Response Evaluation for Horizontally Loaded

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2005; 29:597–625Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.428

Response evaluation for horizontally loaded fixed-head pilegroups using 3-D non-linear analysis

Emilios M. Comodromos1,n,y and Kyriazis D. Pitilakis2

1Department of Civil Engineering, University of Thessaly, Pedion Areos, 383 34 Volos, Greece2Department of Civil Engineering, Aristotle University of Thessaloniki, PO Box 450, 54 006, Greece

SUMMARY

The response of laterally loaded pile foundations may be significantly important in the design of structuresfor such loads. A static horizontal pile load test is able to provide a load–deflection curve for a single free-head pile, which significantly differs from that of a free- or fixed-head pile group, depending on theparticular group configuration. The aim of this paper is to evaluate the influence of the interaction betweenthe piles of a group fixed in a rigid pile cap on both the lateral load capacity and the stiffness of the group.For this purpose, a parametric three-dimensional non-linear numerical analysis was carried out fordifferent arrangements of pile groups. The response of the pile groups is compared to that of the single pile.The influence of the number of piles, the spacing and the deflection level to the group response is discussed.Furthermore, the contribution of the piles constituting the group to the total group resistance is examined.Finally, a relationship is proposed allowing a reasonable prediction of the response of fixed-head pilegroups at least for similar soil profile conditions. Copyright # 2005 John Wiley & Sons, Ltd.

KEY WORDS: horizontally loaded pile groups; soil–pile interaction; 3-D non-linear analysis; pile groupresponse prediction

1. INTRODUCTION

The response of laterally loaded pile foundations may be significantly important in the design ofstructures to such loads. In many cases, the criterion for the design of piles to resist lateral loadsis not the ultimate lateral capacity but the deflection of the piles [1]. In the case of bridges orother structures founded on piles, only a few centimetres of displacement could cause significantstress development on these structures. The load–deflection curve of a single free-head pile canbe determined using numerical methods and/or results from pile load tests, while full-scale pilegroup tests for determining the response of a pile group are very rare due to the extremely highcost required. Furthermore, for single piles, various approaches have been proposed with theaim to take into account non-linearities arising from soil–pile interaction. Within thisframework, Reese [2] proposed the well-known ‘p–y analysis’. This approach is based on the

Received 17 June 2004Revised 15 December 2004Copyright # 2005 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: E. M. Comodromos, Department of Civil Engineering, University of Thessaly, Pedion Areos, 38334 Volos, Greece.

differential equation (1) for solving the problem of the laterally loaded pile and it can be used forboth free- and fixed-head single piles.

EId4y

dx4þ Px

d2y

dx2þ Esy ¼ 0 ð1Þ

where EI is the flexural stiffness of the pile, Px the axial load of the pile, Es the soil modulus,y the deflection and x the length along the pile.

According to the p–y method, the soil response is described by a family of curves giving soilresistance as a function of deflection and depth below the ground surface. The simplicity of themethod in conjunction with the well-defined procedures for establishing the p–y curves [3–6]made the method the most widely used. Although the method is reliable for evaluating theresponse of a single pile under horizontal load, it is questionable if reasonably reliable simplemethods could be applied to assess the response of pile groups. It is however commonlyaccepted that for the same mean load, the piles of a pile group exhibit significantly greaterdeflection than an identical single pile. This behaviour should be attributed to the fact that theresisting zones behind the piles overlap. Clearly the effect of the overlapping becomes larger asspacing between piles decreases. The application of three-dimensional (3-D) numerical analysis,on the other hand, is also rarely utilized because of the complexity in simulating the non-linearities of the interaction between soil and piles, but mainly because such a procedure isextremely computationally demanding. It is however the most powerful tool for pile groupresponse evaluation under horizontal or other loading, since it is able to predict both stiffnessand ultimate resistance reduction factors, particularly in the case of sensitive soils undergoingplastification for even a low level of loading.

The aim of this paper is to use numerical analysis and tools to estimate the interaction levelbetween soil and piles for various layouts of horizontally loaded fixed head pile groups and todetermine the reduction factors for ultimate lateral load capacity and stiffness corresponding tothe working load or any other load level. Moreover, with the objective of estimating the effect ofinteraction on pile groups, numerical results are used to derive a relationship, which could beutilized to predict the response of a fixed-head pile group provided that the load–deflectioncurve of the single fixed-head pile is already known. A curve-fining procedure using the MSEXCEL program and the built-in Visual Basic language, presented in Appendix A, has assistedconsiderably in defining the appropriate equations.

2. THE EFFECT OF SOIL–PILE INTERACTION TO PILE GROUPS

According to Prakash and Sharma [7], and Oteo [8], the lateral group efficiency nL defined byEquation (2) may reach only 40%, depending on the number of piles in a group and the layoutof the group.

nL ¼ultimate lateral load capacity of a group

n� ultimate lateral load capacity of single pileð2Þ

As mentioned in the introduction, the load–deflection curve could be the determining factor forthe design of a project and therefore the group stiffness reduction factor caused by a lateral loadis of greater importance than the group efficiency factor. The widely used p–y method could beconsidered as extremely effective for the prediction of a single pile under horizontal loading and

Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2005; 29:597–625

E. M. COMODROMOS AND K. D. PITILAKIS598

this has been demonstrated by the application of the back-analysis procedure in many cases,where a pile load test was carried out. Even though available data from a single pile test underhorizontal loading exists together with the results from a p–y analysis, further calculations arerequired to establish the response of a pile group due to the effect of pile–soil–pile interaction.Poulos [9, 10] introduced four different kinds of interaction and reduction factors for piles underlateral load, depending on the loading at the pile head and the type of deformation. Moreover,based on the elastic continuum approach, Randolph [11] proposed a relationship for estimatingthe interaction factors in fixed-head piles, demonstrating that the interaction under lateralloading decreases much more rapidly with spacing between piles than for axial loading. Wakaiet al. [12] used 3-D elasto-plastic finite element analysis to estimate the effect of soil–pileinteraction within model tests for free or fixed-head pile groups. In that analysis, thin frictionalelements were inserted between the pile and the soil in order to consider slippage at the pile–soilinterface. It must be mentioned, however, that in many cases, where the pile–soil interaction isgoverned by non-linearities arising from the soil separation behind the pile and the yield of soilin front of the pile, a 3-D analysis including interface elements around the piles can beconsidered more accurate in providing the response of a pile group.

