Computers and Geotechnics 62 (2014) 164–174

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Computers and Geotechnics

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Variation of limiting lateral soil pressure with depth for pile rows in clay

http://dx.doi.org/10.1016/j.compgeo.2014.07.0110266-352X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Address: School of Civil Engineering, Aristotle University of Thessaloniki,Thessaloniki 541 24, Greece. Tel.: + 30 2310 994 227; fax: + 30 2310 995 619.

E-mail address: kgeorg@civil.auth.gr

K. Georgiadis ⇑School of Civil Engineering, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki 541 24, Greece

a r t i c l e i n f o

Article history:Received 27 April 2014Received in revised form 18 July 2014Accepted 19 July 2014

Keywords:PilesPile groupsLateral loadsNumerical analysisBearing capacityClays

a b s t r a c t

Pile group interaction effects on the lateral pile resistance are investigated for the case of a laterallyloaded row of piles in clay. Both uniform undrained shear strength and linearly increasing with depthshear strength profiles are considered. Three-dimensional finite element analyses are presented, whichare used to identify the predominant failure modes and to calculate the reduction in lateral resistancedue to group effects. A limited number of two-dimensional analyses are also presented in order toexamine the behaviour of very closely spaced piles. It is shown that, contrary to current practice, groupeffects vary with depth; they are insignificant close to the ground surface, increase to a maximum valueat intermediate depths and finally reduce to a constant value at great depth. The effect of pile spacing andpile–soil adhesion are investigated and equations are developed for the calculation of a depth dependentreduction factor, which when multiplied by the limiting lateral pressure along a single pile, provides thecorresponding variation of soil pressure along a pile in a pile row. This reduction factor is used to performp–y analyses, which show that, due to this variation of group effects on the lateral soil pressures withdepth, the overall group interaction effects depend on the pile length. Comparisons are also made withapproaches used in practice that assume constant with depth reduction factors.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction where p and p are the limiting load per unit length for a single

A crucial parameter in the design of laterally loaded piles is thelimiting lateral soil resistance and its variation with depth. This iscommonly expressed in terms of the lateral bearing capacity factorNp, which is defined as the limiting lateral load per unit pile length,pu, normalised with respect to the pile diameter, D, and theundrained soil shear strength, su. For single piles in clay, it iswell-established that Np starts form a small value at the groundsurface, increases non-linearly with depth in the upper part ofthe pile, reaches an ultimate value at some depth and remains con-stant in the lower part of the pile [1–3]. Fig. 1 presents the varia-tion of the bearing capacity factor Np with depth for uniform soilstrength and different values of the adhesion factor a (=limitinginterface shear stress/ undrained shear strength).

In the case of pile groups in clay, the limiting lateral soil pres-sures (and consequently also the lateral bearing capacity factor)are generally reduced, due to group interactions effects, comparedto those that act on single piles. A lateral bearing capacityreduction factor can be defined as:

fm ¼pu;g

pu;s¼ Np;g

Np;sð1Þ

u,s u,g

pile and a pile in a pile group, respectively. Np,s (=pu,s/suD) and Np,g

(=pu,g/suD) are the lateral bearing capacity factors for a single pileand a pile in a pile group, respectively.

Group interaction effects for laterally loaded piles have beeninvestigated by several researchers using large scale load tests,model tests and numerical analysis. Large scale load tests havebeen performed on pile groups in clayey [4–7], sandy [8–10] andmulti-layered soils [11,12]. Model tests have been reported forboth 1 g [13,14] and centrifuge conditions [15,16]. Numericalinvestigations of group effects include two-dimensional planestrain simulations of both active and passive loading of pile groups,such as those reported by Chen and Poulos [17], [18], Bransby [19],Bransby and Springman [20], Chen and Martin [21] and Georgiadiset al. [22–24], and three-dimensional studies [11,25,26].

In practice, a single reduction factor (called the p-multiplier inconventional p–y analysis) is assigned to each pile of a pile group[7,27,28], which depends solely on pile spacing and does not takeaccount of the variation of group effects with depth. As a conse-quence, a very wide range of reduction factors can be found inthe literature for the same pile spacing.

