Research ArticleEffect of Cyclic Loading on the Lateral Behavior of OffshoreMonopiles Using the Strain Wedge Model

Keunju Kim1 Boo Hyun Nam2 and Heejung Youn1

1Department of Civil Engineering Hongik University Sangsu-dong 72-1 Mapo-gu Seoul 04066 Republic of Korea2Civil Environmental and Construction Engineering Department University of Central Florida Orlando FL 32816 USA

Correspondence should be addressed to Heejung Youn geotechhongikackr

Received 24 April 2015 Revised 28 July 2015 Accepted 13 August 2015

Academic Editor Alessandro Palmeri

Copyright copy 2015 Keunju Kim et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents the effect of cyclic loading on the lateral behavior of monopiles in terms of load-displacement curves deflectioncurves and p-y curves along the pile A commercial software Strain Wedge Model (SWM) was employed simulating a 75m indiameter and 60m long steelmonopile embedded into quartz sands In order to account for the effect of cyclic loading accumulatedstrains were calculated based on the results of drained cyclic triaxial compression tests and the accumulated strains were combinedwith static strains representing input strains into the SWMThe input strains were estimated for different numbers of cycles rangingfrom 1 to 105 and 3 different cyclic lateral loads (25 50 and 75 of static capacity) The lateral displacement at pile head wasfound to increasewith increasing number of cycles and increasing cyclic lateral loads In order tomodel these deformations resultingfrom cyclic loading the initial stiffness of the p-y curves has to be significantly reduced

1 Introduction

Monopiles have been frequently used to resist offshoreenvironmental loads such as wind wave tidal and ice loadsUnlike onshore loading conditions the offshore environ-mental loads have two main differentiated characteristicsthe magnitude of lateral load is significant and the load iscyclic Cyclic loading may cause serious problems in offshorestructures and surrounding soils depending on the soil typeand properties During the design lifetime the offshorestructure is known to undergo about 108 lateral cyclic loadingevents of varying amplitude [1] During this great numberof cycles the offshore structures need to endure fatigue andsurrounding soils should not alter much to satisfy the lateralcapacity and displacement criteria Engineering propertiesof surrounding soils can deteriorate because of a possibleaccumulation of excess pore water pressure induced by cyclicloading For cohesive soils the deterioration of the claystructure or the forming of a gap between soil and pile couldbe the main concerns causing degradation of engineeringproperties Furthermore an excessive number of cycles may

induce friction fatigue along the pile regardless of soil typewhich is another critical factor causing the deterioration ofside resistance For cohesionless soils the excess pore waterpressure is unlikely to develop as the soils are normally indrained conditions rather both the strength and stiffnesswill increase due to densification of surrounding soils Stillthe lateral displacement and rotation at pile head accumulatepossibly leading to a serviceability problem This is believedto be considerable especially in the case of a one way cyclicloading

The 119901-119910 curves have been widely used in practice toanalyze the lateral behavior of piles Many 119901-119910 curves havebeen proposed chiefly considering soil types (sand or clay)loading types (static or cyclic) and ground water conditions(soil under water or above water) In conventional designprocess the soils supporting piles are replaced by a set ofnonlinear springs and the pile is assumed to act as a beamsupported by the springs A soil spring acts as a reaction 119901per lateral displacement 119910 at a pile elevation The conceptwas first introduced by Winkler [2] and has been updated bymany researchers Reese et al [3] suggested a parabola 119901-119910

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 485319 12 pageshttpdxdoiorg1011552015485319

2 Mathematical Problems in Engineering

Qcyc = 25 50 and 75 of Qu

(cyclic)

20m

40m

10m

10m

10m

10m

75m

D = 75mL = 60mt = 90mmE = 210GPa

120574998400 = 11kNm3

c = 0kPa120601998400 = 375∘

Dr = 75 (dense)

Static capacity 2204MN

Cyclic lateral load

551MN (25)1102MN (50)1653MN (75)

The p-y curves were obtained at5 15 25 and 35m to calculate

qampl

Pile information

Soil properties

Time

Load

N cycles

Figure 1 Material properties of soil and pile and loading conditions

curve and an empirical adjustment factor 119860 with differentdefinitions for static and cyclic loading Later simplifiedapproaches to establish the 119901-119910 curves have been reported[4]

