Variational solution for the effect of vertical loadon the lateral response of offshore piles

Fayun Liang a,n, Hao Zhang a,1, Jialai Wang b,2

a Department of Geotechnical Engineering, Tongji University, Shanghai 200092, Chinab Department of Civil, Construction, and Environmental Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA

a r t i c l e i n f o

Article history:Received 15 April 2014Accepted 3 March 2015

Keywords:Vertical loadLateral loadOffshore pilesVariational methodParameters analysis

a b s t r a c t

Pile foundations supporting offshore and coastal structures are usually simultaneously subjected to verticalloading from the superstructures and lateral loading due to wind or wave actions. An analytical model ispresented to investigate the effects of vertical loads on the lateral responses of piles applied in such cases. Inthis model, the response of the soil is given by the fundamental Mindlin's solution for half-space subjected toboth concentrated horizontal and vertical loads. The deformations and reaction pressures of the pile arerepresented by finite series. The responses of the pile are determined by using the principle of minimumpotential energy. The proposed model is validated by comparison with the results of field load tests andlaboratory load tests. The influence of related parameters including lateral load level, ratio of the vertical tothe horizontal loads, pile slenderness ratio and pile flexibility factor has also been studied in this paper.

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1. Introduction

Pile foundations supporting the offshore and coastal structuressuch as harbor constructions and long-span bridges are usuallysimultaneously subjected to vertical loading from the supportedsuperstructures and lateral loading due to wind or wave actions.Numerous theoretical methods have been developed to analyze theresponse of such foundations, including the elastic continuummethod(Butterfield and Banerjee, 1971; Poulos and Davis, 1980; Chen andChen, 2008), the theoretical load–transfer curve method (Randolphand Wroth, 1978), and the nonlinear subgrade reaction method(Matlock, 1970; Reese and Welch, 1975; Georgiadis and Butterfield,1982). In all these models, either the vertical loads or the horizontalloads are considered independently. However, the effect of the verticalloads on the lateral responses of piles is critical as shown in a numberof experimental studies. For example, Anagnostopoulos andGeorgiadis (1993) investigated the interaction among the axial andthe lateral responses of piles in clay with model tests. Zhang et al.(2002) investigated the effects of the vertical load on the group lateralresistances by centrifuge model test. They pointed out that theinfluence of the vertical loads is closely linked to the pile–soil system.Knappett and Madabhushi (2009) observed the amplifications of the

lateral displacements and the unstable collapse in pile groups underthe action of significant axial load and in liquefiable soils.

Additionally, by means of a three-dimensional numerical modelbased on the finite element program system ABAQUS (version 6.8), theinteraction effects of combined loading for piles and their dependenceon system parameters were further investigated numerically byAchmus and Thieken (2010a, 2010b) and Achmus et al. (2009).Karthigeyan et al. (2006, 2007) also showed the significant influenceof vertical loads on pile's lateral response through a series of three-dimensional finite element analyses (GEOFEM3D) on single pile.Altogether, most of the existing numerical investigations suggest adecrease in lateral deflection due to the presence of the vertical loads.

By using local elasto-plasticity yield surfaces that allows coupling ofthe differential equations for axial and lateral loading, Levy et al.(2005) presented an analytical model that predicts the behavior of asingle vertical pile under combined axial and lateral loading. Areduction was observed in the horizontal load required to displacethe pile head for inclined loading. Recently, the authors (Liang et al.,2012) proposed an integral equation method to examine the effects ofvertical loads on the lateral response of piles. The model shows thatthe bending moment and the horizontal displacement distributionalong the pile increase considerably with the axial load. Altogether, theresults given by numerical analyses and analytical investigations aresomewhat inconsistent with respect to the effects of vertical loads onthe lateral response of piles (Hussien et al., 2014).

This study presents a variational approach to analyze the effects ofvertical loads on the lateral responses of offshore piles and to find thereason for the disagreement existing between the analytical andnumerical solution. This variational approach was used by Shen

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Ocean Engineering

http://dx.doi.org/10.1016/j.oceaneng.2015.03.0040029-8018/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ86 21 6598 2773; fax: þ86 21 6598 5210.E-mail addresses: fyliang@tongji.edu.cn (F. Liang),

zhanghaotumu@163.com (H. Zhang), jwang@eng.ua.edu (J. Wang).1 Tel.: þ86 21 6598 2773; fax: þ86 21 6598 5210.2 Tel.: þ1 205 348 6786; fax: þ1 205 348 0783.

Ocean Engineering 99 (2015) 23–33

et al. (1997) to study the vertical deformation of pile groups in soil. Intheir study, the soil was modeled by the theoretical load–transfercurves, and the vertical deformation of the pile was represented by afinite series. This variational method was then extended to verticallyloaded pile groups in an elastic half-space (Shen et al., 1999), and thelaterally loaded piles (Shen and Teh, 2002). In Shen and Teh's (2002)study, two finite series were used to approximate the lateral displace-ments of piles subjected to a horizontal load and a bending moment atthe pile head. More recently, a numerical solution for laterally loadedpiles in a two-layer soil profile was proposed by Yang and Liang(2006).

In the present study, the soil is modeled as an elastic half-space,whose response is given by Mindlin's solution. Finite series are usedto approximate the horizontal and vertical displacements, thereaction pressures acting on the pile, and the shear stresses of thepile. The principle of minimum potential energy is then triggered toestablish the governing equations of the structure. The details of theparameters study, the verification of the proposed model againstsome field load tests, and the laboratory load tests are discussed.

2. Method of analysis

2.1. Definition of the problem

Consider a pile subjected to both the lateral and the vertical loads,as shown in Fig. 1. In this figure, l and d are the length and thediameter of the pile, respectively; H, M and P are the horizontal load,the bending moment, and the vertical load acting at the head of thepile, respectively; pz and τz are the horizontal reaction pressures andthe vertical shear stresses at depth z, respectively; σb is the normalstress at the pile base. In this study, the pile is assumed to beembedded in a soil modelled as an elastic half-space. The funda-mental Mindlin's solution (1936) for a concentrated horizontal loadand a vertical load is used to simulate the response of the soil.