Comodromos [13] utilized 3-D FDA (Finite Different Analysis) to evaluate the response offree-head pile groups. Based on the results of a full-scale pile load test, he firstly applied back-analysis techniques using the p–y method and a 3-D FDA to verify and adjust the soilparameters. A parametric 3-D analysis was then performed and the results have been comparedwith those of the pile test. The effect of the pile–soil–pile interaction was then estimated forvarious group configurations and, finally, a relationship was proposed allowing the establish-ment of load–deflection curves limited for free-head pile groups. As stated in that paper, theapplicability of the proposed formulae to different soil profiles should be verified or readjustedfor different soil profiles. A similar procedure is applied in this paper to evaluate the response offixed-head pile groups. Various pile group layouts have been analysed using 3-D FDA. Thecurve-fining procedure, given in Appendix A, was then used to define the precise form of arelationship with the ability to predict a pile group response based on that of the single pile.

3. SOIL PROFILE}SINGLE PILE RESPONSE

Numerical analyses for fixed-head single pile and groups correspond to a given soil profile andpile dimension, for which a free-head pile test has been carried out. The first step was to justifyand adjust the soil parameters, and a free-head p–y analysis and a 3-D FDA were carried outand the results were compared to those recorded from the full-scale test. Detailed description ofthe site, the pile test configuration and the soil conditions at which the free-head pile test hasbeen carried out is given by Comodromos [13] while a brief description is given below. Figure 1shows the pile load test arrangement and the soil profile, while Table I summarizes the soilprofile design parameters. The important role of the first layer to the response of piles underhorizontal load is widely accepted, particularly when this layer is of high and variablecompressibility and shear strength. For laterally loaded piles, shear and soil modulus may betaken as a function of undrained shear strength [1, 14, 15]. The proposed correlations in theabove references lie within a large range, 115–250 for shear modulus, and 15–95 for soilmodulus, depending on load level and the value of the shear strength. According to laboratorytests and based on the back analysis performed to adjust the soil parameters, it was found that


RESPONSE EVALUATION FOR HORIZONTALLY LOADED FIXED-HEAD PILE 599

the undrained shear strength of layer A varies with depth according to Equation (3), whileEquation (4) gives the most suitable approximation of shear modulus for layer A.

cu ¼ 5þ 45z

HaðkPaÞ; 54cu450 ð3Þ

G ¼ 90 cu ¼ 450þ 4050z

HaðkPaÞ; 4504G44500 ð4Þ

where Ha is the thickness of layer A, z the depth at a particular point within layer A.

Figure 1. Soil profile and design parameters.

Table I. Geotechnical properties of soil layers.

Layer A B C1 C2

Bottom elevation (m) �36 �48:0 �52:0 �70:0Shear modulus G (MPa) 90 cu 3.35 24.0 24.0Angle of friction j0 (deg) 0 0 40 40Dilation angle c (deg) 10 0 0 12Undrained shear strength cu (kPa) 5–50n 110 0 0Unit weight g ðkN=m3Þ 20.0 20.0 22.0 22.0

nLinearly varying with depth.



Figures 2 and 3 illustrate the deflection and the pile head rotation for the test pile togetherwith the predictions using p–y analysis and the 3-D FDA using FLAC3D [16]. It can be seen thatpredicted and recorded values are in close agreement and consequently the soil parameters andthe p–y curves used in the free-head analysis can be used for a further analysis of fixed-headsingle pile and pile groups.

Even if the present paper focuses on the response of fixed-head piles, it is of interest to presentthe effect of separation for free-head piles, since, in that case, the lateral displacements areessentially greater than in the case of a fixed-head pile. Figure 4 shows the soil–pile separationalong the pile for the maximum load of 1:2 MN predicted by 3-D FDA. The separation is equalto 10:5 cm at the top of the pile and 0:3 cm at a depth of 6 m; while the displacement at the head

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Deflection y (m)

Hor

izon

tal L

oad

H (

MN

)Test Pi le

FLAC 3D

p-y analysis

Figure 2. Load–deflection curves for test pile, p–y analysis and FLAC3D analysis.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1Pile Head Rotation δ (deg)

Late

ral L

oad

H (

MN

)

Test Pile

FLAC 3D

p-y analysis

Figure 3. Lateral load–pile head rotation curves for test pile, p–y analysis and FLAC3D analysis.



of the pile is 17:5 cm: According to the results, the mobilized area is larger in front of the pilethan behind it and this is mainly due to the existence of the interface surface around the pile,which provides the capability of soil–pile separation when the soil tensile strength is reached. Inthe given soil profile, where the clayey soil at the surface allows separation, the interface wasconsidered necessary in order to avoid any hanging of soil elements to those of the pile.

4. PARAMETRIC ANALYSES OF FIXED-HEAD PILE GROUPS

4.1. Group configurations and loadings

Having justified the soil parameters, 3-D parametric analyses were carried out. The three-dimensional finite difference code FLAC3D has been used for a series of parametric analyses offixed-head pile groups. A grid generator subroutine has been implemented using the FISH built-in programming language providing the possibility of mesh refinement and geometry variationaccording to the specific group configuration. Figure 5 illustrates the finite difference grid of a4� 4 pile group, consisting of 19 800 brick elements and 18 903 nodes. The dimension of the gridis 42 m in the x and y directions and 70 m deep. At the bottom plane of the grid, all movementsare restrained. The lateral sides of the mesh were taken far enough from the piles to avoid anyboundary effect. The planes x ¼ �21:0 m and x ¼ þ21:0 m are free to move in the y and zdirections but not in the x direction. Similarly, the planes y ¼ �21:0 m and y ¼ þ21:0 m are freeto move in the x and z directions but not in the y direction. Similar boundary conditions wereapplied for different pile group grids. In order to accelerate calculations, the benefit of symmetryon the vertical plane y ¼ 0 has been made use of and thus the half-grid defined by y50 was

0

10

20

30

40

50

-0.05 0.00 0.05 0.10 0.15 0.20

Displacement conditions (m)

Dep

th (m

)

Soil-pile separation

Displaced soil profile

Figure 4. Soil–pile separation along the pile for the maximum load of 1:2 MN:



finally used. The other half was removed using the ‘model null’ statement and the boundaryconditions were modified accordingly.

