Analysis of Pile Foundations under Seismic Loading
Jayram Ramachandran
CBE Institute 2005: Final Report
1
Intr Introd odu ucti ction
Pile foundations are commonly used to transfer loads from a structure to the ground in cases where the structural loads are very high or at locations where the soil at shallow depths cannot carry the imposed loads. Under Under modera mo derate te and strong strong seismic seismic loading, pile foundations foundations undergo large displacements and the behavior of the pile-soil system can be strongly nonlinear. linear. Such Such pile-soil pile-soil systems systems interac interactt with the supported supported structure; structure; hence hence their nonlinear nonlinear behavior is extremely important in evaluating the seismic response of pile-supported structures. tures. In seismic regions, regions, the analysis analysis and design of pile-supported pile-supported structures structures requires requires an accurate prediction of the pile head response and the load resistance to lateral shaking caused by earthquake ground motions. In order to predict the response of pile groups under seismic loadin loading, g, it is importa important nt to und under ersta stand nd the interac interactio tion n betwe between en the piles piles in a group. group. Th This is research studies the interaction between piles in pile groups accounting for soil nonlinearity and pile-soil gapping effects. Most pile foundations consist of a group of piles rather than a single pile. Thus, an important component of the analysis of pile supported structures is the ability to perform an accurate and efficient efficient analysis analysis of pile groups. Howev However, er, most pile group analysis techniqu techniques es are an extension extension of technique techniquess used to predict the response of single piles. Therefor Therefore, e, Section Section 2 presents presents a brief review of single pile analysis techniques. The Beam on Winkler Foundation model is a computationally efficient tool to estimate the response of single piles accounting for pile and soil nonlinearity nonlinearity.. A number number of researc researchers hers have used this technique technique and hence, hence, this model is briefly briefly described. described. Section Section 3 presen presents ts an overview overview of the current current techniqu techniques es available to perform pile group analysis. These include elasticity based methods, wave propagation agation based methods methods and the p-mult p-multipl iplier ier method. method. Sectio Section n 4 presen presents ts the results results of a numerical study using 3D finite element analysis to investigate pile-to-pile interaction in a pile group, accounting for soil nonlinearity and pile-soil gapping effects.
1
2
Sing Single le Pile Pile Anal Analys ysis is
Most pile group analysis techniques are an extension of methods employed to compute single pile response. Therefor Therefore, e, it is highly instructiv instructivee to review techniq techniques ues used to compute single pile response, including the various numerical and analytical methods that have been developed developed.. Closed Closed form analytical analytical solutions solutions have been obtained obtained by a nu number mber of researc researchers hers for single piles under axial or lateral loading assuming the soil to be linear viscoelastic and the pile pile to be a linear linear elasti elasticc beam [15, 17, 18, 24]. Such Such closed closed form soluti solutions ons are hard to extend to reflect the true nonlinear behavior behavior of the pile-soil pile-soil system. Some researc researchers hers include soil nonlinearity in the analysis using an equivalent viscous dashpot [14]. The most direct method of analysis involves 2D/3D modeling of the pile and the soil continuum using the finite element element method. In such an analysis, analysis, the soil nonlineari nonlinearity ty may be considere considered d explicitly by using appropriate material models [12, 37]. The finite element method, although powerful, is computationally very expensive since it requires discretization of the pile and the large volume volume of surroundin surroundingg soil. In contrast contrast,, the Beam-on-Winkle Beam-on-Winkler-F r-Foundat oundation ion model mo del is a versatile and efficient approach to pile foundation analysis that has been adopted widely. This section reviews the Winkler model.
2.1 2.1
Beam Beam on Win Winkl kler er Fou Found ndat atio ion n Model Model
The Beam on Winkler Foundation Model model or Winkler model is commonly used to study study the respons responsee of single single piles. piles. The pile is modeled modeled as a beam while while the surrou surroundi nding ng soil is modeled modeled using continuo continuously usly distributed distributed springs and dashpots. dashpots. Pile nonlinearit nonlinearity y may be consid consider ered ed in the analysis analysis using using an app approp ropria riate te non nonlin linear ear material material model. model. Hyster Hystereti eticc energy loss in the soil is modeled using nonlinear springs whose parameters are determined experimentally from load-deformation curves [21, 22] or analytically by defining stiffness and strengt strength h parame parameter terss [3, 7, 8, 28]. 28]. The dashpots dashpots place placed d in parall parallel el with the nonline nonlinear ar
2
Figure 1: Beam on Winkler foundation model for a single pile under lateral loading springs account for energy loss due to radiation damping, under dynamic loading conditions. Since radiation damping is a frequency dependent phenomenon, the dashpot coefficients are usually usually frequenc frequency y dependen dependent. t. The Winkler Winkler model is easily easily implemen implemented ted for analysis analysis of nonlinear linear pile-soil pile-soil systems systems using most standard 1-D finite elemen elementt programs. programs. Within Within the finite element framework, the pile is modeled using discrete beam elements and the continuous soil springs/dash springs/dashpots pots are replaced replaced by discrete discrete springs springs and dashpots. Figure Figure 1 shows shows the discretized form of the Winkler model for the analysis of a single pile under dynamic (including seismic) seismic) lateral lateral loading. A similar model may be used for the analysis of single piles under static static lateral lateral loads. loads. Howe Howev ver, und under er static static loads, loads, the dashpots dashpots would would be irrele irrelev vant. ant. Th Thee
3
Winkler model has been shown to provide a good estimate of the pile response by several researchers [3, 32].