Georgiadis et al. [24] performed two-dimensional analyticalupper bound plasticity calculations, numerical limit analyses anddisplacement finite element analyses for the determination of theultimate pu (lower part of a pile) for the case of an infinite numberof laterally loaded piles in a row. This specific case is relevant to a

Nomenclature

D pile diameterEp modulus of elasticity of pileEu undrained modulus of elasticityfm lateral bearing capacity reduction factorfmu ultimate lateral bearing capacity reduction factorHug lateral capacity of pile in pile groupHus lateral capacity of single pilek undrained shear strength increase with depthKo lateral earth pressure coefficient at-restL pile lengthNp lateral bearing capacity factorNpo lateral bearing capacity factor at ground surfaceNp,g pile in pile group lateral bearing capacity factorNp,s single pile lateral bearing capacity factorNpu ultimate lateral bearing capacity factorNpu,g pile in pile group ultimate lateral bearing capacity factorNpu,s single pile ultimate lateral bearing capacity factorNpu(2D) two-dimensional failure ultimate lateral bearing capac-

ity factorp lateral load per unit pile lengthpu ultimate lateral load per unit length

pu,g pile in pile group ultimate lateral load per unit lengthpu,s single pile ultimate lateral load per unit lengths centre-to-centre pile spacingsu undrained shear strengthsuo undrained shear strength at ground levels1 pile spacing beyond which Np = Nps

y lateral pile displacementz depthZ normalised depthzo depth of no pile group interactionzt tension crack deptha adhesion factorb1, b2, b3 functions of pile spacingc soil bulk unit weightcp pile unit weightk non-dimensional factormp Poisson’s ratio of pilemu undrained Poisson’s ratiora active earth pressurerp passive earth pressuresf ultimate shear stress along discontinuity

K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174 165

number of practical problems, such as the analysis of landslide sta-bilising piles and soldier piles for earth retaining structures. Basedon these calculations, a practically exact (very close upper andlower bounds of the exact values were achieved) design chartwas presented for the calculation of the limiting lateral pressureat depth (Fig. 2) and an empirical equation was proposed for theultimate (at depth) lateral bearing capacity factor Npu.

Fig. 1. Variation of Np with z/D for a single pile in homogeneous clay with differentinterface adhesion factors a (after Georgiadis and Georgiadis [2]).

This paper presents three-dimensional finite element analysesof an infinite number of laterally loaded piles in a row in clay.The problem definition is shown in Fig. 3. The variation of groupeffects with depth is investigated and depth dependent lateralbearing capacity reduction factors are calculated. Both homoge-neous strength and strength linearly increasing with depth profilesare considered. A limited number of two-dimensional plane strainfinite element analyses are also presented in order to investigatethe behaviour of pile rows with very closely spaced piles.

2. Finite element analyses

Two sets of finite element analyses are presented. The main setinvolves three-dimensional analyses of laterally loaded rows ofpiles, performed with the finite element program Plaxis 3DFoundation V2.2 [29], in which several different centre-to-centrepile spacings, s, normalised with respect to the pile diameter, D,

Fig. 2. Variation of Npu with s/D for a row of piles in clay (after Georgiadis et al.[24]).

D

L zCLAY

su = constantorsu = suo + k·z

ss

ss

s

H

Fig. 3. Problem geometry.

166 K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174

are considered. The same pile length L = 40 m is maintained in allanalyses, which is long enough to allow the determination of thelimiting load profile down to a sufficient depth. Preliminary analy-sis performed with three different pile diameters proved that theeffect of pile diameter D on the variation of the limiting soilpressure with the normalised depth z/D is negligible. Therefore,the same pile diameter of D = 1 m was adopted in the analysespresented in the following sections. It is noted that for these piledimensions and typical pile and soil Young’s moduli, the pilesexamined are flexible according to the Broms [30] classificationsince bL� 2.5. However, the discussion and equations presentedin the paper are valid for both flexible and rigid piles.

Taking advantage of the loading and geometrical symmetry,only one pile needs to be modelled, as seen in Fig. 4, which showsa typical finite element mesh for s/D = 6. 15-node wedge elementswere used for both the soil and the pile and 8-node quadrilateralinterface elements were used for the pile–soil interface. An averageof approximately 15,000 elements was used in all analyses. Thebottom boundary of the mesh was fixed in all directions andthe vertical boundaries were fixed only in the normal direction.