The elastic perfectly plastic Mohr–Coulomb constitutive model was used, in conjunction witha non-associated flow rule, to simulate the non-linear elasto-plastic material behaviour of soillayers given in Figure 1. Due to the fact that soil has a limited capacity in sustaining tension,interface elements were introduced to allow pile separation from the surrounding soil.Separation occurs near the top and behind the pile generally no deeper than 20% of pile length,depending on pile and soil stiffness. Together with the local yield at the top of the soil wherelarge compressive stresses are developed in front of the soil, separation is considered as the mainreason for the non-linear behaviour. According to Poulos and Davis [1], separation is able tocause an increase in displacements up to the extreme level of 100%, while 30 to 40% appears tobe more reasonable in the case of stiff piles.

The constitutive model of the interface elements in FLAC3D is defined by a linear Coulombshear-strength criterion that limits the shear force acting at an interface node, a dilation anglethat causes an increase in effective normal force on the target face after the shear strengthlimit is reached, and a tensile strength limit. Figure 6 illustrates the components of theconstitutive model acting at an interface node. The interface elements are allowed toseparate if tension develops across the interface and exceeds the tension limit of the interface.Once gap is formed between the pile–soil interface, the shear and normal forces are setto zero.

XY

Z 42.0 m

70.0 m

Job Title: Pile Group 4*4, s = 3.0D

Layer_ALayer_BLayer_C1Layer_C2

42.0 m

Figure 5. Finite difference grid for a 4� 4 pile group.



The normal and shear forces are determined by the following equations:

F ðtþDtÞn ¼ knunAþ snA ð5Þ

FðtþDtÞsi ¼ F

ðtÞsi þ ksDu

ðtþ0:5DtÞsi Aþ ssiA ð6Þ

where Fn and Fsi are the normal and shear force, respectively, kn and ks the normal and shearstiffness, respectively, A the area associated with an interface node, Dusi the incremental relativeshear displacement vector and un the absolute normal penetration of the interface node into thetarget face, sn the additional normal stress added due to interface stress initialization, ssi theadditional shear stress vector due to interface stress initialization.

In many cases, particularly when linear elastic analysis is performed, values for interfacestiffness are defined to simulate the non-linear behaviour of a problem. In the present analysis,where non-linear analysis is carried out and the use of interface element covers the soil–pileseparation, the value for the interface stiffness should be high enough, in comparison with thesurrounding soil, in order to minimize the contribution of those elements to the accumulateddisplacements. To satisfy the above requirement, the guidelines of FLAC3D manual [16] proposevalues for kn and ks of the order of ten times the equivalent stiffness of the stiffest neighbouringzone. The use of considerably higher values is tempting as it could be considered as moreappropriate but in that case the solution convergence will be very slow. Based on this principle,a value of 1000 MPa=m was taken for both kn and ks: According to the results of preliminarysingle pile analysis, this value was sufficient to ensure that no additional horizontaldisplacements were attributed to the pile due to the deformation of springs representing theinterface. The bored piles consisted of class C30/35 concrete and their behaviour was consideredlinearly elastic. The modulus of elasticity of the piles was determined using Equation (7) [17],and was found equal to 42 000 MPa; including the stiffening due to the existence of steelreinforcement bars. A reduction in moment of inertia I of the order of 50% for the upper part ofthe pile was applied, due to the fact that the test was extreme enough to produce concretecracking.

Ei1 ¼ Ei281:11=3 ¼ 12 000 F

1=3c28 1:1

1=3 ð7Þ

S = slider T = tensile strength D = dilation ks = shear stiffnesskn = normal stiffness

P

kn

D

T

S ks

target face

Figure 6. Components of the interface constitutive model in FLAC3D:



A group of nine piles fixed in a pile cap with a 3� 3 arrangement was initially considered. Thespacing, defined as the axial distance between the centres of the piles, was taken equal to 2, 3and 6 times the pile diameter. At a second stage, layouts of 3� 2; 3� 1 and a 4� 4 were alsoexamined. For these cases the spacing between the piles was taken as equal to 3 pile diameters.The case of a fixed-head single pile was also considered since its response is to be used tocompare the group responses with. The loading sequence included the initial step, at which theinitial stress field was established, followed by 10 loading steps from 0.1 to 0:8 MN with aconstant increment of 0:1 MN and two further loading steps up to 1.0 and 1:2 MN: The loadwas applied at the top of the central pile and was equal to the mean load multiplied by thenumber of piles. The direction of loading was always the x-direction. As a consequence of thefact that the piles were considered fixed in a rigid pile head, they have all been forced toundertake the same deflection. To simulate this, the nodes at the pile head were considered to beslaves to the node on which the load was applied.

4.2. Numerical results

Figures 7–9 illustrate the displacement contours along the direction of loading at the plane y ¼ 0for the case of the 3� 3 layout with spacing of 2, 3 and 6 diameters (D). The displacementcontours correspond to a mean load of 0:8 MN at the pile cap. The level of interaction betweenpiles and soil can be seen qualitatively from the unification of the displacement contours. Whenspacing is too small (case of Figure 7), a common displacement is observed at the soil surfacebetween the piles, while from a certain level of loading, the resisting zones behind the pilesoverlap. When these zones are plastified, the lateral load capacity is rather the load capacity ofan equivalent single pile containing the piles than the summation of the lateral load capacity ofthe piles. A comparison between Figures 7–9 demonstrates that as spacing increases, the effectof overlapping between the resisting zones becomes less significant.

Figure 7. Displacement contours along the direction of loading for the case of a 3� 3layout with a spacing of 2:0 D:



A detailed comparison of the results demonstrated that spacing significantly affects the load–deflection curve while the number of rows and the total number of piles also play an importantbut less affecting role. Figure 10 illustrates the load–deflection curves at the top of the pile forvarious pile groups together with those of the fixed-head single pile. The stiffer group is the one





consisting of three piles in a row in the direction of loading (layout 3� 1) with a spacing of3:0 D; followed by the group 3� 2 with the same spacing. When examining the groups in a3� 3 layout, it can be verified that when spacing decreases, the stiffness of the group declines.Finally, the load–deflection curve of the 4� 4 group with a spacing of 3:0 D shows the loweststiffness, indicating that the number of piles affects the response of the group. Despite thevariation of the load–deflection curve of each group, it can be concluded that all curves have aform similar to that of the single pile.