2.2. Potential energy of the pile

The total potential energy πp of the single pile subjected to boththe lateral and vertical loads, as depicted in Fig. 1, can be written as

πp ¼12∭VEp

∂wz

∂z

� �2

dvþ12

ZlEPIP

∂2ρz

∂z2

� �2

dzþ12∬Sτzwzds

þ12∬AσbwbdAþ

12

Zlpzρzddz�Ptwt�Htρt�

∂ρt

∂zMt ð1Þ

In this equation, the first two terms on the right hand side givethe elastic strain energy of the pile, where V is the volume of thepile, Ep and Ip are Young's modulus, and moment of inertia of thepile, respectively, wz and ρz are the vertical and the horizontaldisplacements of the pile at depth z, respectively. The third andfourth terms on the right hand side of Eq. (1) correspond to thework done by the vertical shear stress τz along the pile shaft andthe normal stress σb at the pile base, respectively, where S, and Aare the surface and cross-sectional area of the pile, respectively,and wb is the vertical displacement at the pile base. The fifth termis the work done by the horizontal reaction pressures pz along thepile shaft. The last three terms are the works done by the verticalload Pt , by the horizontal load Ht , and by the applied moment Mt

at the pile head, where wt and ρt are the vertical and horizontaldisplacements of the pile at the pile head, respectively.

By using Gauss integration and assuming uniform σb and wb atthe pile base, Eq. (1) can be rewritten as

πp ¼ Upþ12

Pg� �T wg

� �þ12π

d2

� �2

σbwbþ12

Pq� �T ρq

n o

�Ptwt�Htρt�∂ρt

∂zMt ð2Þ

where Up ¼ 12∭VEp ∂wz

∂z

� �2dvþ12

RlEPIP

∂2ρz∂z2

� 2dz. Pg

� �¼ Pg1; Pg2;⋯;�

PgnggT , and wg� �¼ wg1;wg2;⋯;wgng

� �T are the vertical forces andsettlement of the pile at the Gauss points, respectively.

Pq� �¼ Pq1; Pq2;⋯; Pqnq

� �T and ρq

n o¼ ρq1;ρq2;⋯;ρqnq

n oTare the

horizontal forces and displacements of the pile at the Gauss points,respectively. Here ng and nq are the number of Gauss points chosenfor the pile along its shaft. The coefficients in the vectors Pg

� �and

Pq� �

are given by

Pgi ¼12πdlηiτi i¼ 1;2;…;ngð Þ ð3Þ

Pqi ¼12ldηipi ði¼ 1;2;…;nqÞ ð4Þ

where ηi is the weighting coefficient of the Gauss points, τi is thevertical shear stress at the Gauss points, and pi is the horizontalreaction pressure at the Gauss points. Eq. (2) can be further expressedas

πp ¼ Upþ12Pf gT ρw

� ��Ptwt�Htρt�∂ρt

∂zMt ð5Þ

where

Pf g ¼ Pq1; Pq2;…; Pqnq; Pg1; Pg2;…; Pgng ; Pb� �T

; Pb ¼ πd2

� �2

σb;

ρw� �¼ ρq1;ρq2;…;ρqnq;wg1;wg2;…;wgng ;wb

n oT

For a pile in an elastic half-space, we have

Pf g ¼ ks �

ρw� � ð6Þ

where ks �

is the soil stiffness matrix and is described in a subsequentsection of the paper. Substituting Eq. (6) into Eq. (5) yields

πp ¼ Upþ12ρw

� �T ks �

ρw� ��Ptwt�Htρt�

∂ρt

∂zMt ð7Þ

2.3. Displacement series of the pile

Three finite series are used to approximate the deformation ofthe pile under the applied vertical loads, lateral loads, and bendingmoments. Denoting the horizontal deflection of a pile, ρz due to

M

H

P

σb

τz

pz

pz z

l

d

Fig. 1. Forces acting on the pile.

F. Liang et al. / Ocean Engineering 99 (2015) 23–3324

the combination of a lateral load and a vertical load acting at thepile head as ρzh, we have

ρzh ¼ ahþbhzlþ

Xni ¼ 1;2;3

βhi siniπzl

ð8Þ

Similarly, denoting ρz induced by the combination of a momentand a vertical load applied at the pile head as ρzm, then

ρzm ¼ amþbmzlþ

X2n�1

i ¼ 1;3;5

βmi cosiπz2l

ð9Þ

The vertical displacements of the pile,wz can be expressed as

wz ¼Xki ¼ 1

βgi 1�zl

� i�1ð10Þ

In Eqs. (8–10), ah, am, bh, bm, βhi, βmi and βgi are unknownconstants. n is the number of terms used in the trigonometricfunction, and k is the number of terms used in Eq. (10).

2.4. Minimization of the potential energy

The principle of minimum potential energy requires that πp bean extremum with respect to the admissible displacement fieldcharacterized by the undetermined coefficients ah, am, bh, bm, βhi,βmi and βgi in Eqs. (8)–(10). Hence

∂πp

∂δj¼ 0 ðj¼ 1;2;…;nþ2þkÞ ð11Þ

where δj denotes the undetermined coefficients in Eqs. (8–10).Substituting the total potential energy given by Eq. (7) into Eq. (11)yields

∂Up

∂δjþ ∂ρw

∂δj

� T

ks �

ρw� �¼ ∂wt

∂δj

� T

Ptþ ∂ρt

∂δj

� T

Htþ∂ ∂ρt

∂z

� ∂δj

8<:

9=;

T

Mt

ð12ÞEq. (12) is the governing variational formulation for a pile in a

soil modeled as an elastic half-space.