The pile was modelled as linear elastic with Young’s modulusEp = 2.9�107 kPa, Poisson’s ratio vp = 0.1 and unit weight cp = 25kN/m3. The soil was modelled as linear elastic – perfectly plasticwith a Tresca failure criterion. Analyses were performed witheither uniform undrained shear strength su = 50 kPa and 100 kPaor with linearly increasing with depth undrained shear strength

80D

50D

s

X

YZ

Fig. 4. Typical finite element mesh for s/D = 6.

k = 1, 1.5 and 2 kPa/m. It is noted that the lateral bearing capacityfactor Np (=pu/suD), which is used to subsequently calculate thep-multiplier through Eq. (1), is independent of the value of su. Inthe majority of the analyses in which the undrained shear strengthwas taken to increase linearly with depth, a nominal strengthintercept of suo = 1 kPa was assigned at the ground surface. Analy-ses, however, were also performed with greater values of thestrength intercept, suo. In all cases, an undrained Poisson’s ratioof vu = 0.49 and an undrained Young’s modulus over shear strengthratio, Eu/su = 200 were selected. The bulk unit weight was taken asc = 18 kN/m3. Analyses were also performed with c = 16 kN/m3 and20 kN/m3, which showed that the results are not affected by c forthis range of values. The interface behaviour was modelled as elas-tic – Coulomb plastic with tension cut-off [29]. A zero interfacefriction angle, a limiting shear stress, sf = a�su and zero tensilestrength were assigned to the soil–pile interface. Several valuesof the adhesion factor at the pile–soil interface were considereda = 0.1, 0.3, 0.5, 0.7, 1.

All analyses were performed by first generating initial stressesthrough a Ko procedure. Different values of Ko were initially consid-ered in order to verify that the limiting lateral soil–pile pressureprofile, which is of interest in this study, is independent of theselection of Ko. All results presented in the following sections wereobtained with Ko = 1. After the generation of initial stresses, thepile was ‘‘wished-in-place’’ by changing the material propertiesof the appropriate volume elements from soil to concrete andfinally a horizontal load was applied at the pile head. The distribu-tion of pu along the length of the pile was determined in each casefrom the p–y curves at various depths. These were obtained bycombining the first derivative of the shear force versus depthdiagrams with the deflection versus depth diagrams at severalapplied lateral load levels (more details on this procedure are givenin Georgiadis and Georgiadis [2]).

As mentioned above, a limited number of two-dimensionalplane strain finite element analyses were also performed. The in-plane geometry and material properties of the three-dimensionalanalyses were maintained, with the exception of the pile row(vertical wall in 2D) which was modelled as a plate element withequivalent axial and bending stiffness. Interface elements withvarious adhesion factors were also used in this case. Thetwo-dimensional finite element mesh is shown in Fig. 5.

3. Uniform soil strength profile

The variation of the lateral bearing capacity factor Np withdepth, obtained from the numerical results for adhesion factora = 0.3 and several normalized pile spacings s/D, is presented inFig. 6. Similar results were obtained for all the adhesion factorsconsidered in this section (a = 0.1, 0.3, 0.5, 0.7). It can be seen that,as expected, similar to the pattern observed in single piles (Fig. 1),the lateral resistance increases with depth and reaches an ultimate

80D

50D

Fig. 5. Two-dimensional analyses finite element mesh.

Fig. 6. Variation of Np with z/D for various pile spacings and a = 0.3.

Fig. 7. Variation of Npu with s/D for a = 0.3. Comparison of three-dimensionalanalysis results to Eq. (2).

K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174 167

value at some depth. Pile spacing has a significant influence onboth the calculated ultimate lateral bearing capacity factor Npu

and on the variation of Np at smaller depths. It can also be observedthat the influence of pile spacing on the bearing capacity factorreduces as the pile spacing increases. Group interaction effectson the lateral resistance diminish at s/D = 10.