As was previously stated, the criterion for the design in the majority of cases of piles to resistlateral loads is not the ultimate lateral capacity but the deflection of the piles under a specificload. From the results of the numerical analyses, it may be concluded that the piles in groupsundergo considerably more deflection for a given mean load Hm per pile (defined as the totalload of the group divided by the number of piles in the group) than a single pile under the sameload. A comparison between the deflection of the single pile and that of the pile group for thesame mean load provides the stiffness efficiency factor defined by the following equation:

RG ¼ymLs

ymGð8Þ

in which ymG and ymLs stand for the deflection at the head of the piles and the single pile underthe same horizontal mean load Hm; respectively. The stiffness of a pile group for a given meanload Hm can then be calculated using Equation (9).

KG ¼ RGKS ð9Þ

in which KS is the stiffness of the single pile for a given load and KG denotes the stiffness of thepile group for the same load. The total group stiffness is determined by multiplying KG with thenumber of piles of the group. Figures 11 and 12 illustrate the variation of the stiffness reductionfactor with row numbers and spacing, respectively. It can be seen that the reduction, as defined

0

0.2

0.4

0.6

0.8

1

1.2

0.00 0.02 0.04 0.06 0.08 0.10

Normalized Deflection y/D

Late

ral M

ean

Load

H (M

N)

x. Hd Sng. Pile Fix. Gr. 1*3, d=3D

x. Gr. 2*3, d=3D Fix. Gr. 3*3, d=6D

x. Gr. 3*3, d=3D Fix. Gr. 3*3, d=2D

x. Gr. 4*4, d=3D

Fi

Fi

Fi

Fi

Figure 10. Numerically established lateral load–deflection curves for the fixed-head single pile, and variousconfigurations of fixed-head groups.



by Equation (8), may attain the level of 40% for groups with multiple rows. The effect becomesless significant in the case of the single row group, where the reduction factor was found of theorder of 0.80. The effect of the pile spacing can be seen in Figure 12. Hence, for the samespacing, the greater the number of the piles in a group the greater the stiffness reduction. For thesame layout, the group with the minimum spacing shows also the maximum reduction.

In order to investigate the effect of interaction accurately, the responses of the piles in 3� 3layouts were examined precisely. As anticipated, the central pile carries the lowest load for thesame deflection, presenting the minimum stiffness, while the two corner piles on the direction ofloading (P7 and P9) carry the biggest load, presenting the maximum stiffness. Figures 13–15show the response of the piles in the case of spacing equal to 2.0, 3.0 and 6:0 D; respectively. Itcan be seen that for any level of loading, the corner piles resist more than the others, while the

0.3

0.4

0.5

0.6

0.7

Fix. Gr. 3*3,d=6D

Fix. Gr. 3*3,d=3D

Fix. Gr. 3*3,d=2D

Stif

fnes

s R

educ

tion

Fac

tor

RG

Defl. 1%D Defl. 3%D Defl. 5%D

Figure 12. Variation of stiffness reduction factor with spacing, for a deflection of 1, 3 and 5% D at thehead of a fixed-head pile group.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fix. Gr. 1*3,d=3D

Fix. Gr. 2*3,d=3D

Fix. Gr. 3*3,d=3D

Fix. Gr. 4*4,d=3D

Stif

fnes

s R

educ

tion

Fac

tor

RG

Defl. 1%D Defl. 3%D Defl. 5%D

Figure 11. Variation of stiffness reduction factor with group size for a deflection of 1, 3 and 5% D at thehead of a fixed-head pile group.



internal pile always carries the smaller load. It should also be noted that the front piles (P8 andP9) resist more than the back row (P2 and P3). Figures 16–18 illustrate the normalized loadundertaken by the piles of the group as a function of the normalized deflection. The central pileP5 initially carries the 65, 65 or 69% of the mean load for spacings of 2.0, 3.0 and 6:0 D;

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.00 0.10 0.20


Late

ral L

oad

H (

MN

)

Loading direction

P2 P3

P5 P6

P8 P9

d

P1

P2

P3

P4

P5

P6

P8

P9

P7

Figure 14. Lateral load–deflection curves of piles P2, P3, P5, P6, P8 and P9 in a 3� 3group with an axial distance of 3:0 D:

0

200

400

600

800

1000

1200

1400

1600

0.00 0.05 0.10 0.15 0.20


Late

ral L

oad

H (

kN)

P2 P3

P5 P6

P8 P9

Loading direction

d

P1

P2

P3

P4

P5

P6

P8

P9

P7




respectively. These percentages gradually increase to 78, 75 or 75%, respectively, when thedeflection level becomes greater than a value of 10% of the pile diameter. On the other hand,pile P9 initially carries 120, 120 and 115% of the mean load. This percentage gradually decreaseswith deflection level, becoming 117, 116 or 112% when deflection increases to 10% of the pilediameter. The loads transferred to the other piles of the group remain within the limits of thesetwo piles. It can be observed that the load carried by the piles of the layout with a spacing of6:0 D remains invariant no matter the deflection levels and that the response of the piles is

60%

80%

100%

120%

140%

0.00 0.05 0.10 0.15 0.20 0.25


Nor

mal

ized

Lat

eral

Loa

d H

/Hm

Loading direction

P2 P3 P5

P6 P8 P9

d

P1

P2

P3

P4

P5

P6

P8

P9

P7

Figure 16. Variation of normalized load with normalized deflection for piles P2, P3, P5, P6, P8 and P9 in a3� 3 layout with a spacing of 2:0 D:

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.00 0.05 0.10


Late

ral L

oad

H (

MN

) Loading direction

P2 P3

P5 P6

P8 P9

d

P1

P2

P3

P4

P5

P6

P8

P9

P7




almost linear (Figure 18). Comparable results are shown in Figures 19 and 20 for the 4� 4group with a spacing of 3:0 D: As anticipated, in this case, the effect of interaction is higher. Thecentral pile P10 initially carries 60% of the mean load and the corner pile P13 140%. Thesepercentages gradually change with deflection level, becoming 134 and 70% when deflectionattains a value of 10% of the pile diameter.