2.5. Load-deformation relationship for soil

The point-load solution of Mindlin (1936) is used to determinethe soil load–deformation relationship (Eq. (6)). The horizontaland vertical displacement at any chosen Gauss point i along thepile shaft can be given by

ρqi ¼∬Ap f 1ðzi; zpÞpzdApþ∬Sf 2ðzi; zτÞτzdsþ∬Af 2ðzi; zbÞσbdA

ði¼ 1;2;…;nqÞ ð13Þand

wgi ¼∬Ap f 3ðzi; zpÞpzdApþ∬Sf 4ðzi; zτÞτzdsþ∬Af 4ðzi; zbÞσbdA

ði¼ 1;2;…;ngÞ ð14ÞThe vertical displacement at the base of the pile,wb, can be

written as

wb ¼∬Ap f 3ðzb; zpÞpzdApþ∬Sf 4ðzb; zτÞτzdsþ∬Af 4ðzb; zbÞσbdA ð15Þ

In Eqs. (13)–(15), f 1ðzi; zpÞ, f 2ðzi; zτÞ, f 2ðzi; zbÞ, f 3ðzi; zpÞ, f 3ðzb; zpÞ,f 4ðzi; zτÞ, f 4ðzi; zbÞ, f 4ðzb; zτÞ and f 4ðzb; zbÞ are the coefficients givenby Mindlin's point-load solution (see Appendix B); and S, A and Ap

are the areas of the integration of the pile shaft for τz , σb, and pzrespectively. Approximating pz and τz by finite series, we have

pz ¼Xk1j ¼ 1

αqjzpl

� j�1; τz ¼

Xk2j ¼ 1

αgjzpl

� j�1ð16Þ

In Eq. (16), αqj and αgj are unknown constants; k1 and k2 are thenumbers of the terms in the finite series. These two finite serieshave been successfully used to approximate the shear stress (Shenet al., 1999) and the reaction pressure of the pile (Shen and Teh,2002). Then Eqs. (13)–(15) can be rewritten as

ρqi ¼Xk1j ¼ 1

∬Ap f 1ðzi; zpÞαqjzpl

� j�1dApþ

Xk2j ¼ 1

∬Sf 2ðzi; zτÞαgjzpl

� j�1ds

þ∬Af 2ðzi; zbÞσbdA ði¼ 1;2;…;nqÞ ð17Þ

wgi ¼Xk1j ¼ 1

∬Ap f 3ðzi; zpÞαqjzpl

� j�1dApþ

Xk2j ¼ 1

∬Sf 4ðzi; zτÞαgjzpl

� j�1ds

þ∬Af 4ðzi; zbÞσbdA ði¼ 1;2;…;ngÞ ð18Þ

wb ¼Xk1j ¼ 1

∬Ap f 3ðzb; zpÞαqjzpl

� j�1dApþ

Xk2j ¼ 1

∬Sf 4ðzb; zτÞαgjzpl

� j�1ds

þ∬Af 4ðzb; zbÞσbdA ð19Þ

Rewriting Eqs. (17)–(19) in matrix form, we have

ρw� �¼ f

�αf g ð20Þ

where αf g ¼ αq1;αq2;⋯;αqk1 ;αg1;αg2;⋯;αgk2 ;σb� �T . ρw

� �is defined

in Eq. (5). f �

is amatrix of order ðnqþngþ1Þ � ðk1þk2þ1Þ, and can beexpressed as

f �¼ ½f ρL� ½f ρV �

½f wL� ½f wV �

" #ð21Þ

The coefficients in the sub-matrix ½f ρL�, ½f ρV �, ½f wL� and ½f wV � aregiven in Appendix A.

2.6. Solution for pile response

After some mathematic manipulations, Eq. (12) can be rewrit-ten as

k �

β� �¼ PLAf g ð22Þ

where

β� �¼ ah; bh; :::;βhn; am; bm; :::;βmð2n�1Þ;βg1;βg2; :::;βgk

n oT;

PLAf g ¼ Ht ;0; :::;0;0;Mt ;0; :::;0; Pt ; Pt ; :::; Ptf gT ;and [k] is the pile–soil stiffness matrix and given in Appendix C.

For a pile with a free head boundary condition, the vector β� �

in Eq. (22) can be solved for the horizontal load, the bendingmoment, and the vertical load applied at the pile head. Then,superposing ρzh and ρzm calculated from Eqs. (8) and (9), respec-tively, yields the deflections of a pile ρz . Accordingly, the rotationangles and bending moments of the pile at any depth, θz and Mz ,respectively can also be obtained. Similarly, the vertical displace-ments of the pile at any depth wz , can then be determined usingEq. (10). The vertical shear stresses τz and the horizontal reactionpressures pz along the pile shaft can be determined by Eq. (16).

In the above calculations, the presence of the lateral load, thebending moment, and the vertical load at the pile head willgenerate rotation at the pile head. In such a case, the work doneby the horizontal load Ht and the vertical load Pt at the pile headshould be redefined as shown in Fig. 2. To this end, an iterativeprocedure is required. In this procedure, the initial rotation angleθ0 at the pile head can be obtained using the above calculations atfirst. Then the updated horizontal load H0

t and vertical load P0t at

the pile head can be determined by

H0t ¼Ht cos θ0 þPt sin θ0 ; Pt

0 ¼ Pt cos θ0 �Ht sin θ0j������������������ ð23Þ

F. Liang et al. / Ocean Engineering 99 (2015) 23–33 25

The updated values of H0t and P0

t will be used in Eq. (22) to solvenew rotation angle θ0

0 at the pile head. This iterative procedurewill be repeated until the difference of the rotation angle at thepile head obtained between two successive iterations is within 5%.

3. Validation of the analytical model

The above solution procedure has been coded into Matlab. 16terms are used in the finite series in Eqs. (8) and (9) and firstequation in Eq. (16). Four terms are used in the finite series in Eq.(10), and 3 terms are used in the finite series in the secondequation of Eq. (16). The number of Gauss points, nq and ng in Eq.(2), are chosen as 16 and 3, respectively. Numerical results withthese initial computing parameters are verified through compar-isons with the results of field load tests and laboratory load testsfrom three different published cases, one with respect to anoffshore pile under pure lateral load and the others for laboratorymodel piles under combined vertical and lateral loads. The detailsof these three cases are presented in the following sub-sections.

3.1. Case study 1

Kim et al. (2009) reported the field tests of steel pipe pilesunder lateral loads installed at the Incheon Bridge site in South

Korea. The pipe piles had an outer diameter of 1.016 m and a wallthickness of 16 mm. The bending stiffness of the pile section was1,260,000 kN m2. Each pile was driven using an oil pressurehammer and 1.0 m long pile head remained above the groundsurface. The final depth of the driven piles was recorded as 25.6 m.The mapped local geology is that of a marine deposit. The shallowsurface layers are of great interest for this study because laterallyloaded piles typically receive most of their support from the soil inthe upper 7–10 times pile diameters (Briaud, 1997). Therefore,Young's modulus Es and Poisson's ratio ν of the upper marine clay(Li et al., 2013; Kim et al., 2009) are used in this analysis. Thelateral load was applied at a point 0.5 m above the ground surfaceand increased substantially up to a maximum load of 900 kN.Fig. 3 shows the comparison of the pile head deflection versuslateral load curves from the field testing and computed by thepresented method. Reasonable agreement with testing result hasbeen achieved by the present model at likely working load.