As seen in Fig. 6, with the exception of very closely spaced piles(s/D = 1.1 and 1.2), there are no group interaction effects on thelateral resistance close to the ground surface; the Np–z/D curvescoincide close to the pile head and start deviating at a critical depthwhich increases as the pile spacing increases. For s/D P 4, theNp–z/D curves converge again to a single curve and to the sameultimate lateral bearing capacity factor Npu, which corresponds tothat of a single laterally loaded pile. For s/D < 4, lower Npu valuesare reached which decrease as s/D decreases. The depth at whichNpu is reached can be seen to vary significantly with s/D, from 6Dfor a single pile (s/D P 10) to 16D for s/D = 2. The ultimate lateralbearing capacity factor Npu is further discussed below. It can alsobe seen in Fig. 6, that for closely spaced piles, a maximum isreached at a depth of between 9 and 11 pile diameters and thenNp gradually reduces with depth until it reaches its ultimate valueNpu. This feature is also discussed in more detail below.

4. Ultimate lateral bearing capacity factor

Based on two-dimensional plane strain (in a horizontal planeperpendicular to the piles axis) analytical plasticity calculations,numerical limit analyses and displacement finite element analyses,Georgiadis et al. [24] showed that the following equation (which isplotted in Fig. 2) provides a practically exact theoretical solutionfor the calculation of the ultimate lateral bearing capacity factorof a pile in a row of infinite piles:

Npu;g ¼Npu;s 1þ 0:13lns=D�1s1=D�1

þ0:24�0:02a� �

lns=D�1s1=D�1

� �for s< s1

ð2Þ

where s1 is the pile spacing beyond which group effects on theultimate lateral capacity diminish (Npu,g = Npu,s for s P s1), whichdepends on the value of the adhesion factor a:

s1=D ¼ 3:1þ 1:4a ð3Þ

Npu,g is the ultimate lateral bearing capacity factor for a pile in apile row and Npu,s is the associated factor for a single pile, whichcan be calculated from the following equation [31]:

Npu;s ¼pu;s

suD

¼ pþ 2 arcsin aþ 2 cosðarcsin aÞ

þ 4 cosarcsin a

2

� �þ sin

arcsin a2

� �� �ð4Þ

Eq. (2) is compared to the three-dimensional analysis results fora = 0.3 in Fig. 7. As seen, the 2D analysis-based Eq. (2) comparesexcellently with the 3D numerical results for pile spacings greaterthan approximately two pile diameters. However, for s/D < 2 theNpu,g values calculated from the three-dimensional analyses devi-ate from the values obtained from the 2D-based equation. In thetwo-dimensional problem, it is not possible for a failure mecha-nism to develop in the soil in the limiting case in which the pilesare in contact with each other (s/D = 1). Consequently, Npu,g

increases sharply as s/D decreases and tends to infinity as s/D tendsto unity. In contrast, in the three-dimensional problem, the limit-ing case of s/D = 1 corresponds to a different two-dimensionalplane strain (in the vertical plane) problem of a laterally loadedvertical wall, where the net load per unit length at a sufficientlylarge depth is equal to the difference between the passive andthe active pressure at the same depth. It is straightforward, usingMohr’s circles of stress, to derive the following stress field solutionfor Npu,g (for s/D = 1) as a function of the adhesion factor a:

Npu;gðs=D ¼ 1Þ ¼ puðs=D ¼ 1ÞsuD

¼ rp � ra

suD

¼ 2½1þ arcsinðaÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2p

� ð5Þ

If it is now assumed that for low normalised pile spacings a pile rowbehaves as an equivalent vertical wall, then:

Npu;g ¼ 2½1þ arcsinðaÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2p

�ðs=DÞ ð6Þ

Fig. 9. Variation of reduction factor with depth for various pile spacings and a = 0.3.Comparison of three-dimensional analysis results to Eq. (10).

168 K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174

The ultimate lateral bearing capacity factor for the whole range ofs/D values can therefore be expressed as follows:

Npu;g ¼minðN1;N2Þ 6 Npu;s ð7Þ

where N1 and N2 are the values of Npu,g calculated from Eqs. (2) and(6), respectively. Eq. (7) is compared with the three-dimensionalfinite element analysis results in Fig. 8 for adhesion factorsa = 0.1, 0.3, 0.5 and 0.7. As seen in this figure, the transition fromEq. (6) (equivalent ‘‘vertical wall’’ behaviour) to Eq. (2) (individualpile failure mechanisms with pile–pile interaction) takes place atapproximately s/D = 2; the exact value depends on the adhesionfactor.