2.2 2.2
Model Model para parame mete ters rs
The Winkler model requires definition of parameters for the nonlinear springs as well as viscous viscous dashpots. This section section defines defines the parameters parameters for the Winkler Winkler model under lateral lateral loading. loading. The nonlinear nonlinear spring spring may be represen represented ted using either either the bilinear bilinear or Bouc-W Bouc-Wen [35] models. The bilinear bilinear model is a very very simple representa representation tion of the hysteret hysteretic ic behavior behavior of the soil, but its main attraction is that it can be easily implemented in most finite element programs. The Bouc-Wen model is much more versatile as it allows a smooth transition from linear to nonlinear behavior. However, it is not implemented in a number of commonly used analysis analysis programs. Both the bilinear bilinear and Bouc-W Bouc-Wen models require require definition definition of the initial initial spring stiffness and strength at large displacements (assuming a zero post yield stiffness [3]). Based on a number of analytical studies that compare pile head displacements from Winkler model analyses with 3-D finite element analyses, the stiffness of the soil springs per unit length under lateral loading can be approximated realistically by [14, 17]:
kx = 1.2E s (z)
(1)
where kx is stiffness of the continuously distributed spring and E s (z) is the Young’s modulus of the soil soil as a functi function on of depth depth z. Th Thee value alue in equation equation 1 is multipl multiplied ied by the spacin spacingg between two adjacent springs to obtain the discrete spring stiffness. The maximum force in the spring is equal equal to the ultimate ultimate lateral reaction reaction per unit length of pile. For cohesive cohesive soils, the lateral soil strength based on theoretical studies [3, 8, 28] using the theory of plasticity
4
under plane-strain conditions is given by:
F s,max s,max (z ) = λS u (z )d
(2)
where F s,max s,max (z ) is the maximum lateral force exerted by the soil on the pile (as a function of depth, z), d is the pile diameter, S u (z) is the variation of shear strength with depth and dimensionless less parameter. parameter. Considera Considerable ble research research has been b een done to quantify quantify λ [8, 19]. λ is a dimension λ values between 9 and 12 may be appropriate at depths where plane strain conditions are valid. At shallow shallow depths, plane strain strain conditions conditions are not valid due to vertic vertical al deformati deformation on of the soil during lateral motion of the pile. Hence, λ values of 2 or 3 have been suggested. The following expression for the variation of λ with depth is recommended [19]:
λ(z ) = 3 +
σx z + J S u d
(3)
where σx is the overburden pressure and J is a co-efficient that is obtained by calibrating the model against known experimen experimental tal data. In the absence absence of such such data, the recommended recommended value of J = 0.5 may be used. For cohesionless soils, the following expression for the limiting force of the soil on the pile [7] is used: F s,max s,max (z ) = µγ s d
1 + sin φ z 1 − sin φ
(4)
where F s,max s,max (z ) is the maximum lateral force exerted by the soil on the pile (as a function of depth, z), γ s is the specific weight of the soil, φ is the angle of internal friction and µ is a dimensionless parameter. A value of µ=3 has been suggested [7]. The maximum force (or strength) of the spring is obtained by multiplying the maximum reaction force of the soil in equations 2 and 4 with the spacing between two adjacent springs, s. A number of expressions expressions are availabl availablee for the dashpot coefficient coefficient [14, 34]. The expression expression
5
used in this study defines a frequency dependent dashpot coefficient as used by [3]: c(a0 , z) = Qa0 0.25 ρs V s ds −
(5)
where c is the dashpot coefficient, a0 is a non-dimensional frequency parameter (= ωd/V s), ω is the angular frequency, ρs is the soil density, V s is the shear wave velocity in the soil medium, d is the pile diameter, and s is the spacing between between two two adjacen adjacentt dashpots. The coefficient Q is given by the expression:
3 Q= 2[1 +
if z < 3d, 3.4 1.25 π 0.75 4 π (1−ν s
]
( )
where ν s is the Poisson’s ratio of the soil.
6
if z > 3d.
(6)
3
Pile Pile Grou Group p Anal Analys ysis is
Most pile foundations consist of a group of piles rather than a single pile. Thus, an important component of the analysis of pile supported structures is the ability to perform an accurate and efficient analysis of pile groups. Just as for single piles, various numerical and analytical methods methods have have been developed developed for the analysis analysis of pile groups. groups. Curren Currently tly,, the most accurate method of pile group analysis may be the 3D finite element analysis method. Such analyses have been performed under static [11, 38] as well as dynamic (including seismic) [16] loading conditions conditions.. Howev However, er, solving any problem problem with the 3D finite elemen elementt method method is extremely extremely time consuming and hence numerous attempts have been made to find more computationally efficien efficientt solution solution procedures procedures.. A quasi-thr quasi-three-d ee-dimen imensional sional method is presen presented ted in [36], where nonlinear dynamic analysis of single piles and pile groups is performed in the time domain using strain-depen strain-dependen dentt moduli and damping, damping, and yielding yielding at failure. Since Since most pile groups are made up of identical piles and the soil profile at the locations of all piles in the group may also be assumed to be identical, one of the most common techniques of pile group analysis involves modifying the response of a single pile by a suitable factor to obtain the respo respons nsee of the the grou group p [13, [13, 15, 17, 27]. 27]. Th Thee respo respons nsee of the the sing single le pile pile may may be obta obtain ined ed using any of a wide range of techniques, but, most commonly a Winkler model is used, as discussed in Section 2. The resistance provided by the group under vertical or lateral loading is generally generally not equal equal to the sum of the resistances resistances of the individual individual piles. Most often, often, the group resistance is less than the sum of the individual pile resistances and is a function of the pile group configur configurati ation on as well as pile pile spacin spacing. g. A method method for analysis analysis of pile pile groups groups using interaction factors and based on the theory of elasticity was proposed by Poulos and Davis Da vis [27]. Th This is method method is app applic licabl ablee to pile groups groups in elasti elasticc soils soils und under er static static loadin loadingg conditions. Approximate Approximate analytical closed form solutions for pile-to-pile interaction factors under dynamic loads, based on frequency domain solutions are also available [13, 15, 17].
7
These solution procedures have also been modified to accommodate time domain analysis procedures procedures for nonlinear nonlinear problems [4]. This section section reviews reviews some of the curren currentt procedures procedures used to analyze pile groups.