3.2. Case study 2

Fig. 4 compares the present model with experimental studycarried out by Sastry and Meyerhof (1986). The experimental studywas performed on a single pile with central inclined loads appliedto its head at ground elevation. Two load inclinations used in thetest were α¼901 (horizontal load) and 451 (combined horizontalload and vertical load). The vertical hollow steel model pile used inthe test had an outside diameter of 74 mm and 7 mm wallthickness ðEpIp ¼ 25:9 kN m2Þ, jacked 950 mm into a prepared softclay bed. The full details of soil properties including Young'smodulus Es used in the current analysis are the same as thosereported by Meyerhof and Sastry (1985) and Sastry and Meyerhof(1990). Fig. 4 shows that excellent agreement with experiment hasbeen achieved by the present model for inclined load lower than500 N. Above this load, the present model seems to underestimatethe ground-line displacement of the pile, which may be caused bythe linearly elastic model for the soil. The real behavior of the soilat this load level may be essentially nonlinear.

3.3. Case study 3

This case study pertains to laboratory model tests on aluminumclosed-ended piles of 19 mm outside diameter and 1.5 mm wallthickness, jacked 500 mm into a prepared soft clay bed (wL¼42%,wP¼24%, and cu¼28 kPa). The laboratory tests were performed ona single pile under both vertical and lateral loads applied to thepile head at ground elevation through dead weights. In this test,the lateral load of 130 N was applied incrementally under aconstant vertical load of 160 N. Young's modulus Es of the soil is

θ0

H

P

Fig. 2. Pile segment at depth z under axial and lateral loading.

0

200

400

600

800

1000

0 50 100 150 200

Late

ral l

oad

(kN

)

Lateral displacement (mm)

Kim et al. (2009)

Present method

Fig. 3. Pile head lateral deflection versus lateral load.

0

2

4

0 400 800 1200

Gro

und-

line

dis

plac

emen

t (m

m)

Inclined Load (N)

Sastry & Meyerhof (1986),

Present method,

Sastry & Meyerhof (1986),

Present method,

α=90

α=90

α=45

α=45

Fig. 4. Displacements of pile under inclined loads.

F. Liang et al. / Ocean Engineering 99 (2015) 23–3326

taken as 7 MPa using the relation Es¼250–400cu (Poulos andDavis, 1980). Poisson's ratio ν of the clayey soil is taken as 0.49.The comparison between test data of Anagnostppoulos andGeorgiadis test data and the predicted results are shown inFig. 5. It could be observed that the comparison of the responseof laterally loaded piles under vertical load reveals reasonableagreement, although the theoretical ground-line displacements ofthe pile are smaller above lateral load of 80 N, which may becaused by the linearly elastic model for the soil. The real behaviorof the soil at this load level may be essentially nonlinear.

Thus, by using the present model, parametric studies can becarried out to shed new lights on the effect of vertical compressiveload on the lateral responses of an offshore pile.

4. Parametric studies

To investigate the effect of vertical compressive load on thelateral responses of an offshore pile, several key influence factors,

including the lateral load level, the loading ratio, the pile slender-ness ratio and the pile flexibility factor, are analyzed in thisparametric study using the present model.

4.1. Influence of the lateral load level

To examine the influence of the lateral load level on theinteraction effects, the horizontal load–displacement curves of alaterally loaded pile without vertical load (N/H¼0 in Fig. 6) andone with vertical load (N/H¼20) are compared in Fig. 6. In thisfigure, the pile flexibility factor KR ¼ EpIp=EsL

4 and the pile slen-derness ratio l/d are chosen as 10�3 and 25, respectively. Otherparameters used in the analysis are: pile length l¼25 m, pilediameter d¼1 m, Young's modulus of the soil Es¼2.5 MPa, Pois-son's ratio of the soil υ¼0.25, and the bending moment M¼Hd.

Fig. 6 shows that the horizontal deflection is reduced by thepresence of the vertical load when the lateral load is not high.With the increase of the lateral load, the reduction effect inducedby the axial force is diminishing. Beyond a certain point, the lateraldisplacement starts to increase with the lateral load. This increasein lateral displacement may be resulted from the additionalbending moment in the pile due to the moment arm of the

0

25

50

75

100

125

150

0 10 20 30 40 50 60

Late

ral l

oad

(N)

Lateral displacement (×10-2mm)

Anagnostopoulos & Georgiadis (1993)

Present method

Vertival load = 160 N

Fig. 5. Lateral load–deflection curves of piles.

0

100

200

300

400

500

600

700

0 20 40 60 80 100 120

Late

ral l

oad

(kN

)

Lateral displacement (mm)

N/H=0

N/H=20

l/d=25, K R=10 -3, υ=0.25Horizontal load & Moment

Freehead

Fig. 6. Horizontal load–displacement curves of piles.

0

0.2

0.4

0.6

0.8

1

-0.08 0 0.08 0.16 0.24 0.32 0.4

z/l

ρzEsd/H

N=10H

N=5H

N=0

l/d=25, KR=10 -3, υ=0.25Horizontal load only, Freehead

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2 2.5

z/l

Mz/(Hd)

N=10H

N=5H

N=0

l/d=25, KR=10 -3, υ=0.25Horizontal load only, Freehead

Fig. 7. Influence of vertical loading on lateral responses of pile. (a) Lateraldisplacement distribution and (b) bending moment distribution (low lateral loadlevel).

F. Liang et al. / Ocean Engineering 99 (2015) 23–33 27

vertical load, i.e. from geometrical non-linearity, which is consid-ered in this study. Similar result was obtained by Achmus andThieken (2010a, 2010b) for piles under combined vertical andhorizontal loading. Therefore, we can categorize the lateral loadinto two levels: a low lateral load level at which the presence ofthe vertical load reduces the lateral displacement of the pile and ahigh lateral load level at which the presence of the vertical loadincreases the lateral deformation of the pile.