5. Lateral bearing capacity reduction factor

Similarly to the ultimate lateral bearing capacity factor Npu, areduction factor constant with depth can also be defined below acertain depth:

fmu ¼minðN1;N2Þ=Npu;s 6 1 ð8Þ

Closer to the ground surface, where Np varies with depth, groupeffects also vary with depth, contrary to the common assumptionof a unique reduction factor fm for each pile in a pile group. Fig. 9shows the variation of the reduction factor fm with depth, obtainedfrom the numerical results through Eq. (1) for a = 0.3. It can beobserved that in accordance with the observations made inFig. 6, there is no pile group interaction (fm = 1) at the ground sur-face for any pile spacing. In fact, for s/D P 2 the reduction factorremains equal to unity up to a critical depth, zo. Below this deptha sharp drop is observed, fm reaches a minimum value at an inter-mediate depth of 4–6.5 diameters and gradually tends towards theultimate value which can be determined through Eq. (8). Based onthe numerical results, the critical depth zo can be computed asfollows:

zo=D ¼ 0:5ðs=D� 1Þ ð9Þ

As seen in Fig. 9, pile group interaction effects on the lateralresistance are more pronounced at intermediate (relatively small)depths of 4–6.5 pile diameters, for all pile spacings. For large pilespacings of s/D P 4 no group effects are detected at depths greaterthan 12 pile diameters.

It is also seen in Fig. 9 that fm reaches the final value fmu at adepth that depends on s/D.

0

2

4

6

8

10

12

14

16

1 2 3 4 5 6

Npu

s/D

Equa�on (7)

α = 0.1 (FEA)

α = 0.3 (FEA)

α = 0.5 (FEA)

α = 0.7 (FEA)

α = 0.7

α = 0.1

α = 0.5

α = 0.3

Fig. 8. Variation of Npu with s/D for various pile–soil adhesion factors. Comparisonof three-dimensional analysis results to Eq. (7).

Based on the numerical results of Fig. 9, the following equationis proposed for the calculation of the distribution of fm along thelength of a pile as a function of pile spacing:

fm ¼ 1� ð1� fmuÞ 1� e�ð0:075�Z=DÞ2h i

� b1ðZ=DÞb2 eb3Z=D ð10Þ

where

b1 ¼ 0:6ðD=s� 0:1Þ ð11Þ

b2 ¼ 0:6s=D ð12Þ

b3 ¼ �b2 0:25ð1þ e�s=DÞ þ ð0:07s=DÞ3h i

ð13Þ

Z ¼ z� zo ð14Þ

Eq. (10) is compared to the numerical results in Fig. 9.The effect of the adhesion factor, a, on the ultimate reduction

factor fmu is examined in Fig. 10, which compares the fmu valuesobtained from the numerical analysis results to Eq. (8). The effectof a on the distribution of fm along the length of a pile is examinedin Fig. 11 for s/D = 2 and 4. It is evident from these figures thatalthough, as discussed above, the adhesion factor has an importanteffect on the lateral bearing capacity factor, this is not reflected onthe reduction factors. The adhesion factor has a small influence onfmu (it mainly affects the critical pile spacing beyond which groupeffects diminish), while the variation of fm along the length of a pileis largely independent of the pile–soil adhesion. Noting that Eq.(10) accounts for the effect of a on fmu through Eqs. (2)–(8), itcan be considered valid for any value of a.

As noted in the introduction, available p-multiplier approachesassign a single fm value to the whole length of a pile and thereforedo not take account of the variation of group interactions with

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6

f mu

s/D

Equa�on (8)

α = 0.1 (FEA)

α = 0.3 (FEA)

α = 0.5 (FEA)

α = 0.7 (FEA)

α = 0.7

α = 0.1

α = 0.5

α = 0.3

Fig. 10. Variation of ultimate reduction factor with pile–soil adhesion and pilespacing.

Fig. 11. Variation of reduction factor with depth for various pile–soil adhesions ands/D = 2 and 4.

K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174 169

depth. Fig. 12a and b compares the fm values calculated with Eq.(10) to the constant with depth values proposed by Reese andVan Impe [28], Mokwa and Duncan [27] and Rollins et al. [7], forpile spacings s/D = 2 and 4, respectively. It can be seen in these fig-ures that appropriate choice of an average p-multiplier, to beapplied over the whole length of a pile, depends on the pile length.