As the aforementioned 119901-119910 curves were developed notfor offshore wind turbines but for offshore oil platformsthere have been concerns that the offshore monopiles withtheir large lateral load compared to dead weight may behavedifferently Moreover the available 119901-119910 curves are validatedfor small piles with diameter of 1-2m under small numberof cycles (119873 lt 100) only while offshore monopiles have5ndash7m diameter and are subject to much more cycles Theexisting 119901-119910 curves may not be applicable to the offshoremonopiles without appropriate validation Achmus et al [1]instead developed design charts for offshore wind turbinefoundation installed in sandy soils based on a stiffness degra-dation model Cyclic triaxial tests and series of numericalanalyses were carried out in order to investigate the long-term behavior under cyclic loading Wichtmann et al [5 6]suggested a High Cycle Accumulation (HCA) model whichcaptures the effect of cyclic loading and performed seriesof finite element analyses The proposed HCA model wasbased on series of drained cyclic triaxial compression andextension tests [7 8] Instead of using complicated FEMpractical software analysing the lateral behavior of a pileby means of the Strain Wedge Model (SWM) was appliedin [9] to quantify the deformation of an offshore wind

turbine illustrating an increase in the lateral displacementwith increasing number of cycles Recently nonlinear 119901-119910curves in cohesive soils were proposed accounting for soil-pile interaction degradation in soil stiffness and strengthdue to cyclic loading soil-pile gap forming and radiationdamping [10]

The numerical analyses were often supported by smallor full scale test results Roesen et al [11 12] investigatedthe soil-pile interaction of a monopile embedded in sandsusing small scale tests The monopile was subject to a highnumber of one way cyclic loads (up to 60000 cycles) withvarying amplitude mean load and loading period It wasreported that the increase of the accumulated rotation withgrowing number of cycles could be approximated by a powerfunction Moller and Christiansen [13] conducted small scaletests to evaluate the existing 119901-119910 methods for static andcyclic loading and compared them with numerical analysisresults

This paper investigates the lateral behavior of a monopilesubjected to a lateral cyclic loading focusing particularly onthe influences of the number of cycles and magnitude ofcyclic lateral load The effect of cyclic loading was assessedin terms of accumulated strains which were later modifiedinto the input strains used in the SWM [14] The load-displacement curves deflection curves and cyclic 119901-119910 curveswere obtained from SWM and the lateral behavior of themonopile was analyzed

Mathematical Problems in Engineering 3

F

B

M

C

L

P 120591120591

Δ120590h

F120601m 120601m

120601m

120601m

120601m

120573m

(h minus x)tan 120573m tan120601m

D

D

h

x

hminusx

C

B

Figure 2 Basic strain wedge in uniform soil (redrawn from [17])

2 Material Properties and Loading Conditions

Figure 1 illustrates the offshore monopile embedded inquartz sands used in this study along with the pile and soilproperties The monopile which has to resist relatively largelateral load was a 60m long and 90mm thick steel tubethe diameter of which was 75m The pile was assumed tobe linear elastic with Youngrsquos modulus of 210GPa the soilswere assumed to be homogeneous through the layer Thematerial properties of the soil were adopted from a previousresearch which investigated the effect of cyclic loading on theaccumulated displacement of pile embedded into quartz sand[1] The investigated sands have a peak internal friction angleof 375∘ effective unit weight of 11 kNm3 and a cohesion of0 kPa The used soil was dense thus the relative density ofthe soil was assumed to be 75 throughout the layer and themaximum andminimum void ratio were assumed to be 0874and 0577 as used in the literature [7]

The information of cyclic lateral load is difficult to quan-tify because the load is site specific and variable in directionand magnitude Therefore the static lateral capacity (119876

119906)

of the monopile was calculated using the load-displacementcurve at seabed from the SWM analysis and the cycliclateral loads were chosen as 25 50 and 75 of the staticcapacity The static lateral capacity was determined to be2204MN at 381mm of lateral displacement at seabed [15]Consequently the used cyclic lateral loads 119876cyc were 5511102 and 1653MN respectively In the simulations afterapplying a certain number of cyclic loads the pile head was

loaded with the lateral load of 2204MN to obtain the load-displacement relationship at pile head and the 119901-119910 curveswere obtained at 5 15 25 and 35m depth along the pile belowseabed