4.2. Influence of the ratio of the vertical to the Horizontal loads

Fig. 7 examines the synergistic effects of a horizontal force and avertical load applied simultaneously at the head of a pile at low lateralload level. Three different ratios of N/H¼0, 5 and 10 are considered inthis figure. The pile flexibility factor KR and the pile slenderness ratiol/d are chosen as 10�3 and 25, respectively. Other parameters used inthe analysis are: pile length l¼25m, pile diameter d¼1m, Young'smodulus of the soil Es¼2.5 MPa, Poisson's ratio of the soil υ¼0.25,

and the lateral load H¼100 kN. The effect of the vertical load on thelateral responses of the pile is revealed in Fig. 7. Both the lateraldisplacement and the bending moment of the pile, particularly withinits upper part, decrease with the vertical load.

Fig. 8 illustrates the combined effects of a horizontal force, abending moment, and a vertical load on the pile's responses at lowlateral load level. In this figure, the numerical results withKR ¼ 10�3, l/d¼25 and M¼Hd for three different ratios of N/H¼0, 5, 10 are presented. Other parameters used in the analysisare: pile length l¼25 m, pile diameter d¼1 m, Young's modulus ofthe soil Es¼2.5 MPa, Poisson's ratio of the soil υ¼0.25, and thelateral load H¼100 kN. Both the lateral displacement and thebending moment have similar distributions along the pile as thoseshown in Fig. 7. However, the deflection and the maximummoment (Fig. 8) of the pile with the bending moment applied atits head are higher by up to 17% than the one without the bendingmoment (Fig. 7) for large N/H. Its interaction effects (Fig. 8) are lesssignificant than the one without the moment (Fig. 7).

Fig. 9 examines the synergistic effects of a horizontal force and avertical load applied simultaneously at the head of a pile at high lateralload level. Three different ratios of the vertical to the horizontal loadsN/H¼0, 5 and 10, are considered in this figure. The pile flexibility

0

0.2

0.4

0.6

0.8

1

-0.08 0 0.08 0.16 0.24 0.32 0.4

z/l

ρ zE sd/H

N=10H

N=5H

N=0

l/d=25, KR=10 -3, υ=0.25Horizontal load & Moment

Freehead

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2 2.5 3

z/l

Mz/(Hd )

N=10H

N=5H

N=0

l/d=25, KR=10 -3, υ=0.25Horizontal load & Moment

Freehead

Fig. 8. Influence of vertical loading on responses of pile under both horizontal forceand moment loading. (a) Lateral displacement distribution and (b) bendingmoment distribution (low lateral load level).

0

0.2

0.4

0.6

0.8

1

-0.08 0 0.08 0.16 0.24 0.32 0.4

z/l

ρzEsd/H

N=10H

N=5H

N=0

l/d=25, KR=10 -3, υ=0.25Horizontal load only, Freehead

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2 2.5

z/l

Mz/(Hd)

N=10H

N=5H

N=0

l/d=25, KR=10 -3, υ=0.25Horizontal load only, Freehead

Fig. 9. Influence of vertical loading on lateral responses of pile. (a) Lateraldisplacement distribution and (b) bending moment distribution (high lateral loadlevel).

F. Liang et al. / Ocean Engineering 99 (2015) 23–3328

factor KR and the pile slenderness ratio l/d are chosen as 10�3 and 25,respectively. Other parameters used in the analysis are: pile lengthl¼25m, pile diameter d¼1m, Young's modulus of the soilEs¼2.5 MPa, Poisson's ratio of the soil υ¼0.25, and the lateral loadH¼1000 kN. The effect of the vertical load on the lateral responses ofthe pile can be seen clearly from Fig. 9. Both the lateral displacementand the bending moment of the pile, particularly within its upper part,increase with the vertical load.

Fig. 10 compares the present solution with Liang et al.'s (2012)solution of a free-head pile under vertical and horizontal loads withthree different loading ratios: N/H¼0, 5 and 10. Close agreementshave been achieved by these two models on the deflection at the pilehead and the maximum moment of the pile. Liang et al.'s (2012)solution seems to over predict the deflection and the maximummoment (Fig. 10) for high loading ratio of N/H, as shown in Fig. 10.This is because Liang et al.'s (2012) solution ignored the vertical shearstress along the pile shaft and the normal stress at the pile base.Therefore, the interactions among the horizontal reaction pressures,the vertical shear stress, and the normal stress were not accountedfor. As a matter of fact, if we remove those interactions and onlyconsider the horizontal reaction pressure in the present model, the

present model produces almost identical results as Liang et al.'s(2012) solution for both the deflection at the pile head and themaximum moment of the pile, as shown in Fig. 10. Fig. 10 also showsthat ignoring the vertical shear stress and the normal stress can causeover estimation of the lateral deflection at the head and themaximum moment of the pile up to 10% for a relatively large N/H.By considering these interactions, the present solution is believed toproduce more accurate predictions than Liang et al.'s (2012) model.

The combined effects of a horizontal force, a bending moment, anda vertical load on the pile's responses at high lateral load level areshown in Fig. 11. In this figure, the numerical results with KR ¼ 10�3,l/d¼25 and M¼Hd for three different ratios of N/H¼0, 5, and 10 arepresented. Both the lateral displacement and the bending momentalong the pile have similar distributions as those shown in Fig. 9.However, with the bending moment applied at the head of the pile, itsdeflection and the maximummoment (Fig. 11) are higher by up to 15%than the one without the bending moment (Fig. 9) for large N/H value,and its interaction effects (Fig. 11) are more significant than the onewithout the moment (Fig. 9). Such a significant increment in deflec-tion and the maximum moment is usually ignored in the existingmethods, leading to an underestimation of the lateral response of a

0

0.1

0.2

0.3

0.4

0.5

0 5 10

ρ 0E s

d/H

N/H

Liang et al. (2012)

Present method

Present method (only considering horizontal reaction pressures)

l/d=25, KR=10 -3, υ=0.25Horizontal load only, Freehead

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10

Mm

ax/(H

d)

N/H

Liang et al. (2012)

Present method

Present method (only considering horizontal reaction pressures)

l/d=25, KR=10 -3, υ=0.25Horizontal load only, Freehead

Fig. 10. Comparisons with Liang et al.'s method for free-head pile under verticaland horizontal loading. (a) Deflection at the pile head and (b) maximum moment.