60%

80%

100%

120%

140%

0.00 0.03 0.06 0.09 0.12 0.15


Nor

mal

ized

Hor

izon

tal L

oad

H/H

m

P2 P3 P5P6 P8 P9

Loading direction

d

P1

P2

P3

P4

P5

P6

P8

P9

P7


60%

80%

100%

120%

140%

0.00 0.05 0.10 0.15 0.20 0.25


Nor

mal

ized

Hor

izon

tal L

oad

H/H

m

Loading direction

P2 P3 P5P6 P8 P9

d

P1

P2

P3

P4

P5

P6

P8

P9

P7




40%

60%

80%

100%

120%

140%

160%

0.00 0.05 0.10 0.15 0.20 0.25 0.30


Nor

mal

ized

Lat

eral

Loa

d H

/Hm

P1 P2 P5 P6P9 P10 P13 P14

P1

P2

P4

P3

P5

P16

P13P9

P6 P14P10

Figure 20. Variation of normalized lateral load with normalized deflection of piles P1, P2, P5, P6, P6, P9,P10, P13 and P14 in a 4� 4 group with an axial distance of 3:0 D:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.00 0.05 0.10 0.15 0.20


Lat

eral

Loa

d H

(M

N) P1

P2

P4

P3

P5

P16

P13P9

P6 P14P10

P1 P2

P5 P6

P9 P10

P13 P14

Figure 19. Lateral load–deflection curves of piles P1, P2, P5, P6, P6, P9, P10, P13 and P14 in a 4� 4 groupwith an axial distance of 3:0 D:



The bending moment in each pile was computed from the displacement field at itscentre using Equation (10), where the curvature of the pile is obtained by numericaldifferentiation.

M ¼ �EId2y

dx2ð10Þ

Figure 21 illustrates the bending moment in piles P2, P5 and P9 corresponding to an appliedmean load of 0:4 MN for a 3� 3 group with a spacing of 3:0 D: Pile P2 is in the middleof the rear row, P5 is the central pile, while P9 is at the corner of the front row. It may be notedthat differences between bending moment of these piles of the group are less than the 10%,despite the fact that the load carried by the corner piles is almost double the load of thecentral pile. This can be attributed to the fact that the resistance of soil zones in front of thepiles carrying higher loads is larger, since the effect of interaction at these zones is small.The bending moment curve predicted for an identical fixed-head single pile is essentiallydifferent. At the head of the pile, the bending moment predicted for the single pile is threetimes less than the values predicted for the piles of the 3� 3 group. The difference of thepredicted values for the maximum bending moment along the piles between the single pile andthe piles of the group, on the other hand, does not exceed 20%. It may be noticed, however, thatwhile the bending moment of the single pile approaches zero at the mid-depth of the pile, thepiles of the group are subjected to bending moment for significantly greater depth. The sameconclusions can be drawn when the applied mean load is increased to 0:8 MN; as illustrated inFigure 22.

0

10

20

30

40

50

-0.20 -0.10 0.00 0.10 0.20 0.30 0.40

Moment (MN.m)

Dep

th (

m)

P2

P5

P9

Fxd Single

Loading direction

d

P1

P2

P3

P4

P5

P6

P8

P9

P7

Figure 21. Numerically established distribution of bending moments along piles P2, P5 and P9 of a 3� 3layout with a spacing of 3:0 D; compared to the predicted curve of an identical fixed-head single pile, for a

mean lateral load of 0:4 MN:



5. RESPONSE PREDICTION FOR FIXED-HEAD PILE GROUPS

The response of any type of group could be established numerically, based on the methodologyapplied in the previous paragraphs. It must be recognized, however, that the procedure isextremely computationally demanding. The calculation time for a pile group including 10 loadincrements was approximately 20 h on a Pentium IV 1900 MHz with 512 MB RAM. Takinginto account that a 3-D grid preparation is also a difficult and time-consuming task, one candeduce that this kind of analysis could be applied only for a limited number of cases. Moreover,the database files of the results for each layout including 10 load increments requiredapproximately 250 MB of disc space.

Based on the fact that the load–deflection curves of each group have a similar form to that ofthe single pile, it seems essential to derive a relationship giving the ability to define the load–deflection curve of a given pile group using that of a single pile. The latter can be establishedusing three-dimensional analysis, an in situ test or even an accurate p–y analysis. It is evidentthat such a relationship would be eventually affected by the load–deflection curve of a single pilein a given soil profile, by the spacing and the number of columns and rows in the pile group, andthe total number of piles.

An extensive effort has been undertaken for the determination of a formula, which could bevalid at least for pile groups in similar soil conditions as the one examined in this paper. It isevident that such a relationship would eventually be affected at least by the load–deflectioncurve of a single pile in a given soil profile, by the spacing and the number of columns and rowsin the pile group, and the total number of piles. Equation (11) was established to calculate thevariable amplification factor in order to allow the response prediction of pile groups with a rigidcap. To determine the aforementioned equation, a curve-fining procedure, of which the code is

0

10

20

30

40

50

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80

Moment (MN.m)

Dep

th (m

)

P2

P5

P9

Fxd Single

Loading direction

d

P1

P2

P3

P4

P5

P6

P8

P9

P7

Figure 22. Numerically established distribution of bending moments along piles P2, P5 and P9 of a 3� 3layout with a spacing of 3:0 D; compared to the predicted curve of an identical fixed-head single pile, for a

mean lateral load of 0:8 MN:



given in Appendix A, was applied using the MS EXCEL program and the built-in Visual Basiclanguage. The main variables determining the deflection amplification factor for groups are thedeflection of the single pile, the spacing between the piles, the number of rows and columns in apile group and the total number of piles, which are included in Equation (11). The deflection isprofoundly non-linear and for this reason, at least three components were needed inEquation (11), in which the deflection of the single pile is introduced with a different weightingfactor (a; b and g).