In order to investigate the effect of the variation of fm withdepth on the overall behaviour of a pile in a row of piles, p–y anal-yses were performed with the finite difference beam on non-linearsprings computer code LPILE [32], using the single pile p–y curves

Fig. 12. (a) Reduction factor: comparisons for s/D = 2. (b) Reduction factor: comparisons for s/D = 4.

Fig. 13. Influence of pile length on the reduction of lateral pile capacity due togroup interaction effects for: (a) s/D = 2 and (b) s/D = 4.

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8

z/D

Np

FEA

Equa�on (15)

α= 1α= 0.1

α= 0.3

α= 0.5

α= 0.7

Fig. 14. Variation of Np with depth for various pile–soil adhesions from two-dimensional analyses. Comparison of numerical and analytical results.

170 K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174

developed by Georgiadis and Georgiadis [2] for clay and the reduc-tion factor of Eq. (10). Fig. 13a and b presents the reduction oflateral capacity of rigid piles due to group interaction effects,expressed in terms of the calculated ratio of the lateral pile capac-ity of a pile in a pile row, Hug, to the lateral capacity of a single pile,Hus, for different pile length to diameter ratios, adhesion factora = 0.3 and pile spacing s/D = 2 and 4. It can be seen that the reduc-tion in lateral capacity is small for short piles, which is consistentwith the small pile group interaction effects observed in the reduc-tion factor plot of Fig. 9. The reduction in lateral capacity becomesmore significant as the normalised pile length L/D increases.Fig. 13a and b also compares the p–y analysis results to the lateralcapacity reduction according to Reese and Van Impe [28], Mokwaand Duncan [27] and Rollins et al. [7]. As expected, the constantwith depth p-multiplier approach fails to capture the variation ofgroup interaction with pile length, especially for short piles. Theaccuracy of each of the constant multiplier approaches examineddepends on the L/D ratio.

6. Two-dimensional analyses

In the previous section it was shown that for s/D < 2 the ulti-mate lateral capacity (at depth) is associated with a laterallyloaded ‘‘vertical wall’’ deformation mode and that the ultimate

lateral bearing capacity factor can be determined from a lateralearth pressure calculation. A similar calculation can be performedto determine the variation of the lateral resistance with depth andsubsequently the reduction factors fm for closely spaced piles.

Fig. 14 shows the Np–z/D curves determined through two-dimensional finite element analyses for various values of the adhe-sion factor. It can be seen that the normalised lateral resistanceincreases linearly with depth up to a certain depth (zt), then dropssharply and gradually stabilises at a constant value. This behaviouroccurs due to the development of a tension crack along the backside of the wall. Because of this tension crack, only passivepressures act on the upper section of the wall (z < zt) and thereforethe normalised lateral resistance is a function of soil weight andconsequently of depth. In the lower part of the wall (z > zt), bothactive and passive pressures act on the wall and the contributionof soil weight cancels out. The following relationship is proposedfor the determination of the normalised lateral resistance(Np = p/su), based on conventional earth pressure calculations:

Np ¼czsuþ Npuð2DÞ

2 for z 6 zt

Npuð2DÞ for z > zt

(ð15Þ

where Npu(2D) is given from Eq. (5) and zt is the depth of the tensioncrack, which depends on the adhesion factor (Fig. 14). The followingequation provides a good fit of the numerically obtained zt values:

zt=D ¼ 4:3þ 0:35 ln a ð16Þ

As seen in Fig. 14, Eq. (15) provides an excellent lower bound ofthe numerical results. It is noted that, according to Fig. 6, the abovetwo-dimensional calculations are valid for small pile spacings ofs/D < 1.5. For greater pile spacings the pile row behaviour deviatesfrom that of a continuous vertical wall.

Fig. 15. Variation of Np with depth for single piles with a = 0.5 and various suo/kDratios.

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10 12 14

z/D

Np

FEA

Equa�on (17)

α = 1 α = 0.5

Fig. 16. Comparison of Eq. (17) to numerical results for suo/kD = 0.5.