0

0.2

0.4

0.6

0.8

1

-0.1 0 0.1 0.2 0.3 0.4 0.5

z/l

ρzEsd/H

N=10H

N=5H

N=0

M=Hd, υ=0.25, l/d=25,KR=10 -3Horizontal load & Moment

Freehead

0

0.2

0.4

0.6

0.8

1

-0.4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

z/l

Mz/M

N=10H

N=5H

N=0

Horizontal load & Moment Freehead

M=Hd, υ=0.25, l/d=25,KR=10 -3

Fig. 11. Influence of vertical loading on responses of pile under both horizontalforce and moment loading. (a) Lateral displacement distribution and (b) bendingmoment distribution (high lateral load level).

F. Liang et al. / Ocean Engineering 99 (2015) 23–33 29

pile. Clearly, the present method provides a more accurate analysis forsingle pile subjected to both the horizontal and vertical loads. Fig. 12compares the present model with that of Liang et al.'s (2012) on a pilesubjected to a horizontal load, a vertical load, and a bending momentsimultaneously. Similar to Fig. 10, ignoring the vertical shear stress andthe normal stress can lead to 7% of over estimation of the lateraldeflection and the maximum moment of the pile for large N/H value.

4.3. Influence of pile slenderness ratio

To study this effect, a series of analyses with the present modelhave been carried out considering five different l/d ratios (15, 20,25, 30 and 35). All these analyses are performed for piles underthree typical vertical loads (N¼5H, N¼8H and N¼10H) and thepile flexibility factor KR is taken as 10�3. Additionally, both thepile–soil system (pile diameter d¼1 m, Young's modulus of thesoil Es¼2.5 MPa, Poisson's ratio of the soil υ¼0.25) and two lateralload levels (including low lateral load level of 100 kN and the highlateral load level of 1000 kN) are also considered in these analyses.

Based on the numerical results obtained, the PVD is calculatedfor various l/d ratios. The PVD is defined as follows in terms of thelateral pile displacement with vertical load (LDWV) and the lateralpile displacement under pure lateral loading (LDPL):

PVD¼ LDWV�LDPLLDPL

� 100% ð24Þ

The influence of the pile slenderness ratio (l/d) on the PVDvalues observed for the three typical vertical loads and two lateralload levels is shown in Fig. 13(a) and (b). It can be noted that theinfluence of vertical load decreases with an increase in slendernessratio of piles at all typical vertical load levels and lateral loadlevels.

4.4. Influence of pile flexibility factor

By using the present model, the influence of the pile flexibilityfactor under combined loading is also studied by performing theanalysis of a pile with three different KR (0.0001, 0.0005 and0.001). Three typical vertical loads (N¼5, N¼8H and N¼10H), thepile-soil system (pile diameter d¼1 m, Young's modulus of the soilEs¼2.5 MPa, Poisson's ratio of the soil υ¼0.25), two lateral loadlevels (including low lateral load of 100 kN and the high lateralload of 1000 kN) and the pile slenderness ratio (l/d¼25) areconsidered in this analysis.

The influence of the pile flexibility factor (KR) on the PVDvalues (see Eq. (24)) observed for the typical vertical loads and twolateral load levels is shown in Fig. 14(a) and (b). It can be foundthat the influence of vertical load increases with the flexibilityfactor of piles at all typical vertical load levels (Fig. 14(a)) when thelateral load level is low. However, an opposite result is obtained forhigh lateral load level as shown in Fig. 14(b).

0

0.1

0.2

0.3

0.4

0.5

0 5 10

ρ 0E s

d/H

N/H

Liang et al. (2012)

Present method

Present method (only considering horizontal reaction pressures)

M=Hd, υ=0.25, l/d=25,KR=10 -3Horizontal load & Moment

Freehead

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10

Mm

ax/M

N/H

Liang et al. (2012)

Present method

Present method (only considering horizontal reaction pressures)

M=Hd, υ=0.25, l/d=25,KR=10 -3Horizontal load & Moment

Freehead

Fig. 12. Comparisons with Liang et al.'s method for free-head pile under verticalloading, horizontal force and moment loading. (a) Deflection at the pile head and(b) maximum moment.

- 14

- 12

- 10

-8

-6

-4

-2

014 16 18 20 22 24 26 28 30 32 34 36

PVD

(%)

Pile slenderness ratio (l/d)

N=5H Small lateral load)

N=8H Small lateral load)

N=10H Small lateral load)

KR=10 -3, υ=0.25Horizontal load only, Freehead

0

10

20

30

40

50

60

14 16 18 20 22 24 26 28 30 32 34 36

PVD

(%)

Pile slenderness ratio (l/d)

N=5H Lager lateral load)

N=8H Lager lateral load)

N=10H Lager lateral load)

KR=10 -3, υ=0.25Horizontal load only, Freehead

Fig. 13. PVD at various pile slenderness ratios l/d. (a) Low lateral load level and(b) high lateral load level.

F. Liang et al. / Ocean Engineering 99 (2015) 23–3330

5. Conclusions

An analytical model using variational approach is proposed inthis paper to analyze the effect of vertical loads on the lateralresponses of offshore piles. This model can easily determine the pileresponses under any combination of vertical compressive loads andlateral loads. A few conclusions can be drawn based on this study:

1. Finite series can be used to approximate the response of a pileunder combined vertical and lateral loads, including the hor-izontal deflection and the bending moment along the pile shaft.This has been verified by reasonable agreements of the presentsolution with the field load tests and laboratory load tests.

2. The proposed method is more accurate than the existing elasticanalytical solution for the pile under both the axial and lateralloads. It accounts for the combined effect of the horizontalreaction pressure, the vertical shear stress along the pile shaft,and the normal stress at the pile base.

3. Numerical results indicate that the effect of vertical compressiveloads on the lateral responses of offshore piles is not negligible.Both the lateral displacement and the bending moment of thepile, in particular within its upper part, decrease with the verticalloading at low lateral load level. However, if the additionalbending moment predominates in the pile due to the momentarm of the vertical load, an opposite trend may be observed whenthe lateral load applied at the head of the pile is at high level.