3.1 3.1.1 3.1.1
Use Us e of pile-t pile-to-p o-pile ile inter interact action ion factor factorss in pile pile group analy analysis sis Elasticit Elasticity y based based solution solution
Poulos and Davis [27] developed a method of analysis of pile groups using single pile analysis results along with pile-to-pile interaction factors. These solutions are based on elasticity and apply to pile groups under static loading. Separate but similar procedures were developed for the analysis of pile groups under both axial and lateral loading conditions. In this method, the response of each pile in a group is a sum of the response of the pile if it were a single pile and the additional displacements induced in the pile due to its interaction with each of the other piles in the group. group. In computing computing the response response of the single pile under lateral loads, the pile is assumed to be a thin rectangular vertical strip of constant flexibility and the soil is considered to be a continuum. The pile is divided into a number of elements, each acted upon by a uniform horizontal stress, which is assumed to be constant across the width of the pile. The soil is assumed to be an ideal, homogeneous, isotropic, semi-infinite elastic material with properties properties unaffected unaffected by the presenc presencee of the pile. pile. A solution solution is obtained obtained by imposing compatibility between the displacements of the pile and the soil for each element of the pile. The soil displacement influence factors, relating the soil displacements with the horizontal stress, are evaluated by integration over a rectangular area of the Mindlin equation for the horizontal displacement of a point within a semi-infinite mass caused by a horizontal point load within the mass. This solution is also available for piles in soils with Young’s modulus varying arying with depth. depth. The soil displacemen displacementt influence influence factors are a function function of the pile flexibility factor which measures the relative pile to soil stiffness, pile length to diameter ratio
8
and the Poisson’ Poisson’ss ratio of the soil. These These influence influence factors have have been computed computed consideri considering ng the effect effect of the differen differentt paramete parameters rs [25, 27]. The procedures procedures used in the elasticit elasticity y based analysis of single piles may be found in great detail in the references cited. A major advantage of the elasticity based method of analysis is that since the soil is treated as a contin continuum uum,, it is possibl possiblee to consider consider the effec effects ts of inte interac ractio tion n betw between een piles. piles. The method employed to compute the pile group response is an extension of the procedure used to compute the response response of single piles piles [26]. The interactio interaction n between between two identical identical piles is considere considered d first and the analysis is then extended extended to general pile groups. The two two identiidentical, equally equally loaded loaded piles piles are each divided divided into a certain certain number number of elements. elements. While elastic elastic conditions prevail, the horizontal displacements of the soil and pile at each element may be equated, and together with the equilibrium equations may be solved for the unknown pressure pressures. s. In this analysis, analysis, the only interact interaction ion effect that is considere considered d is the horizont horizontal al move moveme ment nt of one pile that that result resultss from from loadin loadingg on anothe anotherr pile. pile. Th Thee inter interact action ion factor factorss are expressed as a ratio of the additional displacement caused by the adjacent pile to the displacem displacemen entt of the pile caused by its own loading. loading. Interac Interaction tion factors as a function function of the spacing between piles, pile flexibility factor, departure angle (for laterally loaded piles – the angle between the horizontal line joining the two piles and the line of action of the load), pile head fixity condition, pile length to diameter ratio and the Poisson’s ratio of the soil are avail availabl ablee [27]. [27]. Th Thee method method of analys analysis is of pile pile groups groups with two two piles piles is exten extended ded to a gengeneral pile group using the principle of superposition. The additional horizontal displacement caused in a particular pile in the group, due to the loading on every other pile in the group is computed by considering the interaction between piles in pairs and summing the additional displacements caused by each of the adjacent piles in the group. One of the major disadvantages of this method is that it is hard to extend this method to compute the load-deformation response of pile foundations with significant nonlinear demands. mands. It is also hard to analyze the foundation foundation under complex complex loading scenarios scenarios such such as 9
seismic or other dynamic loads.
3.1.2 3.1.2
Wav ave e propag propagatio ation n based based soluti solution on
Wave propagation based methods use wave propagation theory to compute the interaction between between piles in groups under dynamic loads. These interaction interaction factors are combined with the single pile response computed using any procedure (usually Winkler model type procedure) to comput computee the response response of pile pile groups groups.. Th These ese procedure proceduress were original originally ly develo developed ped in the frequency domain and applied to foundation systems in which the pile response may be considered elastic and the soil is modeled as a linear viscoelastic medium. Dobry and Gazetas developed a simple analytical solution for the dynamic stiffness and damping of floating pile groups [13] subjected to harmonic loading at the pile head. The vibration of each pile causes the generation of waves that propagate in all directions, causing additional displacements in the adjacent piles. The vibrating pile is referred to as the active pile, while the adjacent pile is referred to as the passive pile. Dobry and Gazetas introduced some simplifying assumptions on the nature of the generated wave field and the consequent displacement of adjacent piles. They used closed form expressions for the generated wave field (in frequency domain) and assumed that the additional displacement caused in the passive pile is equal to this dynamic displacem displacemen entt field. field. For piles under under axial loading, loading, the interacti interaction on factor factor is derived derived from the asymptotic cylindrical wave expression [23] that was assumed for the generated wave field around a vibrating pile:
αv (S ) ≈ (
−βωS −iωS d 0.5 βω S ) exp( ) exp( exp( ) 2S V s V s
(7)