K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174 171

7. Non-homogeneous soil strength profile

7.1. Single piles

In order to extend the study to consider normally consolidatedsoils, a series of three-dimensional finite element analyses wasperformed in which the soil shear strength increased linearly withdepth. In the majority of the analyses, a nominal strength interceptof suo = 1 kPa was assumed at the ground surface, while the adhe-sion factor was set equal to one, which is typical for normallyconsolidated soils. However, in order to first investigate the effectof strength non-homogeneity on the limiting lateral pressures on asingle pile, a limited number of analyses were performed withhigher values of suo and a = 0.5. Fig. 15 illustrates the Np–z/D rela-tionship for a single pile with pile–soil adhesion factor a = 0.5 andfour different strength uniformity ratios suo/kD = 0.5, 5, 10, (infinitefor uniform strength). It can be seen that the ultimate lateralbearing capacity factor Npu is the same in all four cases and thatthe effect of increasing strength with depth is that Npu is reachedat a smaller depth. Consequently, the transition from the surfacewedge-type failure mechanism to the deeper two-dimensionalflow around mechanism in the ground (according to the commonlyused conceptual behavioural model) takes place closer to theground surface as the shear strength profile becomes less uniform.This feature can be attributed to the different depth of the tensioncrack that develops at the back of the piles. If no tension crackdevelops (which is the case for suo/kD = 0), then both an activeand a passive wedge form and both active and passive pressures

act even at very small depths. It is straightforward to show usingsimple limit equilibrium calculations that in the case of a simpli-fied 2D laterally loaded ‘‘vertical wall’’ problem, this leads to aconstant with depth normalized load (Np), i.e. the ultimate valueof Np is reached at z = 0. This was also verified using two-dimen-sional finite element analyses. On the other hand, if the maximumtension crack depth develops (which is the case for uniformstrength suo/kD = ), then only a passive wedge forms and passivepressures act at the top part of the pile, which leads to a lateralresistance that increases with depth and consequently so does Np.

Fig. 16 shows the calculated Np–z/D relationship for a single pilein normally consolidated soil (linearly increasing strength withdepth, nominal 1 kPa strength at ground surface and a = 1). Thisis compared to the relationship proposed by Georgiadis andGeorgiadis [2] for uniform soil strength:

Np ¼ Npu � ðNpu � NpoÞe�kðz=DÞ ð17Þ

where Npo is the bearing capacity factor at the ground surface:

Npo ¼ 2þ 1:5a ð18Þ

Npu is the ultimate bearing capacity factor calculated from Eq. (4)and k is a non-dimensional factor set equal to 1 for normally consol-idated clay, while for uniform clay is a function of a.

Also shown for comparison is the curve for a = 0.5. In bothcases, k = 2 kPa/m. Fig. 17 compares the Np–z/D relationships forthree values of k = 1, 1.5 and 2 kPa/m. As seen, the influence of kon the lateral bearing capacity factor is small and therefore is notinvestigated further.

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10 12 14

z/D

Np

k = 1 kPa/m

k = 1.5 kPa/m

k = 2 kPa/m

Fig. 17. Normally consolidated soil: Influence of k on the lateral bearing capacityfactor.

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6

f mu

s/D

FEA

Equa�ons

Equa�on (19)

Equa�on (2)

Equa�on (6)

Fig. 18. Normally consolidated soil: variation of ultimate reduction factor with pilespacing.

0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2

z/D

fm

FEA

Equa�on (20)

s/D = 1.5 2 2.5 3 4 6,10

Fig. 19. Normally consolidated soil: variation of reduction factor with depth forvarious pile spacings. Comparison of three-dimensional FEA results to Eq. (20).

Fig. 20. Wedge-type failure mechanisms (incremental displacement plots) for s/D = 4: (a) uniform shear strength and (b) linearly increasing strength with depth.

172 K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174

7.2. Lateral bearing capacity reduction factors for linearly increasingstrength with depth

The variation of the ultimate reduction factor with s/D obtainedfrom the three-dimensional analysis results for the normally

consolidated soil profile with k = 2 kPa/m is illustrated in Fig. 18.Also shown is the theoretical variation according to Eqs. (2) and(6) with a = 1. It can be observed that the numerically obtainedresults compare excellently with the ultimate reduction factorscalculated using the Npu values that correspond to the two two-dimensional failure modes considered in the previous section:laterally loaded vertical wall (for very closely spaced piles:Eq. (6)) and flow around failure for a laterally translating row of