4. The lateral response of piles under combined loading is alsodependent on the pile slenderness ratio (l/d) and the pile flexibilityfactor (KR). At low lateral load level, compressed piles with a smalllength-to-diameter ratio and piles with almost rigid behavior (largeKR value)show particularly strong interaction effects. However, athigh lateral load level, as the l/d ratio and KR factor increases, theinfluence of vertical compressive load on the lateral response ofpiles reduces.

The analytical model described in this paper can be used as astarting point from which more rigorous models may be devel-oped. With the aid of the results of previous investigations andphysical test results, for example, by introducing the concept oflocal elasto-plasticity yield surfaces or appropriate elastic–per-fectly plastic p–y curves into the present analytical model, somelimitations of the model (the soil is assumed to be linear-elasticand the model does not allow for cyclic loading) could potentiallybe addressed and the pile behavior could be modeled moreaccurately.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no. 41172246), and National Key BasicResearch Program of China (Grant no. 2013CB036304). Financialsupport from these organizations is gratefully acknowledged.

Appendix A

The coefficients in Eq. (21) are given as follows:

f ρLði; jÞ ¼∬Ap f 1ðzi; zpÞzpl

� j�1dAp ði¼ 1;2;…;nqÞðj¼ 1;2;…; k1Þ

ðA1Þ

f ρV ði; jÞ ¼∬Sf 2ðzi; zτÞzpl

� j�1ds ði¼ 1;2;…;nqÞðj¼ 1;2;…; k2Þ

ðA2Þ

f ρV ði; jÞ ¼∬Af 2ðzi; zbÞσbdA ði¼ 1;2;…;nqÞ j¼ k2þ1ð Þ ðA3Þ

f wLði; jÞ ¼∬Ap f 3ðzi; zpÞzpl

� j�1dAp ði¼ 1;2;…;ngÞðj¼ 1;2;…; k1Þ

ðA4Þ

f wLði; jÞ ¼∬Ap f 3ðzb; zpÞzpl

� j�1dAp i¼ ngþ1ð Þðj¼ 1;2;…; k1Þ ðA5Þ

f wV ði; jÞ ¼∬Sf 4ðzi; zτÞzpl

� j�1ds ði¼ 1;2;…;ngÞðj¼ 1;2;…; k2Þ

ðA6Þ

f wV ði; jÞ ¼∬Af 4ðzi; zbÞσbdA ði¼ 1;2;…;ngÞ j¼ k2þ1ð Þ ðA7Þ

f wV ði; jÞ ¼∬Sf 4ðzb; zτÞzpl

� j�1ds i¼ ngþ1ð Þðj¼ 1;2;…; k2Þ ðA8Þ

f wV ði; jÞ ¼∬Af 4ðzb; zbÞσbdA ði¼ ngþ1Þ j¼ k2þ1ð Þ ðA9ÞThe integration involved in Eqs. (A1)–(A9) can be performed asdescribed in Appendix B.

Soil stiffness matrix ks �

is given by

ks �¼ T½ � f ��1 ðA10Þwhere

T½ � ¼TL½ �

TV½ �

" #ðA11Þ

The coefficients in sub-matrix TL½ � and [TV] are given by Eqs.(C11)–(C12) in Appendix C.

Appendix B

The horizontal displacement at a point with depth coordinate zcaused by a unit horizontal point load acting at another point with

-9

-8

-7

-6

-5

-4

-3

-2

-1

00 0.0002 0.0004 0.0006 0.0008 0.001 0.0012

PVD

(%)

Pile flexibility factor (KR)

N=5H

N=8H

N=10H

l/d=25, υ=0.25Horizontal load only, Freehead

0

10

20

30

40

50

60

70

80

90

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012

PVD

(%)

Pile flexibility factor (KR)

N=5H

N=8H

N=10H

l/d=25, υ=0.25Horizontal load only, Freehead

Fig. 14. PVD at various pile flexibility factors KR. (a) Low lateral load level and(b) high lateral load level.

F. Liang et al. / Ocean Engineering 99 (2015) 23–33 31

depth coordinate zp, and the horizontal displacement at the samepoint caused by a unit vertical point load acting at another point,are given by Mindlin's equations as follows:

f 1ðz; zpÞ ¼1

16πGð1�υÞð3�4υÞ

R1þ 1R2

þ x2

R31

þð3�4υÞx2R32

þ2zpzð1�3x2=R22Þ

R32

(

þ4ð1�υÞð1�2υÞ½1�x2=ðR2þzpþzÞR2�R2þzpþz

ðB1Þ

f 2ðz; zpÞ ¼r

16πGð1�υÞðz�zpÞR31

þð3�4υÞðz�zpÞR32

�4ð1�υÞð1�2υÞR2ðR2þzþzpÞ

þ6zpzðzþzpÞR52

( )

ðB2Þin which G is the soil modulus of the soil, υ is Poisson's ratio of soil,

R1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þðz�zpÞ2

q, R2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þðzþzpÞ2

q, and x and r are, respec-

tively, the distance along the x axis and the horizontal distancebetween the point z at which the displacement is evaluated andthe point zp at which the point load is applied.

Similarly, the vertical displacement at a point with depthcoordinate z caused by a unit horizontal point load acting atanother point with depth coordinate zp, and the vertical displace-ment at the same point caused by a unit vertical point load actingat another point, are given by Mindlin's equations as follows:

f 3ðz; zpÞ ¼x

16πGð1�υÞz�zpR31

þð3�4υÞðz�zpÞR32

�6zpzðzþzpÞR52

þ4ð1�υÞð1�2υÞR2ðR2þzþzpÞ

( )