where αv (S ) is the pile interaction factor, S is the pile spacing, d is the pile diameter, β is the soil damping, ω is the angular frequency of the input loading, and V s is the shear wave velocit velocity y in the soil medium. medium. For laterally laterally loaded loaded piles, piles, the interacti interaction on factor factor is a function function of
10
the angle θ between the line of the two piles and the direction of the horizontal force applied. However, it is sufficient to compute the interaction factor for θ = 0 and 90 , and then use ◦
◦
the following expression to compute the interaction factor for any angle θ [26]: αh (S, θ) ≈ αh (S, 0 )cos2 θ + αh (S, 90 )sin2 θ ◦
(8)
◦
To compute the interaction factors for θ = 0 and 90 , it was assumed that the 90 pile ◦
◦
◦
is affected only by S-waves which emanate from the active pile with a velocity V s , while the 0 pile is affected only by compression-extension waves that emanate from the active ◦
pile and travel with a velocity V La La . V La La is the so called Lysmer’s analog velocity, V La La = 3.4V s /[π(1 − ν )]. )]. Thus, the following expressions were proposed: −βωS −iωS d 0.5 βω S ) exp( )exp( ) 2S V s V s −βωS −iωS d βω S )exp( ) αh (S, 0 ) ≈ ( )0.5 exp( 2S V La V La La La
αh (S, 90 ) ≈ ( ◦
(9)
◦
(10)
The work by Dobry and Gazetas was carried forward by Gazetas and Makris [15, 17]. They retained the expressions for the wave field generated by the vibration of the active pile. Howev However, er, they assumed that the adjacent adjacent pile modifies the arriving wave wave field ( us(ω, z )) depending upon its relative flexural rigidity and the vertical fluctuations of the arriving wave ave field. field. The additiona additionall displa displace cemen ments ts in the passiv passive pile pile are approxim approximate ately ly equal equal to the arriving wave field for long flexible piles and smoothly varying us (ω, z ). The additional displacements are nearly zero for rigid piles and rapidly varying us (ω, z). However, in general, the additional additional response response will be something something in between between these two extremes. extremes. These These additional additional displacements may be computed by applying us (ω, z ) as a support excitation to the passive pile. This may be accomplished using the Winkler model [3, 17]. The use of this technique as suggested above is restricted to linear systems, where nonlinear
11
effects may be included indirectly in the form of equivalent viscous damping to account for energy energy dissipatio dissipation. n. Howev However, er, the wave wave propagation propagation based technique technique has been modified to obtain time domain expressions expressions for the wa wave ve field generate generated d by the active active pile. This allows the use of time time domain domain procedu procedures res in comput computing ing pile pile groups groups respons response, e, thereb thereby y allow allowing ing nonlinear nonlinear effects effects to be included included directly directly using nonlinear nonlinear material material models [3, 4]. Although Although the time domain procedures allow the solution of a larger class of problems, they still have some limitations. The consideration of nonlinear effects is restricted to computing the single pile response. response. The pile-to-pile pile-to-pile interact interaction ion factors are based on Fourier Fourier transforms transforms of the closed closed form expressions expressions obtained in frequenc frequency y domain. domain. Th Thus, us, the effect of soil nonlinearit nonlinearity y on pile-to-pile interaction is not considered.
3.2
Pile Pile group group analy analysis sis usin using g p-y multi multipli pliers ers
Another Another class of pile group analysis techniqu techniques es involv involves es the use of p-y multiplie multipliers. rs. These These procedures are an extension of the Winkler model technique used for single piles (Section 2). In the Winkler model, the soil is replaced by discrete springs (Figure 1) and soil nonlinearity is represented using a nonlinear load-deformation curve referred to as the p-y curve for lateral load response. response. These These p-y curves curves may be generate generated d using analytical analytical expressions expressions for stiffness stiffness and strength of the p-y curve along with standard hysteretic models such as the bilinear and Bouc-Wen models, as suggested in Section 2. However, p-y curves most commonly are obtained from back calculations based on experimental or 3D finite element response data [12, 19, 21, 30, 31, 37]. The Winkler model for single piles may be extended for analysis of pile groups. groups. In such a model, mo del, each each pile is modeled as a beam supported supported on horizont horizontal al springs that account account for soil response. The nonlinear nonlinear response of each each soil spring is represen represented ted by a p-y curve. The p-y curves used in pile group analysis are obtained by modifying p-y curves for single pile analysis using p-y multipliers. The effect of pile-to-pile interaction on soil p-y curves curves has been studied by a number number of researc researchers hers [9, 10, 11, 38]. This techniqu techniquee of pile 12
group analysis is widely used and is also incorporated into software programs. However, the major drawback of this pile foundation model is that though it is computationally much more efficient than 3D finite element models, it still requires extensive modeling and may take considerable analysis time, particularly for large pile groups under dynamic loads. Thus, the use of such models may not be very efficient for certain applications such as vulnerability analysis.
13
4
Pile-t Pile-to-P o-Pile ile Inter Interac actio tion n Analy Analysis sis
Some of the currently available techniques of pile group analysis were reviewed in the previous section. Pile group analysis using pile interaction factors is common. These procedures are applicable applicable to elastic elastic systems as they use the frequenc frequency y domain. domain. Although Although these procedures were extended to the time domain in order to incorporate pile and soil nonlinearity effects, they only consider nonlinearity effects in the computation of the single pile response. Currently available techniques for computing pile interaction factors do not account for nonlinearity. Since soil and pile nonlinearity effects do have a significant effect on pile foundation response, it is necessary to understand the effect of these nonlinearities on pile-to-pile interaction, particularly in groups with closely spaced piles. Pile-to-pile interaction interaction studies were conducte conducted d using 3-D finite elemen elementt analysis. analysis. A single pile model was first calibrated calibrated against experimental response data. The parameters used in this model were then used to perform pile group analysis. analysis. Soil nonlinearit nonlinearity y and pile-soil pile-soil gapping gapping effects effects were considere considered d in the analysis.