Ra ¼

3

nx

� �0:2yaDd

þ1:1

ffiffiffid

pd

lnðnx þ nyÞybD log

ðn4x þ nyÞ0:8

yD

" #exp

1

d

� �

ð0:7ygDd3Þ0:03

ð11Þ

where Ra is the deflection amplification factor, yD the normalized deflection of the fixed-headsingle pile defined as y=D; d the relative pile spacing defined as s=D; nx; ny the number of piles inthe direction of loading and the perpendicular one, respectively, and a; b; g the parameters to bedetermined by the curve-fining procedure.

Using the deflection amplification factor from Equation (11) for a given mean horizontalload, Equation (12) provides the group deflection yG:

yG ¼ Rayd ð12Þ

The most suitable values for a; b and g were automatically defined by the curve-finingprocedure as a ¼ 0:8; b ¼ 0:2; and g ¼ 0:1: In Figure 23, the bold lines represent the pile group

0

0.2

0.4

0.6

0.8

1

1.2

0.00 0.05 0.10 0.15 0.20

Deflection y (m)

Late

ral M

ean

Load

Hm

(M

N)

Fix. Gr. 1*3, d=3D

Prediction for Layout 1*3, d=3D

Fix. Gr. 2*3, d=3D


Fix. Gr. 3*3, d=3D


Fix. Gr. 4*4, d=3D


Figure 23. Comparison between numerically established load–settlement curves using FLAC3D andthose predicted by Equations (11) and (12), for various fixed-head pile groups configurations

with spacing s ¼ 3:0 D:



load–deflection curves calculated using FLAC3D; while the dashed lines correspond to thepredicted curves using Equations (11) and (12). The calculated and predicted curvesdemonstrate notable agreement.

The validity of Equation (11) has been also verified using the experimental results given byWakai et al. [12] for a fixed-head single pile and fixed-head pile groups. The model consists ofaluminium piles with an outside diameter of 50 mm and a pile length of 1500 mm in a sandy soil(Onahama sand) arranged in a 3� 3 layout with a spacing of 2:5 D: Figure 24 illustrates theresponse of the fixed-head single pile and that of the fixed-head pile group. In the same figure,the prediction by Wakai et al. [12] resulting from a 3-D non-linear analysis is shown togetherwith the prediction using Equation (11) in which the determined values a ¼ 0:8; b ¼ 0:2; andg ¼ 0:1 were used. It can be seen that this equation provides a prediction sufficiently close toboth the measured curve and to that curve provided by the 3-D analysis of Wakai et al. [12].

The verification of the methodology to full or large-scale fixed-head pile groups was notfeasible since no measurements for such tests subjected to lateral loading were available.However, in order to examine the validity to comparable conditions, the data from a pile groupwith moment-free connection [18] and a free-head pile group [19] were used. Brown et al. [18]carried out tests on a large-scale pile group subjected to lateral loading. Their model consists ofclosed-end steel piles with an outside diameter of 273 mm and a pile length of 13:1 m: The pileswere driven in a preconsolidated clay formation arranged in a 3� 3 layout with a spacing of3:0 D: The equal deflection level was applied to all piles using a loading frame with moment-freeconnections. Figure 25 illustrates the response of the single pile and that of the moment-freepile group. The prediction provided by Equation (11) using the previously determined valuesa ¼ 0:8; b ¼ 0:2 and g ¼ 0:1 is satisfactorily close to the measured deflection values.

0

0.4

0.8

1.2

1.6

0.00 0.02 0.04 0.06 0.08 0.10


Hor

izon

tal M

ean

Load

Hm

(kN

)

Measured Fix. Head Single Pile (Wakai et al.)

Measured Fix. Gr. 3*3, d=2.5D (Wakai et al)

Calculated by Wakai et al., Fix. Gr. 3*3, d=2.5D

Prediction for Fix. Gr. 3*3, d=2.5D (Equation 11)

Figure 24. Comparison between measured, calculated [12] and predicted by Equation (11)load–deflection curve.



Rollins et al. [19] performed a lateral loading test on a large-scale free-head pile group.Their model consists also of closed-end steel piles with an outside diameter of 324 mmand a pile length of 9:1 m: The piles were driven in a composite soil profile consistingof gravel fill at the top while layers of clayey, silty and sandy soil were encountered down thedepth of 11:0 m: The pile arrangement was in a 3� 3 layout with a spacing of 3:0 D:Load was applied to every pile using different load cells connected to a common loadingframe. Thus, for the same central frame loading each pile was able to carry different load anddeflection (free-head pile), according to the resistance of the surrounding soil. Figure 26illustrates the response of the free-head single pile and that of the free-head pile groupafter averaging the group load and deflection. The prediction using Equation (11) is shownby the dotted line with triangle markers. This prediction is based on values a ¼ 0:8; b ¼ 0:2and g ¼ 0:1; which it should be reminded are found for fixed-head pile groups. In case that aprecise prediction for this free-head pile group is needed, the application of the curve-finingsubroutine of Appendix A suggests the use of a ¼ 1:05; b ¼ 0:25 and g ¼ 0:10 for which theprediction presented by the dashed line with circular markers is very close to the measuredpoints.

Even though Equation (11) seems to predict the response of a pile group suitably incompletely different soil profiles, for pile sizes other than the one from which it was originallyderived and for different methods of construction, it would be unwise to be used in any soilprofile. The applicability of the proposed formulae to a different soil profile must be verified orthe proposed equations be readjusted by numerical analyses preferably in conjunction within situ test results.

0

10

20

30

40

50

60

70

80

90

0.00 0.05 0.10 0.15 0.20 0.25


Hor

izon

tal M

ean

Load

Hm

(kN

)

Measured Moment-Free Single Pile (Brown et al.)

Measured Moment-Free 3*3, d=3.0D (Brown et al.)

Prediction Fix Hd 3*3, d=3.0D (Eq. 11: α=0.80, β =0.20, γ =0.10)

Figure 25. Comparison between measured load–deflection curve [18] and prediction by Equation (11).



6. CONCLUSIONS

In this paper, the effects of the interaction to the response of a pile group fixed in a rigid capwere examined for various group configurations under horizontal loading. Based on the resultsof the parametric three-dimensional non-linear numerical analysis, the response of particularpiles in the group was investigated and their contribution to the entire group behaviour wasquantified.