3 Input Parameter Calculation

31 SWM Analysis The Strain Wedge Model was initiallysuggested by Norris [16] and has been used to predict thebehavior of flexible piles under lateral loading Figure 2presents the configuration of the Strain Wedge Model inuniform soil The soil resistance against lateral loading isdeveloped by a 3-dimensional passive wedge of soil at thefront of the pile and the configuration of the passive wedge isdetermined by themobilized friction angle and pile diameter120573119898= 45∘+ 1206011198982 119863 is the pile diameter ℎ is the height of

the passive wedge Δ120590ℎis the variation in horizontal stress at

wedge face and 120591 is the shear stress at pile sideThe advantageof the SWM is the transfer of the stress-strain-strengthbehavior of soil observed in element tests to the 1-dimensionalBeam on Elastic Foundation (BEF) parameters While theconventional 119901-119910 curves were experimentally derived fromstrains developed along the laterally loaded pile the SWMprovides a theoretical link between soil properties and lateralbehavior of pile The horizontal strain (120576

ℎ) in the passive

wedge is used to determine the deflection of the pile (119910) andthe variation of horizontal stress at the passive wedge is usedto determine the soil resistance (119901) associated with the BEF

4 Mathematical Problems in Engineering

120576accN=cyc

fp

f120587

fe

fY

12057650static12057650N=cyc

Pile information(D L E t)

Soil properties(120574998400 120601998400p 12057650N=cyc)

SWM analysis

Lateral behavior of monopile

fampl(25 50 and 75 of Qu)

fN

(N = 1 minus 105)

Figure 3 Flow chart of SWM analysis accounting for cyclic loading

Details of the theory and the assumptions used in the SWMare provided for example in [17]

Figure 3 shows a flow chart illustrating the numericalprocess proposed in this paper to analyze the monopileunder cyclic lateral loading The key input parameters intothe SWM include the static strain at 50 peak strength(12057650static) dimension and Youngrsquos modulus of the steel pile

and the material properties of the surrounding soils TheSWM requires the strain at 50 peak strength (120576

50) in order

to assess the lateral behavior of the pile under the staticloading condition On the other hand the strain under thecyclic loading condition accumulates as the number of cyclesincreases (120576acc119873=cyc) which means the strain will not recoverto the original state of the pile upon unloading [7]Thereforethe strain under cyclic loading needs to be modified into aninput strain (120576

50119873=cyc) considering key factors such as themagnitude of cyclic lateral loads average mean stress voidratio and number of cyclic loadsThe proposedmethodologyto assess the key factors will be presented in later sections

32 Input Strain under Cyclic Loading (12057650119873=cyc) Figure 4

shows the concept of calculating the accumulated straindeveloped by the applied number of cycles in the 119902-120576 planeAt the end of 119873 cycles the plastic strain has accumulated

Peak strength① ②

120576accN=cyc 12057650static

12057650N=cyc 12057650N=cyc

Dev

iato

ric st

ress

q

Axial strain 120576

120576accN=cyc + 12057650static①

② 12057650N=cyc

2qampl qf

Figure 4 Accumulated strain after119873 cycles and input strain in 119902-120576plane

to some extent which is denoted by 120576acc119873=cyc Duringa subsequent monotonic loading the deviatoric stress ofthe soil would start from the accumulated strain at zerodeviatoric stress to peak strength (as shown by line A)

Mathematical Problems in Engineering 5

03 04 05 06 07 08 09 10 11 1200

02

04

06

08

10

12

14

1000

600

200

118

120576 50

()

Void ratio (e)Cu

Figure 5 Relation between static strain (12057650static) and void ratio (119890)

for different 119862119906(redrawn from [16])

The probable strain at failure after the considered numberof cycles was calculated by summing twice the accumulatedstrain (120576acc119873=cyc) and twice the static strain at 50 peakstrength (120576

50static) Then the strain at failure under cyclicloading was modified into input strain at 50 peak strength(12057650119873=cyc) which means the corresponding line for a mono-

tonic loading incorporating the cumulative effects followslineB Even though this approach does not perfectly capturethe real behavior of soils and the real accumulated strain thecalculation procedure seems to be simple and reasonableTheinput strains used in this study were calculated by