4.1
3D Finite Finite Eleme Element nt Model Model for Pile Pile Foun Foundat datio ion n Analysis Analysis
Two sets of experimental tests performed at the same site were used to calibrate the finite elemen elementt model. model. Later Lateral al load load tests tests were were perform performed ed on a group group of nine nine steel steel pipe piles piles and also on an isolated isolated single pile [10]. The location of the tests was an overc overconsoli onsolidated dated clay clay site in Houston, TX. In another experimental program at the same site, dynamic lateral load tests were conducted on a cantilevered mass supported by a single steel pipe pile embedded in stiff, overconsolidated clay [6]. 3D models were created using the program ABAQUS [1]. Due to the symmet symmetry ry of the proble problem, m, only only half half the mesh mesh was generate generated d (Figur (Figuree 2). The pile was assumed assumed to respond elastically elastically.. The steel pipe pile was modeled as an equivalen equivalentt cylinder cylinder.. Th Thus, us, the moment moment of inertia inertia and the mass density density were were modified so that flexural flexural
14
Property Value Pile properties Pile diameter 0.273m 273m Pile length 13.1m Flexural stiffness 1.34X 34X 10 104 kN − m2 Lineal pile density 60.3kg/m Soil properties Young’s mo dulus 100M P a Density 2100kg/m3 Poisson’s ratio 0.49 Undrained shear strength 47kP a Table 1: Pile and soil properties used in the 3D finite element model stiffness (EI) and total mass of the cylinder were equal to the flexural stiffness and total mass respectiv respectively ely of the steel pipe. The soil was assumed assumed to be homogeneou homogeneous. s. The model was meshed meshed using 3D brick brick (denoted (denoted C3D8 in ABAQUS) ABAQUS) elements. elements. A number of laboratory laboratory and in-situ field tests were performed during both the mentioned experimental programs, to determine determine soil properties. properties. More details details may be b e found in the cited reference references. s. Howev However, er, the tests tests showe showed d consider considerable able scatter in the results. results. The value value of Young’s modulus for the soil was back calculated so that the stiffness of the load deformation curve in an elastic analysis matched the initial stiffness of the experimental curve. The obtained value was found to fall within the range of experimentally determined values. A constant value of undrained shear strength strength was assumed based on experimen experimental tal evidence evidence.. Soil and pile properties properties used in the model are listed in Table 1. Checks were performed to verify the finite element mesh, assuming the soil to be linear elastic. elastic. The soil-pile soil-pile model was fixed at the base (and free on all sides) sides) and subjected subjected to a lateral lateral load at the ground level. level. The obtained obtained response response was compared with the deflection deflection of a cantilever beam subjected to a tip load, as obtained from beam theory. The difference between the two results was about 8% (Table 2), which is acceptable since beam theory is not exact exact in this case due to the presence presence of shear deformati deformations. ons. The second check check performed, performed,
15
Figure 2: 3D finite element mesh for single pile analysis 16
Analys Analysis is proce procedu dure re Applie Applied d load load (kN) (kN) Pile Pile hea head d Beam theory 50 Beam theory 300 3D FEA 50 3D FEA 300
displ displace acemen mentt (m) (m) Error Error (%) 0.0144 0.00 0.0863 0.00 0.0155 7.73 0.0933 8.08
Table 2: 3D FE model check: verification of pile head response as a cantilever beam compared with beam theory
Analysis proce ocedure Pile head response (m) 3D mo del 0.00128 Anal An aly ytical ical solu soluttion ion usin usingg Wink inkler ler mode modell [29 [29] 0.00 0.0020 2044 Table 3: 3D FE model check: comparison of elastic response with analytical solution compared the response of the 3D model under a static lateral pile head load of 100kN, with solutions predicted by a beam-on-elastic-foundation (Winkler model) approach (Table 3), obtained analytically analytically from the solution solution of the differenti differential al equation equation [29]. The observed observed difference between the 3D model and Winkler model solution is due to the fact the Winkler model assumes that the modulus of subgrade reaction provided by the soil is k = 1.2E s (equation 1) while the 3D model explicitly considers the Young’s modulus of the soil.
4.1.1 4.1.1
Modeling Modeling of soil nonlin nonlineari earity ty
Since the aim of this study is to understand the effect of nonlinearity on interaction factors, nonlinear nonlinear soil models were were used in the 3D finite element element analyses. analyses. Abaqus Abaqus provide providess a large number number of plasticity plasticity models to incorporate incorporate soil nonlineari nonlinearity ty in the analysis. analysis. These These include include the Extended and Modified Drucker-Prager models, the Mohr-Coulomb plasticity model and the Critical state (clay) plasticity model [1]. These are sophisticated plasticity models that require require calibration calibration based based on experimen experimental tal data. In this study, study, the Mohr-Coulom Mohr-Coulomb b model is used since it can be calibrate calibrated d based based on the standard standard Mohr-Coulom Mohr-Coulomb b parameters parameters.. It uses
17
1
60
50
0.8
) a P40 k ( τ , s s e30 r t s r a e h20 S
0.6
x a m
G / G
0.4
0.2
10
0 −3 10
−2
0
−1
10
10
10
0 0
Strain, γ (%)
(a) Modulus reduction reduction curve curve (PI=30) (PI=30) [33]
0.2
0.4 0.6 Shear strain, γ (%)
0.8
1
(b) Shear stress-strain curve
Figure 3: Nonlinear soil model for cohesive soil used in the 3D FEM the classical Mohr-Coulomb yield criterion:
τ = c − σ tan φ
(11)
where τ is the shear stress, σ is the normal stress (negative in compression), c and φ are the Mohr-Coul Mohr-Coulomb omb parameters. parameters. The definition definition of material beha b ehavior vior using the Mohr-Coulom Mohr-Coulomb b model model in Abaqus Abaqus includ includes es the elasti elasticc propert propertie iess (Youn (Young’s g’s modulus modulus and Po Poiss isson’ on’ss ratio) ratio),, density (for dynamic analysis), angle of friction, angle of dilation (which governs the flow potential), potential), and cohesiv cohesivee yield yield stress vs. plastic plastic strain. Since Since the soil being modeled in the presented set of analyses was cohesive, the angle of friction ( φ) was assumed to be zero. The dilation angle (ψ (ψ) was also assumed to be zero. The cohesive stress-strain response (Figure 3(b)) was derived from the modulus reduction curve for cohesive soils with plasticity index, ( e), P I = 30 [33] (Figure 3(a)). The parameters eccentricity ( ) and deviatoric eccentricity (e which determine the shape of the flow potential function, assumed default values: = 1 and e = (3 − sin φ)/(3 + sin φ).
18
4.1.2 4.1.2
Modeling Modeling of pile-soi pile-soill int interfa erface ce
Modeling of the pile-soil interface is critical under strong loads that cause separation between the pile and the soil. Abaqus Abaqus provide providess a number of advance advanced d models for contact contact beha b ehavior. vior. Surface-based contact, which allows modeling of contact between two deformable bodies that undergo undergo small or finite sliding, sliding, was used in the analyses analyses presented presented here. Abaqus Abaqus uses the concept of contact pairs - a master surface and a slave surface. The pile surface was chosen as the master and the surrounding soil surface was chosen as the slave. The mechanical contact simulation of the interaction between two bodies includes a constitutive model for the contact pressure-overclosure relationship (i.e. behavior normal to the contact surfaces) and a friction model that defines the force resisting resisting relativ relativee tangent tangential ial motion of the surfaces. surfaces. The most common common contact contact pressure-o pressure-ove verclos rclosure ure relationsh relationship ip is “hard” contact. contact. When surfaces surfaces are in contact, any contact pressure can be transmitted between them. The surfaces separate if the contact pressure reduces to zero. Separated surfaces come into contact when the separation between them reduces to zero. In addition to the pressure-overclosure relationship, frictional behavior may or may not be included to model the shear stress transmitted across the interfa interface. ce. Abaqus Abaqus has a wide range of friction friction models to choose from. The classical classical isotropic isotropic Coulomb friction model, which defines a friction coefficient relating shear stress to the contact pressu pressure, re, was used. used. A value alue of 0.7 was assumed assumed for the coefficien coefficientt of friction friction,, as used by Bentley and Naggar [5], based on the recommendations of the American Petroleum Institute [2]. [2]. The 3D finite finite elemen elementt model model of the single single pile was compared compared with the experi experime ment ntal al response with and without the friction model.
