.3×107 200 Case II (sands) S4 Loose sand, saturated 35 32 0.3 0.25 40 6 300 1.2×107 70 S5 Medium sand, saturated 50 34 0.3 0.25 40 3 800 2.0×107 100 S6 Medium sand, saturated 50 34 0.3 0.50 20 3 800 2.0×107 100 S8 Dense sand, saturated 90 38 0.3 0.25 40 1 580 4.7×107 150 S9 Dense sand, unsaturated 90 38 0.3 0.25 40 790 9.7×107 220 Note: All values represent those calculated at a depth of 1.5 m. Table 1. Description of parameters used for each test case. β Soil type Description α ao < 0.025 ao > 0.025 κ n Soft clay cu < 50 kPa;Vs < 125 m/s 1 –180 –200 80 0.18 Medium clay 50 <cu < 100 kPa; 125 <Vs < 175 m/s 1 –120 –360 84 0.19 Stiff clay cu > 100 kPa;Vs > 175 m/s 1 –2900 –828 100 0.19 Medium dense sand, saturated 50 <Dr < 85%; 125 <Vs < 175 m/s 1 3320 1640 –1000.1 Medium dense sand, unsaturated 50 <Dr < 85%; 125 <Vs < 175 m/s 1 1960 960 –20 0.1 Dense sand, saturated Dr > 85%; Vs > 175 m/s 1 6000 1876 –100 0.15 Note: See text and eq. [23]. Table 2. Dynamicp–y curve parameter constants for a range of soil types (d = 0.25; L/d = 40; 0.015 <ao < 0.225, whereao = ωro /Vs). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:46 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. of 0–10 Hz (2 Hz intervals) for different classifications of sand and clay based on standard laboratory and field mea- surements (standard penetration test value, relative density, cu, etc.). All results were obtained after one or two cycles of harmonic loading. Results and discussion The results from the computational model showed a gen- eral trend of increasing soil resistance with an increase in the load frequency. The dynamicp–y curves obtained seem to have three distinct stages or regions. The initial stage (at small displacements) shows an increase in the soil resistance (compared with the staticp–y curve) that corresponds to in- creasing the velocity of the pile to a maximum. This in- crease in the soil resistance is larger for higher frequencies. In the second stage, the dynamicp–y curves have almost the same slope as the staticp–y curve for the same displace- ment. This stage occurs when velocity is fairly constant and consequently the damping contribution is also constant. The third stage of the dynamicp–y curve is characterized by a slope approaching zero as plastic deformations start to occur (similar to the staticp–y curve at the same displacement). There is also a tendency for the dynamic curves to converge at higher resistance levels approaching the ultimate lateral resistance of the soil at depthx, Pu (determined from Ameri- can Petroleum Institute 1991). The overall relationship between the dynamic soil resis- tance and loading frequency for each test was established in the form of a generic equation. The equation was developed from an regression analysis relating the staticp–y curve, fre- quency, and apparent velocity,ωy, so [23] P P a a y d n d s o 2 o= + + α β κ ω , P P xd u at depth≤ © 2000 NRC Canada El Naggar and Bentley 1177 Fig. 15. Pile head displacement for Statnamic test with peak load of (a) 350 kN, and (b) 470 kN. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:52 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. wherePd is the dynamicp–y curve at depthx (N/m); Ps is the static soil reaction (obtained from the staticp–y curve) at depthx (N/m); ao is the dimensionless frequency =ωro/Vs; ω is the frequency of loading (rad/s);d is the pile diameter (m); y is the lateral pile deflection at depthx when soil and pile are in contact during loading (m); andα, β, κ, andn are constants determined from curve fitting eq. [23] to the com- puted dynamicp–y curves from all cases considered in this study. A summary of the best-fit values for the constants is provided in Table 2. The constantα is taken equal to unity to ensure thatPd = Ps for ω = 0. For large frequencies or dis- placements, the maximum dynamic soil resistance is limited to the ultimate lateral resistance of the soil,Pu. Figures 17 and 18 show dynamicp–y curves established using eq. [23] and the best-fit constants (as broken lines). The approximate dynamicp–y curves established from eq. [23] represented soft medium clays and loose medium dense sands reasonably well. However, the accuracy is less for stiffer soils (higherVs values). The precision of the fitted curves also increases with an increase in frequency (ω ≥ 4 Hz) where the dynamic effects are important. The low accuracy at a lower frequency (ao < 0.02) may be attributed to the ap- plication of the plane strain assumption in the dynamic anal- ysis. This assumption is suitable for higher frequencies, as the dynamic stiffness of the outer field model vanishes for ao < 0.02 due to the assumption of plane strain. Test case C1 was also used to obtain dynamicp–y curves at depths of 1.0 and 2.0 m to examine the validity of eq. [23] to describe the dynamic soil reactions at other depths along the soil profile. The results showed that eq. [23] (using the constants in Table 2) predicted the dynamic soil reactions reasonably well. Development of a simplified model For many structural dynamic programs, soil–structure in- teraction is modeled using staticp–y curves to represent the soil reactions along the pile length. However, the use of static p–y curves for dynamic analysis does not include the effects of velocity-dependent damping forces. The dynamic p–y curves established using eq. [23] and the parameters given in Table 2 allow for the generation of different dy- namicp–y curves based on the frequency of loading and soil profile. Substituting dynamicp–y curves in place oftraditional static p–y curves for analysis should result in better esti- mates of the response of structures to dynamic loading. Alternatively, the