K. Georgiadis / Computers and Geotechnics 62 (2014) 164–174 173

piles (for larger pile spacings: Eq. (2)). However, the transitionbetween the two mechanisms appears to be gradual in this case,unlike the case of uniform soil strength. This transition can becomputed through the following empirical equation:

fmu ¼ 1� 1:5e�s=D ð19Þ

Fig. 19 shows the variation of the calculated reduction factorswith depth for several normalised pile spacings. It is evident thatthe ultimate reduction factor is reached at a much smaller depth(approximately 4 diameters) than in the case of uniform strength.As discussed above, this indicates that the depth at which the tran-sition from wedge-type failure (where Np and fm vary with depth)to two-dimensional flow-around failure (where Np and fm reachtheir constant final values, Npu and fmu, respectively) takes places,depends on the strength profile. This is illustrated in Fig. 20, whichcompares the final incremental displacements obtained with thetwo different strength profiles for s/D = 4. The depth of thewedge-type failure mechanism is approximately equal to 12D inthe case of uniform strength (Fig. 20a), which is consistent withthe fm–z/D plot of Fig. 9, and close to 4D in the case of linearlyincreasing shear strength (Fig. 20b), which is consistent withFig. 19. It can also be observed in Fig. 20 that while in the case ofuniform strength a wedge type mechanism develops on the frontside of the pile, in the case of linearly increasing strength withdepth a wedge also forms on the back side of the pile. This is dueto the different depth of the tension cracks that develop at the backof the piles, which as discussed above, is significant in the case ofuniform strength and close to zero in the case of linearly increasingstrength with depth.

It can also be seen in Fig. 19 that pile group interaction dimin-ishes at a smaller pile spacing (approximately 6 pile diameters).

The following modification of Eq. (10) provides a good approx-imation (Fig. 19) of the numerical results:

fm ¼ 1� ð1� fmuÞh1� e�ð0:7ðz�zoÞ=DÞ2

ið20Þ

where fmu is the minimum of the values calculated through Eqs. (8)and (19). Reduction factors were calculated for three differentvalues of k = 1, 1.5 and 2 kPa/m. It was found that the influence ofk on the fm–z/D relationship is very small, as it was found to beon the Np–z/D relationship for single piles (which was discussedabove). Therefore Eq. (20) is considered to be valid for any valueof k.

8. Conclusions

Three-dimensional finite element analyses of laterally loadedrows of piles in clay were presented. Both uniform strength andlinearly increasing strength with depth profiles were considered.The numerical study investigated the limiting lateral soil pressuresthat act on a pile in a row of piles and the reduction of thesepressures due to group interaction effects.

The finite element results showed that pile group interactioneffects are not constant along the pile length. No group effects onthe lateral resistance exist close to the ground surface. The reduc-tion in lateral resistance starts below a certain depth, which is afunction of pile spacing, and reaches a maximum at depths rangingfrom 4D (for s/D = 1.1) to 6.5D (for s/D = 8) for uniform soil strengthand 3.5D (for s/D = 1.5) to 5.5D (for s/D = 6) for soil strengthlinearly increasing with depth. The ultimate constant with depthvalues of the lateral resistance are reached at a greater criticaldepth that is also a function of pile spacing. Below this depth,two distinct two-dimensional failure mechanisms in the groundare identified: a laterally loaded ‘‘vertical wall’’ failure mode forclosely spaced piles and a flow around failure mode for greater pilespacings.

The critical pile spacing beyond which there are no groupeffects on the lateral resistance was found to be at approximately10 pile diameters for uniform shear strength and 6 pile diametersfor shear strength linearly increasing with depth. For smaller pilespacings, group interaction effects were found to increase withthe decrease in pile spacing. For very closely spaced piles, a two-dimensional laterally loaded ‘‘vertical wall’’ failure mode wasidentified for the whole length of the piles.

Reduction factors were calculated for all cases analysed and itwas shown that they are not constant with depth, as commonlyassumed in practice. Equations were proposed for the calculationof the reduction factor and its variation with depth for both uni-form undrained shear strength and linearly increasing with depthshear strength profiles. P–y analyses performed using these equa-tions showed that, because of the variation of the reduction factorwith depth, the overall group interaction effects, depend on thepile length. Comparisons were also made with three constant withdepth p-multiplier approaches, which are adopted in practice.

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