4.1.3 4.1.3
Respon Response se of single single pile pile usin using g 3D FE model model
The experimental response of the single pile [10] under a lateral pile head load was compared with the static response of the 3D finite element model (Figure 4) assuming linear elastic pile response and nonlinear soil response using the Mohr-Coul Mohr-Coulomb omb model. Three Three differen differentt 19
100
80 ) N k ( d a o l d a e h e l i P
60
40
Experimental data FEM (perfect soil−pile contact) FEM (hard contact, no friction) FEM (hard contact, friction)
20
0 0
0.01 0.02 0.03 Pile head displacement (m)
0.04
Figure 4: Comparison of single pile response using 3D FEM with experimental response pile-soil contact models were considered: considered: (i)perfect contact (ii)“hard” contact contact with friction (iii)“hard” contact without friction. The analyses showed that the use of the Mohr-Coulomb model for soil nonlinearity along with the “hard” contact model with no friction gives the best match with experimental data. Therefore, it was decided to use this model in the pile group analysis. The 1D FE model (or Winkler model) for the single pile was also compared with the 3D model and experimental response. Figure 5 shows that by using the p-y material for soft clay [19] to model the nonlinear spring in the Winkler model, it is possible to obtain a good matc match with with the experim experimen ental tal respons response. e. Th This is model model has been implem implemen ente ted d in the curren currentt version version of Opensees Opensees [20]. Howev However, er, using the bilinear and Bouc-W Bouc-Wen model for the nonlinear springs does not compare quite as well with the experimental response.
4.2
Developm Developmen entt of Improve Improved d Pile-to-Pile Pile-to-Pile Interac Interaction tion Facto Factors rs
To study the effect of soil nonlinearity and soil-pile gapping on the pile interaction factors, 3D finite element element analyses were perfo p erformed rmed on two-pile two-pile groups. The Mohr-Coulom Mohr-Coulomb b model used in the single pile analysis was used in the pile group model, along with the “hard” contact
20
100
80 ) N k ( d a o l d a e h e l i P
60
40 Experimental data 3D FE model Winkler (soft clay p−y model) Winkler (bilinear) Winkler (Bouc−Wen)
20
0 0
0.01 0.02 0.03 Pile head displacement (m)
0.04
Figure Figure 5: Compar Compariso ison n of single single pile respons responsee using using 1D and 3D FEM with with experi experimen mental tal response (no friction) friction) model used for pile-soil contact. contact. The soil and pile properties properties are listed listed in Table Table 1. The factor factorss affecti affecting ng pile-t pile-to-pi o-pile le inte interac ractio tion n includ includee the pile pile to soil soil relat relativ ivee stiffne stiffness, ss, departure angle and pile length to diameter ratio [26]. However, the most important factor is the pile spacing. spacing. Tw Twoo finite element element models were created created with pile spacing, S = 2d and pile diamet diameter. er. The finite finite eleme element nt meshes meshes for the pile groups groups are S = 5d, where d is the pile shown shown in Figures Figures 6 and 7. The pile groups were were subjected subjected to static lateral lateral pile head loads and the interaction factors at the pile head level were computed by comparing the pile group response response with the single pile response. response. The single pile model was similar similar to what was used in section 4.1, with the only difference that the pile head was assumed to be fixed. This was achieved by restraining the vertical degrees of freedom on the top surface of the single pile. The pile-to-pile interaction factors were computed as:
αh =
∆u plgrp usglpl
21
(12)
Figure 6: 3D finite element mesh for pile group analysis ( S = 2d)
22
Figure 7: 3D finite element mesh for pile group analysis ( S = 5d)
23
2 1.8 1.6 h
α
, r o t c a f n o i t c a r e t n I
1.4 1.2
Elastic soil Nonlinear soil w/o pile−soil gapping Nonlinear soil with pile−soil gapping
1 0.8 0.6 0.4 0.2 0
50
100 Pile group load (kN)
150
200
Figure 8: Effect of soil nonlinearity and pile-soil gapping on interaction factors where αh is the inte interac ractio tion n factor factor,, ∆ u plgrp is the add additi itiona onall displa displace ceme ment nt caused caused in one pile due to the loading on the other pile, and usglpl is the single pile displacem displacemen ent. t. Since Since the system is nonlinear, αh is a funct function ion of the applied applied load. load. To observ observee the effect effect of pile pile nonlinearity and the pile-soil contact model on the interaction factors, for the case of S = 2d, αh was computed for three different cases: (i)assuming the soil to be linear elastic and soilpile contact to be perfect, (ii)modeling soil nonlinearity using the Mohr-Coulomb model with no soil-pile gapping (iii)using the Mohr-Coulomb model for soil nonlinearity and the “hard” contact contact (no friction) friction) model for the pile-soil pile-soil interf interface. ace. The results results of the analysis analysis are shown shown in Figure 8. Figure Figure 9 shows shows the effect of pile spacing on the interac interaction tion factors. factors. The figure clearly clearly shows shows that as the spacing spacing increases, increases, the interac interaction tion between between piles decreas decreases. es. If the piles in a group are very closely closely spaced, then interac interaction tion between between the piles is larger. This implies that the induced additional displacements are also larger. Thus, as the pile spacing is decreased, decreased, the pile group undergoes larger deformati deformations ons for the same load level. level. At very very large spacings, there would be no interaction between piles, in which case, the pile group would carry twice the load of a single pile. Figure 10 shows the effect of pile spacing on the
24
2 S/d = 2
1.8
S/d = 5
1.6 h
α
, r o t c a f n o i t c a r e t n I
1.4 1.2 1 0.8 0.6 0.4 0.2 0
50
100 Pile group load (kN)
150
200
Figure 9: Effect of pile spacing on interaction factors
200 180 160 140 d a120 o l
d a100 e h e l i 80 P
60 40
2Xsingle pile response Pile group response: S/d = 2 Pile group response: S/d = 5
20 0 0
0.005 0.01 0.015 Pile head displacement (m)
0.02
Figure 10: Lateral load response of a two-pile group
25
lateral load-deformation response of pile groups. As the pile spacing decreases, the efficiency of each pile in the group decreases. Further analyses need to be conducted to study the effect of other factors on the interaction factors. These include the effect of departure angle, relative pile-to-soil stiffness, soil strength and dynamic dynamic loading. loading.