dynamic soil reactions can be repre- sented using a simple spring and dashpot model. This model can still capture the important characteristics of the nonlin- ear dynamic soil reactions. A simplified dynamic model that can be easily implemented into any general finite element program is proposed herein. Complex stiffness model As discussed previously, eq. [23] can be used directly to represent the dynamic relationship between a soil reaction and a corresponding pile displacement. The total dynamic soil reaction at any depth is represented by a nonlinear spring whose stiffness is frequency dependent. A more conventional and widely accepted method of cal- culating dynamic stiffness is through the development of the complex stiffness. The complex stiffness has a real partK1 and an imaginary partK2, i.e., [24] Pd = Ky = (K1 + iK2)y The real part,K1 represents the true stiffness,k, and the imaginary part of the complex stiffness,K2, describes the out-of-phase component and represents the damping due to the energy dissipation in the soil element. Because this damping component generally grows with frequency (resem- bling viscous damping), it can also be defined in terms of the constant of equivalent viscous damping (the dashpot constant) given byc = K2 /ω . Then the dynamicp–y curve relation can be described as [25] P k i c y ky cyd = + = +( ) &ω in which bothk and c are real and represent the spring and dashpot constants, respectively; and&y = dy/dt is the velocity. © 2000 NRC Canada 1178 Can. Geotech. J. Vol. 37, 2000 Fig. 16. Description of soil and pile properties for cases I and II. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:55 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. Using eq. [23], the dynamicp–y curve can be written in the form of eq. [24], i.e., [26] P K iK yd = + =( )1 2 P y i P a a y d y n s s o 2 o α β κ ω + + y The stiffness and damping constants are then calculated as [27] k K P y = =1 sα [28] c K P a a y d y n = = + 2 ω β κ ω ω s o 2 o The complex stiffness can be generated at any depth along the pile using the staticp–y curves and eqs. [27] and [28]. Complex stiffness constants: soft clay example The complexstiffness constants were calculated for test case C1 (see Table 2) using the method described in the pre- vious section. The values of the true stiffness,k, were ob- tained for the range of displacements experienced by the pile for the frequency range from 0 to 10 Hz. The stiffness pa- rameter (S1)py was defined as [29] (S1)py = k G py max © 2000 NRC Canada El Naggar and Bentley 1179 Fig. 17. Dynamic p–y curves and staticp–y curve for test case C1 (depth = 1.0 m). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:57 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. http://nrc.literatumonline.com.nrc.literatumonline.com/action/showImage?doi=10.1139/t00-058&iName=master.img-006.png&w=356&h=434 The constant of equivalent damping,c, was obtained by averaging the value from eq. [28] for the range of velocities experienced by the pile for each frequency of loading. Then the equivalent damping parameter (S2)py was defined as [30] (S2)py = cV G r s o ω max Figure 19 shows the true stiffness calculated from the static p–y curve and this stiffness is identical at all loading frequencies considered. There is a definite trend of decreas- ing stiffness with increased displacement due to the soil nonlinearity. The constant of equivalent damping presented in Fig. 20 decreases with a decrease in frequency which can be attributed to separation at the pile–soil interface. The val- ues from Figs. 19 and 20 can be directly input into a finite element program as spring and dashpot constants to obtain the approximate dynamic stiffness of a soil profile similar to that of test case C1. Implementing dynamic p–y curves in ANSYS A pile and soil system similar to that of test case C1 was modeled using the commercial finite element program ANSYS (1996) to verify the applicability and accuracy of the dynamicp–y curve model in a standard structural analy- sis program. A dynamic harmonic load with peak amplitude of 100 kN at a frequency of 6 Hz was applied to the head of the same steel pipe pile used in test case C1. The soil stiff- ness was modeled using three procedures: (1) staticp–y curves; (2) dynamicp–y curves using eq. [23]; and (3) a complex stiffness method using equivalent dampingconstants. The pile head response for each test was obtained and com- pared with the results from the two-dimensional (2D)p–y curve model. The pile was modeled using two-noded beam elements and was discretized into 10 elements that increased in length with an increase in depth. At each pile node, a spring or a spring and a dashpot was attached to both sides of the pile to represent the appropriate loading condition at the pile–soil © 2000 NRC Canada 1180 Can. Geotech. J. Vol. 37, 2000 Fig. 18. Dynamic p–y curves and staticp–y curve for test case S5 (depth = 1.5 m). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:49:59 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. http://nrc.literatumonline.com.nrc.literatumonline.com/action/showImage?doi=10.1139/t00-058&iName=master.img-007.png&w=365&h=440 interface. The pile and soil remained connected and had equal displacement for compressive stresses. The spring or the spring and dashpot model disconnects if tensile stress is detected in the soil, allowing a gap to develop. The soil was first modeled using nonlinear springs with force–displacement relationships calculated directly from staticp–y curves. The soil stiffness was then modeled using the approximate dynamicp–y curve relationship calculated for test case C1 using Table 2. The last test considered a spring and a dashpot in parallel. The pile head response for each test is shown in Figs. 21 and 22 along with the calculated response from the 2D analyticalp–y model. Figure 21a shows that the static p–y curves model computed larger displacements with increasing amplitudes as the number of cycles in- creased. Figure 21b shows that the response computed us- ing the dynamicp–y curve model was in good agreement with the response computed using the 2D analytical model. The results obtained using the complex stiffness model are presented in Fig. 22a and show a decrease in displacement amplitude. The overdamped response can be attributed to using an average damping constant, which overestimates the damping at higher frequencies and large nonlinearity. Figure 22b shows the response of the 2D model compared to the complex stiffness approach with the average damping constant reduced by 50%. The re- sults show that the response in this case is in good agree- ment with the response computed using the 2D analytical model. Conclusions A simple 2D analysis was developed to model the re- sponse of piles to dynamic loads. The time domain was © 2000 NRC Canada El Naggar and Bentley 1181 Fig. 20. Equivalent damping parameter for test case C1 (soft clay) with dimensionless frequency,ao. Fig. 19. True stiffness parameter for test case C1 (soft clay). I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:50:05 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. chosen to efficiently model the transient nonlinear response of the pile–soil system. Staticp–y curves were used to gen- erate the nonlinear soil stiffness in the frame of a Winkler model. The piles were assumed to be vertical and circular and were modeled using standard beam elements. A practi- cally accurate and computationally efficient model was de- veloped to represent the soil reactions. This model accounted for soil nonlinearity, slippage and gapping at the pile–soil interface, and viscous and material damping. Dynamic soil reactions (dynamicp–y curves) were gener- ated for a range of soil types and harmonic loading with varying frequencies applied at the pile head. Closed-form solutions were derived from regression analysis relating the static p–y curve, dimensionless frequency, and apparent © 2000 NRC Canada 1182 Can. Geotech. J. Vol. 37, 2000 Fig. 21. Calculated pile head response using 2D analytical model compared with ANSYS using (a) static p–y curves, and (b) dynamic p–y curves. Fig. 22. Calculated pile head response using 2D analytical model compared with ANSYS using (a) complex stiffness, and (b) modified complex stiffness. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:50:51 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. © 2000 NRC Canada El Naggar and Bentley 1183 velocity of the soil particles. A simple spring and dashpot model was also proposed whose constants were established by splitting the dynamicp–y curves into real (stiffness) and imaginary (damping) components. This model could be used in equivalent linear analysis for harmonic loading at the pile head. The proposed dynamicp–y curves and the spring and dashpot model were incorporated into a commercial finite element program, ANSYS, that was used to compute the re- sponse of a laterally loaded pile. The computed responses compared well with the predictions of the 2D analysis. The following conclusions were drawn from the study: (1) The developed pile–soil model is capable of analyzing the response of piles to lateral transient loading accountingfor the nonlinear behaviour of the soil and energy dissipa- tion. (2) The soil resistance to the pile motion increases with the frequency of the pile head loading (inertial loading) for single piles. (3) The developed dynamicp–y curve can represent the dynamic soil reactions using a predefined staticp–y curve. (4) The pile head response under harmonic loading can be approximately modeled using dynamicp–y curve functions in most of the available structural analysis programs. (5) The implementation of the dynamicp–y curves in time-domain analyses to model the soil reactions is preferred because it accounts for the variation in damping with the displacement level. The damping constant in the complex stiffness model must be chosen accounting for the level of nonlinearity expected. Acknowledgements This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) to the first author and research contract 24-9 from the National Cooperative Highway Research Program (NCHRP). Both sources of support are greatly appreciated. References Abendroth, R.E., and Greimann, L.F. 1990. Pile behavior estab- lished from model tests. 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Reese, L.C., Cox, W.R., and Koop, F.D. 1975. Field testing and analysis of laterally loaded piles in stiff clay.In Proceedings of the 7th Annual Offshore Technology Conference, Houston, Tex., Vol. 2, pp. 671–690. I:\cgj\Cgj37\Cgj06\T00-058.vp Thursday, November 30, 2000 11:50:51 AM Color profile: Generic CMYK printer profile Composite Default screen C an . G eo te ch . J . D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y IN D I N ST O F T E C H o n 04 /2 7/ 14 Fo r pe rs on al u se o nl y. This article has been cited by: 1. Mehdi Heidari, Hesham El Naggar, Mojtaba Jahanandish, Arsalan Ghahramani. 2014. Generalized cyclic p–y curve modeling for analysis of laterally loaded piles. Soil Dynamics and Earthquake Engineering 63, 138-149. [CrossRef] 2. Swagata Bisoi, Sumanta Haldar. 2014. Dynamic analysis of offshore wind turbine in clay considering soil–monopile–tower interaction. Soil Dynamics and Earthquake Engineering 63, 19-35. [CrossRef] 3. V. Zania. 2014. 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