12057650119873=cyc = 12057650static + 120576acc119873=cyc (1)

The static strain (12057650static) varies with the void ratio (ie

relative density) and coefficient of uniformity of the sand asshown in Figure 5 With the chosen relative density (75)and coefficient of uniformity (18) from previous research [7]the static strain (120576

50static) of the studied soil was calculatedto be 042 at the reference confining pressure of 425 kPaThe static strain needs modification to account for the effectof confining pressure which varies along the pile lengthThus the static strain under the reference confining pressure(425 kPa) is modified into an input value for the effectiveoverburden pressure at desired depth using (2) proposed in[16] Consider the following

(12057650)119894= (12057650)425

((120590V0)119894

425)

02

(2)

33 Accumulated Strain (120576acc119873=cyc) The accumulated strainsas a result of a large number of cycles were calculated basedon test data from cyclic drained triaxial tests in compressionand could be assessed using the following proposed equation[7]

120576acc = 119891ampl119891119901119891119884119891119890119891119873119891120587 (3)

Peak

stren

gth

Critica

l state

line

q

qav

Qcycle

p

120578av =

qav p

av

pav

qampl

Δ1205901

1205901

1205903

Δ1205901Cyclic axial

stress+

This graph is not scaledΔy

Δy puD (= 2qampl)

Figure 6 Stress path in 119901-119902 plane from a cyclic triaxial test incompression

Table 1 Summary of the partial functions f i and a list of thematerial constants Ci for the tested sand [7]

Function Material constants

119891ampl = (120576ampl

120576amplref

)

2

120576amplref 10minus4

119891119873= 1198621198731[ln(1 + 119862

1198732119873) + 119862

1198733119873]

1198621198731

34 times 10minus4

1198621198732

0551198621198733

60 times 10minus5

119891119901= exp [minus119862

119901(119901av

119901refminus 1)]

119862119901

043119901ref 100 kPa

119891119910= exp(119862

119884119884av) 119862

11988420

119891119890=(119862119890minus 119890)2

1 + 119890

1 + 119890ref

(119862119890minus 119890ref)

2

119862119890

054119890ref 0874

119891120587= 1 if polarization = constant

where 119891ampl is a coefficient for strain amplitude 119891119901is a

coefficient for average mean stress (119901av) 119891119884is a coefficient

for average stress ratio (119884av) 119891119890is a coefficient for void

ratio 119891119873

is a coefficient for number of cycles and 119891120587is a

factor associated with polarization change Equation (3) wasoriginally developed for the norm of the strain tensor butwithin this paper the assumption is made for simplificationthat the total strain predicted by (3) equals the horizontalstrain in the soil at the front of the pile Table 1 summarizesthe partial functions for the coefficients and the suggestedmaterial constants used in this study

Figure 6 illustrates the stress path in the 119901-119902 planeapplied in a cyclic triaxial test in compression When cycliclateral load is applied to the pile head the soils at the pilefront are compressed and released repeatedly which can be

6 Mathematical Problems in Engineering

1 10 100 1000 10000 1000000

1

2

3

4

5

9485746454

44342412

Stra

in am

plitu

de

(104)

Number of cycles Nqampl (kPa)

pav = 200 kPa120578av = 075

120576ampl

Figure 7 Development of strain amplitude with number of cyclesfor different deviatoric stress amplitudes (redrawn from [7])

simulated by cyclic triaxial test in compression In fact theSWM recommends using triaxial compression test results[17] thus the results from cyclic triaxial test in compressionwere adopted in this study Increase in axial (horizontal)stress (Δ120590

1) causes an increasing deviatoric stress from the

initial in situ stress on the 120578avndash line to the desired deviatoricstress amplitude (119902ampl) above the 120578avndash line and then theaxial stress decreased causing a decrease in deviatoric stressto the desired amplitude under the 120578avndash line In one wayloading condition the cyclic lateral load (119876cyc) induces thedeviatoric stress amplitude (119902ampl) above and under the 120578avndashline indicating that the increase in the axial stress (Δ120590