26
5
Conc onclusi lusio ons
A nu numbe mberr of tech techniq niques ues curre current ntly ly used used to analyz analyzee pile pile founda foundatio tions ns were review reviewed ed.. The Beam Beam on Winkle Winklerr Foun oundat dation ion model, model, common commonly ly used used in the analys analysis is of single single piles piles,, was review reviewed. ed. Tw Twoo major classes classes of pile group analysis analysis techniq techniques— ues—the the pile-to-pi pile-to-pile le interac interaction tion factor factor method and the p-multip p-multiplier lier method were also reviewed. reviewed. The p-multiplie p-multiplierr method method while highly efficient (when compared to 3D finite element analysis) and widely used may still be very time consuming for large pile-supported structures which have a number of large pile groups. The pile-to-pile interaction factor method is based on modifying the response of a single pile to account for interactions among different piles in a group. The available techniques to evaluate these interactions are generally applicable to linear elastic systems and do not incorporate incorporate the effect of soil and pile nonlinearit nonlinearity y. 3D finite element element analyses analyses were performed to compute pile-to-pile interaction factors accounting for soil nonlinearity and pile-soil gapping effects. Interaction factors were computed for two-pile groups under lateral static loading. The analyses showed that both nonlinearity in the soil and gapping between the pile and the soil have a very significant effect on the interaction. The pile spacing is the most important parameter influencing the interaction interaction factors. The interaction interaction factors were computed computed for two two differen differentt pile spacings. spacings. A number number of other factors factors may have have a significan significantt effect on the interaction—departure angle, relative pile-to-soil stiffness, soil strength, pile nonlinearity, and load amplitude and frequency (under dynamic loading). A very extensive set of finite element analyses need to be conducted to understand the full effect of all these factors.
27
Acknowledgment This work was supported by the National Science Foundation through the Mid-America Earthquake Earthquake Center under award number number EEC-9701785. This work was partially supported by National Computational Science Alliance under BCS040006N and utilized the IBM P690 at the National Center for Supercomputing Applications.
28
References [1] Abaqus Inc., Pawtuc Pawtucket, ket, RI. ABAQUS Analysis User’s Manual , 2003. [2] American Petroleum Petroleum Institute, Washington, Washington, D.C. Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms. API Recommended Practice 2A (RP 2A), 2A), 1991. [3] D. Badoni and N. Makris. Makris. Nonlinear Nonlinear response response of single single piles under lateral inertial inertial and seismic loads. Soil Dynamics and Earthquake Engineering , 15:29–43, 1996. [4] D. Badoni and N. Makris. Pile-to-p Pile-to-pile ile interacti interaction on in the time domain – non-linear non-linear axial group response under under harmonic harmonic loading. loading. G´eotec eo techn hniq ique ue,, 47(2):299–317, 47(2):299–317, 1997. [5] K. J. Bentley and M. H. El Naggar. Numerical analysis of kinematic response of single piles. Canadian Geotechnical Journal , 37:1368–1382, 2000. [6] G. W. Blaney and M. W. O’Neill. O’Neill. Measured Measured lateral lateral response of mass on single pile in clay. Journal of Geotechnical Engineering, ASCE , 112(4):443–45 112(4):443–457, 7, 1986. [7] B. B. Broms. Lateral Lateral resistance resistance of piles in cohesionl cohesionless ess soils. Journal of the Soil Mechanics and Foundations Division, ASCE , 90:123–156, 1964. [8] B. B. Broms. Lateral resistance of piles in cohesive soils. Journal of the Soil Mechanics and Foundations Division, ASCE , 90:27–63, 1964. [9] D. A. Brown, Brown, C. Morrison, Morrison, and L. C. Reese. Lateral Lateral load behavior behavior of pile group in sand. Journal of Geotechnical Engineering, ASCE , 114(11):1261– 114(11):1261–1276, 1276, 1988. [10] [10] D. A. Brown, Brown, L. C. Reese, Reese, and M. W. O’Neil O’Neill. l. Cycli Cyclicc later lateral al loading loading of a large large -scale -scale pile group. Journal of Geotechnical Engineering, ASCE , 113(11):1326–13 113(11):1326–1343, 43, 1987.