ðB3Þ

f 4ðz; zpÞ ¼1

16πGð1�υÞð3�4υÞ

R1þ8ð1�υÞ2�ð3�4υÞ

R2

(

þðz�zpÞ2R31

þð3�4υÞðzþzpÞ2�2zpz

R32

þ6zpzðzþzpÞ2R52

)ðB4Þ

To conduct the integration in Appendix A, we can express∬Ap f 1ðz; zpÞ zp

l

� �j�1dAp as

∬Ap f 1ðz; zpÞzpl

� j�1dAp ¼

Zd

Zlf 1ðz; zpÞ

zpl

� j�1dzpdB ðB5Þ

The integration with respect to the pile length, l, can be performedand expressed asZlf 1ðz; zpÞ

zpl

� j�1dzp ¼ 1

16πGð1�υÞlj�1½ð3�4υÞF 0ajþFaj

þx2F 0bjþð3�4υÞx2Fbjþ2zðFcj�3x2FdjÞ

þ4ð1�υÞð1�2υÞðFej�x2Ff jÞ� ðB6Þ

in which

F 0aj ¼Xj

i ¼ 1

�ηj�1i zj� if 0ai; Faj ¼

Xj

i ¼ 1

ð�1Þj�1ηj�1i zj� if ai

F 0bj ¼Xj

i ¼ 1

�ηj�1i zj�1f 0bi; Fbj ¼

Xj

i ¼ 1

ð�1Þj�1ηj�1i zj� if bi;

Fcj ¼Xj

i ¼ 1

ð�1Þjηjizj� iþ1f ci; Fdj ¼Xj

i ¼ 1

ð�1Þjηjizj� iþ1f di;

Fej ¼1r2

Xj

i ¼ 1

ð�1Þj�1ηj�1i zj� if ei; Ff j ¼

1r2ð2Fej–FajÞ

In the above expressions ηj�1i takes the individual coefficients

of an expand binomial expression ða�bÞj�1. For example, ða�bÞj�1

is ða�bÞ2 when j¼ 3, and η21, η22 and η23 are then 1,–2, and

1 respectively. f 0ai, f0bi, and f ai to f ei all are terms of integration

and can be performed analytically. These terms of integration are

as follows:

f 0ai ¼Z z� l

z

hi�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þh2

p dh; f ai ¼Z zþ l

z

hi�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þh2

p dh;

f 0bi ¼Z z� l

z

hi�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr2þh2Þ3

q dh; f bi ¼ f ci

¼Z zþ l

z

hi�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr2þh2Þ3

q dh; f di ¼Z zþ l

z

hi�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr2þh2Þ5

q dh;

f ei ¼Z zþ l

zðhi�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þh2

q�hiÞdh:

The integral with respect to the pile diameter, d, in Eq. (B5) canbe evaluated using the Gaussian integration method. The otherthree integrations in Appendix A can be done in a similar fashion.

Finally, the integration ∬Af 2ðz; zpÞσbdA, ∬Af 4ðz; zpÞσbdA canfurther be described as

∬Af 2ðz; zpÞσbdA¼ σb

Z 2π

0

Z d=2

0f 2ðz; zpÞrbdrbdθ ðB7Þ

∬Af 4ðz; zpÞσbdA¼ σb

Z 2π

0

Z d=2

0f 4ðz; zpÞrbdrbdθ ðB8Þ

where rb is the base radius of the pile. The analytical solution ofthe integration with respect to rb has been given by Poulos andDavis (1980), and the integration with respect to θ is againevaluated by numerical means.

By using the above solution, the integration involved in Eqs.(A1)–(9) can be determined.

Appendix C

Pile–soil stiffness matrix [k] is given by

k �¼ kp

�þ ZLA½ �T ksLA �

ZLA½ � ðC1Þ

where

½kp� ¼½kH�

½kM �½kG�

264

375; ½ZLA� ¼

½ZH �½ZM�

½ZG�

264

375; ksLA

�¼ TLA½ � f LA ��1

ðC2Þ

[kH] and [kM] are given as

½kH � ¼EpIpπ4

2l3

00

14

⋱n4

26666664

37777775; ½kM� ¼

EpIpπ4

2l3

00

12

� �4⋱

2n�12

� �4

26666664

37777775:

ðC3Þ

The coefficients of k� k matrix kG �

are given by

kGij ¼ Epπd2

� �2ði�1Þðj�1Þlðiþ j�3Þ i¼ 1;2;…; kð Þ j¼ 1;2;…; kð Þ ðC4Þ

The coefficients in sub-matrix ½ZH � are given by

½ZH � ¼

1 z1l sin πz1

l

� �⋯ sin nπz1

l

� �1 z2

l sin πz2l

� �⋯ sin nπz2

l

� �⋮ ⋮ ⋮ ⋮ ⋮1 znq

l sin πznql

� �⋯ sin nπznq

l

� �

266664

377775 ðC6Þ

F. Liang et al. / Ocean Engineering 99 (2015) 23–3332

The coefficients in sub-matrix ½ZM � are given by

½ZM � ¼

1 z1l cos πz1

2l

� �⋯ cos ð2n�1Þπz1

2l

� 1 z2

l cos πz22l

� �⋯ cos ð2n�1Þπz2

2l

� ⋮ ⋮ ⋮ ⋮ ⋮

1 znql cos πznq

2l

� �⋯ cos ð2n�1Þπznq

2l

�

266666664

377777775

ðC7Þ

The sub-matrix ½ZG� is of order ðngþ1Þ � k and the coefficientsare given by

ZGij ¼ 1�zil

� j�1i¼ 1;2;…;ngþ1ð Þ j¼ 1;2;…; kð Þ ðC8Þ

The matrix f LA �

is given by

f LA �¼

½f ρL� ½f ρV �½f ρL� ½f ρV �

½f wL� ½f wL� ½f wV �

264

375 ðC9Þ

where coefficients in sub-matrix ½f ρL�, ½f ρV �, ½f wL� and ½f wV � aregiven by Eqs. (A1)–(A9) in Appendix A.

The matrix TLA½ � can be expressed as

TLA½ � ¼TL½ �

TL½ �TV½ �

264

375 ðC10Þ

where coefficients in sub-matrix TL½ � and [TV] are given by

TLij ¼12ldηi

zil

� j�1i¼ 1;2;…;nqð Þ j¼ 1;2;…; k1ð Þ ðC11Þ

and

TVij ¼12πldηi

zil

� j�1i¼ 1;2;…;ngð Þ j¼ 1;2;…; k2ð Þ;

TVij ¼ πd2

� �2

i¼ ngþ1ð Þ j¼ k2þ1ð Þ ðC12Þ

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