For the particular soil profile and large diameter piles embedded in soft to medium clay withtheir tips resting in very dense sandy gravel, it was found that the interaction significantly affectsthe stiffness efficiency factors of the pile groups. It was revealed that the stiffness efficiencyfactor depends on the pile arrangement. At low deformation levels, the interaction has itsmaximum influence with the central piles taking approximately 50% of the load of the cornerpiles for a 3� 3 group with a spacing of 3:0 D: This percentage reduces to 42% in the case of a4� 4 group with the same spacing. When deflection increases and plastification occurs, theinfluence of the interaction gradually decreases, without however any significant variation effecton both the lateral group efficiency and the stiffness reduction factor.

Even though 3-D non-linear analysis is able to establish the load–deflection curve for any typeof pile group with the appropriate accuracy, it is recognized that the procedure is verydemanding computationally. To overcome this drawback and aid prediction of the load–deflection curve of a fixed-head pile group, when neither the finances nor the time for a 3-Danalysis is available, Equations (11) and (12) are proposed. In this work, the validity of theproposed relationships was also examined in the case of different soil profiles (sandy and clayeysoils) as well as for small experimental size and large-scale single pile and pile group. The resultswere extremely encouraging even for large-scale pile groups with different boundary conditions(moment-free and free-head). However, it is still believed that any unverified use could be

0

50

100

150

0.00 0.05 0.10 0.15 0.20 0.25


Hor

izon

tal M

ean

Load

Hm

(kN

)

Measured Free Hd Single Pile (Rollins et al.)

Measured Free Hd Gr. 3*3, d=3.0D (Rollins et al.)

Prediction for Free Hd 3*3, d=3.0D (Eq. 11: α = 0.80, β = 0.20, γ = 0.10)

Prediction for Free Hd 3*3, d=3.0D (Eq. 11: α = 1.05, β = 0.25, γ = 0.10)

Figure 26. Comparison between measured load–deflection curve [19] and prediction by Equation (11).



unwise. Since no analyses have been carried out for different soil profiles, the applicability of theproposed formulae to a different soil profile or constitutive models must be verified or theproposed equations be readjusted by numerical analyses preferably in conjunction with in situtest results. It should be also noted that the proposed methodology is valid for monotonicloading and that for more complex cases, such as cyclic or dynamic loading, the use of moreadvanced constitutive models for the simulation of the soil between piles is warranted. Such agoal is beyond the scope of the present work or even of a single research team and for thisreason the validity of the proposed relationships should be examined by the combined effort ofthe international research community.

APPENDIX A: CODE OF CURVE-FINING PROCEDURE

Sub curv fin()0curv fin Macro0Macro recorded 8/12/2003 by Emilios Comodromos

Dim n y, n x, n val, n val single As Integer

Dim n a, n b, n c As Integer

Dim measur val(1 To 20) As Double

Dim measur single(1 To 20) As Double

Dim estimat val(1 To 20) As Double

Dim ra fin(1 To 20) As Double

Dim rowNum As Integer, colNum As Integer, currCell As Range

Dim ra(1 To 10, 1 To 100, 1 To 100, 1 To 100) As Double

Dim deflection(1 To 10, 1 To 100, 1 To 100, 1 To 100) As Double

Application.Calculation = xlManual

On Error Resume Next

Sheets(00curv fin00).Select

Range(00x1800).Select

colNum = ActiveCell.Column

rowNum = ActiveCell.Row

Set currCell = ActiveSheet.Cells(rowNum, colNum)

colNum = currCell.Value

colNum or = colNum

0insert data0n y: number of columns in the group0n x: number of rows in the group0spacing: pile spacing0numblr of points to be taken into account0lower and upper bound values for a; b and g with0corresponding step increment

rowNum = 2


n y = currCell.Value



rowNum = 3


n x = currCell.Value

rowNum = 4


spacing = currCell.Value

rowNum = 5


n val = currCell.Value

rowNum = 20


par a min = currCell.Value

rowNum = 21


par a max = currCell.Value

rowNum = 22


par a step = currCell.Value

rowNum = 24


par b min = currCell.Value

rowNum = 25


par b max = currCell.Value

rowNum = 26


par b step = currCell.Value

rowNum = 28


par c min = currCell.Value

rowNum = 29


par c max = currCell.Value

rowNum = 30


par c step = currCell.Value

Range (00j200).Select

colNum = ActiveCell.Column



n val single = currCell.Value



0get the measured deflection values for the single pile

For k = 1 To n val single

rowNum = 5 + k


measur single(k) = currCell.Value

Next k

0get the measured deflection values for the piles group

Range(00x1800).Select

colNum = colNum or



For k = 1 To n val

rowNum = 5 + k


measur val(k) = currCell.Value

Next k

0Find the amplification factor ra for all combinations

n a = CInt((par a max - par a min) / par a step) + 1

n b = CInt((par b max - par b min) / par b step) + 1

n c = CInt((par c max - par c min) / par c step) + 1

For i = 1 To n val single

For k a = 1 To n a

For k b = 1 To n b

For k c = 1 To n c

par a = par a min + k a n par a step

par b = par b min + k b n par b step

par c = par c min + k c n par c step

d1 = measur single(i)

w1 = (3 / n x) ^ 0.2 / spacing n d1 ^ par a

w2 1 = 1.1 n spacing ^ 0.5 / spacing n Log(n x + n y) n d1 ^ par b

w2 2 = Log((n x ^ 4 + n y) ^ 0.8 / d1) / Log(10)

w2 3 = w2 2 n Exp(1 / spacing)

w3 = (0.7 n d1 ^ par c n spacing ^ 3) ^ 0.03

ra(i, k a, k b, k c) = (w1 + w2 1 n w2 3) / w3

Next k c

Next k b

Next k a

Next i

0Estimate the deflection of the group for all combinations


For k a = 1 To n a



For k b = 1 To n b

For k c = 1 To n c

deflection(i, k a, k b, k c) = ra(i, k a, k b, k c) n measur single(i)

Next k c

Next k b

Next k a

Next i

0Find the most accurate values for a; b and g within the limit given0Estimate the delection of the group0Find the deviation between calculated and estimated deflection0for the above values A,B and C

n val i = n val single

If n val i > n val Then n val i = n val

resid ini = 100000

For k a = 1 To n a

For k b = 1 To n b

For k c = 1 To n c

resid = 0

For i = 1 To n val i

resid = resid + deflection(i, k a, k b, k c) - measur val(i)