29
[11] D. A. Brown Brown and C.-F. Shie. Numerical Numerical experimen experiments ts into into group effects on the response response of piles to lateral loading. Computers and Geotechnics, Geotechnics, 10:211–230, 1990. [12] D. A. Brown and C.-F. Shie. Three Three dimensional dimensional finite elemen elementt model of laterally laterally loaded piles. Computers and Geotechnics, Geotechnics, 10:59–79, 1990. [13] R. Dobry and G. Gazetas. Gazetas. Simple Simple method for dynamic stiffness stiffness and damping damping of floating pile groups. G´eotec eo techn hniq ique ue,, 38(4):557–574, 38(4):557–574, 1988. [14] [14] G. Gazetas Gazetas and R. Dob Dobry ry.. Horizo Horizont ntal al respons responsee of piles piles in laye layere red d soils. soils. Journal of Geotechnical Engineering, ASCE , 110(1):20–40, 1984. [15] G. Gazetas and N. Makris. Makris. Dynamic Dynamic pile-soil-pile pile-soil-pile interac interaction. tion. Part I: Analysis Analysis of axial vibration. Earthquake Engineering and Structural Dynamics, Dynamics , 20:115–132, 1991. [16] B. K. Maheshwari, Maheshwari, K. Z. Truman, Truman, M. H. El Naggar, and P. P. L. Gould. Three-dimensional Three-dimensional finite element nonlinear dynamic analysis of pile groups for lateral transient and seismic excitations. Canadian Geotechnical Journal , 41:118–133, 2004. [17] [17] N. Makris Makris and G. Gazeta Gazetas. s. Dynami Dynamicc pile-s pile-soil oil-pi -pile le inter interact action ion.. Part Part II: Latera Laterall and seismic response. Earthquake Engineering and Structural Dynamics, Dynamics , 21:145–162, 1992. [18] N. Makris and G. Gazetas. Gazetas. Displacement Displacement phase differences in a harmonically harmonically oscillating pile. G´eotec eo techn hniq ique ue,, 43(1):135–150, 43(1):135–150, 1993. [19] H. Matlock. Correlations for design design of laterally loaded piles piles in soft clay. clay. In Proc., Second Offshore Technology Conference, Conference, pages 577–594, Houston, TX, 1970. [20] S. Mazzoni, F. McKenna, M. H. Scott, G. L. Fenves, Fenves, and B. Jeremi´c. c. Command Language guage Manual: Manual: Op Open en System System for Eart Earthqua hquake ke Engine Engineering Simulation Simulation (OPENSEE (OPENSEES) S).. Pacific Earthquake Earthquake Engineering Research Research Center. website: website: http://peer.berkeley http://peer.berkeley.edu/ .edu/ silvia/OpenSees/manual/doc/OpenSeesManual.doc. 30
[21] M. C. McVay McVay and L. Niraula. Development Development of p-y curves for large diameter piles/drilled shafts in limestone for FBPIER. Technical report, Florida DOT, 2004. [22] P. J. Meymand. Meymand. Shaking table scale model tests of nonlinear soil-pile-superstructure interaction in soft clay . PhD thesis, University of California, Berkeley, 1998. [23] P. M. Morse and K. U. Ingard. Theoretical Acoustics. Acoustics. McGraw-Hill, New York, 1968. [24] G. Mylonakis and G. Gazetas. Gazetas. Kinematic K inematic pile response response to vertical P-wav P-wavee seismic excitaexcitation. Journal of Geotechnical and Geoenvironmental Engineering, ASCE , 128(10):860– 128(10):860– 867, 2002. [25] [25] H. G. Po Poulo ulos. s. Behav Behavior ior of laterall laterally y loaded loaded piles piles:: I – single single piles. piles. Journal of the Soil Mechanics and Foundations Division, ASCE , 97(SM5):711–731, 1971. [26] [26] H. G. Poulos. Poulos. Behav Behavior ior of laterall laterally y loaded loaded piles: piles: II – pile pile groups. groups. Journal of the Soil Mechanics and Foundations Division, ASCE , 97(SM5):733–751, 1971. [27] [27] H. G. Poulos Poulos and E. H. Davis. Davis. Pile Foundation Analysis and Design . John Wiley and Sons, New York, 1980. [28] [28] M. F. Randol Randolph ph and G. T. Houlsb Houlsby y. Th Thee limiti limiting ng pressu pressure re on a circul circular ar pile loaded loaded laterally in cohesive soil. G´eotec eo techn hniq ique ue,, 34(4):613–623, 1984. [29] [29] L. C. Reese Reese.. Behav Behavior ior of piles piles and pile pile groups groups under under later lateral al load. load. Techni echnical cal report, report, FHWA, 1983. [30] L. C. Reese, Reese, W. R. Co Cox, x, and F. D. Koop. Ko op. Analysis Analysis of laterally laterally loaded piles piles in sand. In Proc., Fifth Offshore Technology Conference, Conference , pages 473–483, Houston, TX, 1974. [31] L. C. Reese, Reese, W. R. Cox, Cox, and F. D. Koop. Field testing testing and analysis analysis of laterally laterally loaded piles piles in stiff clay clay.. In Proc., Proc., Seventh Offshore Offshore Technology echnology Conferenc Conferencee, pages 672–690, Houston, TX, 1975. 31
[32] A. Trocha Trochanis, nis, J. Bielak, Bielak, and P. Christiano. Christiano. Simplified Simplified model for analysis analysis of one or two two piles. Journal of Geotechnical Engineering, ASCE , 117(3):448–466 117(3):448–466,, 1991. [33] M. Vucetic Vucetic and R. Dobry. Dobry. Effect of soil plasticity on cyclic response. Journal of Geotechnical Engineering, ASCE , 117(1):89–107 117(1):89–107,, 1991. [34] S. Wang, Wang, B. L. Kutter, Kutter, M. J. Chacko Chacko,, D. W. Wilson, R. W. Boulanger, Boulanger, and A. Abghari. Nonlinear Nonlinear seismic seismic soil-pile soil-pile structur structuree interac interaction. tion. Earthquake Spectr 14(2):377–396, Spectra a , 14(2):377–396, 1998. [35] Y.-K. Wen. Wen. Method Method for random vibration vibration of hystere hysteretic tic systems. systems. Journal of the Engineering Mechanics Division, ASCE , 102(EM2):249 102(EM2):249–263, –263, 1976. [36] [36] G. Wu Wu and W. D. Liam Finn. Dynami Dynamicc non nonlin linear ear analysis analysis of pile pile founda foundatio tions ns using finite element method in the time domain. Canadian Geotechnical Journal , 34:44–52, 1997. [37] Z. Yang Yang and B. Jeremi Jeremi´´c. c. Numerical Numerical analysis analysis of pile behavior behavior under lateral lateral loads in layered layered elastic-plastic soils. International Journal for Numerical and Analytical Methods in Geomechanics Geomechanics,, 26:1385–1406, 2002. [38] Z. Yang Yang and B. Jeremi´ Jeremi´c. c. Numerical Numerical study of group effects effects for pile groups in sands. sands. International Journal for Numerical and Analytical Methods in Geomechanics, Geomechanics , 27:1255– 1276, 2003.
32