Next i

If Abs(resid) 5 resid ini Then

resid ini = Abs(resid)

k a fin = k a

k b fin = k b

k c fin = k c

End If

Next k c

Next k b

Next k a

par a fin = par a min + k a fin n par a step

par b fin = par b min + k b fin n par b step

par c fin = par c min + k c fin n par c step0Chose of alternative values for a; b and g if required0Estimate the delection of the group for these values0Find the deviation between calculated and estimated deflection0for the above values a; b and g

rowNum = 34


currCell.Value = par a fin

rowNum = 35


currCell.Value = par b fin



rowNum = 36


currCell.Value = par c fin

rowNum = 40


par a fin = currCell.Value

rowNum = 41


par b fin = currCell.Value

rowNum = 42


par c fin = currCell.Value

par a = par a fin

par b = par b fin

par c = par c fin


d1 = measur single(i)

w1 = (3 / n x) ^ 0.2 / spacing n d1 ^ par a

w2 1 = 1.1 n spacing ^ 0.5 / spacing n Log(n x + n y) n d1 ^ par b

w2 2 = Log((n x ^ 4 + n y) ^ 0.8 / d1) / Log(10)

w2 3 = w2 2 n Exp(1 / spacing)

w3 = (0.7 n d1 ^ par c n spacing ^ 3) ^ 0.03

ra fin(i) = (w1 + w2 1 n w2 3) / w3

estimat val(i) = ra fin(i) n measur single(i)

Next i

resid p = 0

For i = 1 To n val i

resid p = resid p + estimat val(i) - measur val(i)

Next i

rowNum = 46


rowNum = rowNum + 1


currCell.Value = ra fin(i)

Next i

rowNum = 64


rowNum = rowNum + 1


currCell.Value = estimat val(i)

Next i



0Print the deviation for the proposed by the curve fining procedure0parameters a; b and g

rowNum = 77


currCell.Value = resid ini

0Print the deviation for the alternative (if any) values of a; b and g

rowNum = 79


currCell.Value = resid p

Application.Calculation = xlAutomatic

End Sub

APPENDIX B: NOMENCLATURE

A area associated with an interface nodec cohesioncu undrained shear strength of clayey soilsd relative centre to centre pile spacing defined as s=DD pile diameterEs soil modulusEI flexural stiffness of pileFc28 strength of the concrete corresponding to 28th dayEi28 Young modulus corresponding to 28th dayEi1 Young modulus corresponding to age significantly greater than 28 daysFn normal force (interface element)Fsi shear force (interface element)G shear modulusK bulk moduluskn normal stiffness (interface element)ks shear stiffness (interface element)M bending momentnL lateral load group efficiencynx number of piles in the direction of loading of a pile groupny number of piles in the direction perpendicular to the loading direction of a pile groupPx axial load of pileRa deflection amplification factorRf reduction factors centre to centre pile spacingun absolute normal penetration of the interface node into the target facex length along piley deflectionyD normalized deflection



ys deflection of a single pilez depth below soil surface

Greek letters

b angle giving the direction of loading of pile rowd pile head rotationDusi incremental relative shear displacement vectorsn normal stressj0 angle of internal friction

REFERENCES

1. Poulos HG, Davis EH. Pile Foundation Analysis and Design. Wiley: Singapore, 1980.2. Reese LC. Laterally loaded piles: program documentation. Journal of Geotechnical Engineering Division 1977;

103:287–305.3. Matlock H. Correlations for design of laterally loaded piles in soft clay. Proceedings of 2nd Offshore Technology

Conference, Houston, 1970; 577–594.4. Reese LC, Cox WR, Koop FD. Analysis of laterally loaded piles in sand. Proceedings of 6th Offshore Technology

Conference, Houston, 1974; 473–483.5. Reese LC, Welch RC. Lateral loadings of deep foundations in stiff clay. Journal of Geotechnical Engineering Division

1975; 101:633–649.6. Georgiadis M. Development of p–y curves for layered soils. In Geotechnical Practice in Offshore Engineering, Wright

SG (ed.). American Society of Civil Engineers: New York, 1983; 536–545.7. Prakash S, Sharma D. Pile Foundation in Engineering Practice. Wiley: New York, 1990.8. Oteo CS. Displacement of a vertical pile group subjected to lateral loads. Proceedings of 5th European Conference of

Soil Mechanics and Foundation Engineering, Madrid, 1972; 397–405.9. Poulos HG. Behaviour of laterally loaded piles: I-single pile, and II-pile group. Journal of Soil Mechanics and

Foundation Division 1971; 97:711–751.10. Poulos HG. Pile behaviour}theory and application. G !eeotechnique 1989; 39(3):366–415.11. Randolph MF. The response of flexible piles to lateral loading. G !eeotechnique 1981; 31(2):247–259.12. Wakai A, Gose S, Ugai K. 3-D Elasto-plastic finite element analyses of pile foundations subjected to lateral loading.

Soils and Foundations 1999; 39(1):97–111.13. Comodromos E. Response prediction of horizontally loaded pile groups. Geotechnical Engineering Journal 2003;

34(2):123–133.14. Fleming WG, Weltman AJ, Randolph MF, Elson WK. Piling Engineering. E&FN Spon.: New York, 1992.15. Bransby MF, Springman SM. Selection of load-transfer functions for passive lateral loading of pile groups.

Computer and Geotechnics 1999; 24(3):155–184.16. Itasca. FLAC3D; Fast Lagrangian Analysis of Continua. Itasca Consulting Group; User’s Manual, Version 2.1.

Minneapolis, 2002.17. Covec J. M !eemento d ’ emplois du b !eeton aux !eetat limites et r !eeglements annexes. Technique et Documantation-

Lavoisier: Paris, 1980.18. Brown DA, Reese LC, O’Neill MW. Cyclic lateral loading of a large-scale pile group. Journal of Geotechnical

Engineering Division 1987; 113(11):1326–1343.19. Rollins KM, Peterson KT, Weaver TJ. Lateral load behaviour of full-scale group in clay. Journal of Geotechnical and

Geoenvironmental Engineering 1998; 124(6):468–478.



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Response Evaluation for Horizontally Loaded