1) is

twice the deviatoric stress amplitude It should be noted thatAshour et al [17] recommend using conventional triaxialcompression test where the confining pressure is isotropicHowever the cyclic triaxial test has anisotropic confiningpressure before applying cyclic axial stress The anisotropicconfining pressure as shown in the 119901-119902 plane is appropriatefor in situ stress condition and the cyclic lateral load inducesaxial stress always greater than zero which occurs in one waycyclic loading condition

The coefficient for the amplitude (119891ampl) could be cal-culated from strain amplitude (120576ampl) which decreased withthe number of cycles in the cyclic triaxial tests with constantstress amplitude Figure 7 shows the development of strainamplitude with the number of cycles for various deviatoricstress amplitudes at an average stress with 119901av = 200 kPa and120578av= 075 In order to calculate deviatoric stress amplitude

along the pile resulting fromcyclic lateral load SWManalyseswere conducted using static strains From the static analysesthe 119901-119910 curves and pile deflection curves were obtained The119901-119910 curves provide the soil resistance per unit length (119901) at aspecific lateral displacement (119910) and the lateral displacement

Table 2 Strain amplitudes along the pile derived from Figure 7for different lateral cyclic loads of the pile and different numbers ofcycles (unit 10minus4)

Depth (m)120576ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 208 205 416 367 549 45515 187 163 349 297 511 42725 084 083 200 177 268 22835 030 031 097 085 201 167

Table 3 Calculated coefficients 119891ampl for strain amplitude

Depth (m)119891ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 432 421 1729 1350 3013 206815 349 265 1218 881 2616 182525 071 069 400 312 720 52135 009 010 094 072 406 278

1 10 100 1000 10000

Little and Briaud [18]Hettler [19]

Achmus et al [1]SWM

Number of cycles N

00

05

10

15

20

25

30

35

40

Accu

mul

atio

n ra

te o

f lat

eral

disp

lace

men

t at s

eabe

dysNy

s1

Figure 8 Comparison of accumulation rates of monopile displace-ment at seabed

can be obtained from pile deflection curves The estimatedsoil resistance per unit lengthwas divided by the pile diameter(75m) resulting in the increase in cyclic axial stress (Δ120590

1)

The deviatoric stress amplitude is half the cyclic axial stressIn the same way the deviatoric stress amplitudes along thepile at 5 15 25 and 35m depth were obtained

Using the obtained deviatoric stress amplitudes along thepile the strain amplitudes were estimated using Figure 7Linear interpolation technique was used when exact devia-toric stress amplitude is unavailable As the strain amplitudewas measured with an average mean pressure of 200 kPa thestrain amplitude needs correction to account for the effectof pressure dependency of the secant stiffness With simple

Mathematical Problems in Engineering 7

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

(c) 75 of119876119906

Figure 9 Load-displacement curves at pile head for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

Table 4 Calculated coefficients for number of cycles average stress ratio void ratio polarization changes and average mean stress

CoefficientDepth(m) 119901

av (kPa) 119902av (kPa) 120578

av119891119873(10minus4)

119891119884

119891119890

119891120587

119891119901

119873 = 1 119873 = 102

119873 = 103

119873 = 104

119873 = 105

5 3667 2750 1401525

1100018333

825013750 075 149 637 1371 2166 5751 153 013 1 096

07035 25667 19250 051

Independent of depth Constant valueswere used

Independentof119873

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Qcyc = 25 50 and 75 of Qu

(cyclic)

20m

40m

10m

10m

10m

10m

75m

D = 75mL = 60mt = 90mmE = 210GPa

120574998400 = 11kNm3

c = 0kPa120601998400 = 375∘

Dr = 75 (dense)

Static capacity 2204MN

Cyclic lateral load

551MN (25)1102MN (50)1653MN (75)

The p-y curves were obtained at5 15 25 and 35m to calculate

qampl

Pile information

Soil properties

Time

Load

N cycles

Figure 1 Material properties of soil and pile and loading conditions

curve and an empirical adjustment factor 119860 with differentdefinitions for static and cyclic loading Later simplifiedapproaches to establish the 119901-119910 curves have been reported[4]

As the aforementioned 119901-119910 curves were developed notfor offshore wind turbines but for offshore oil platformsthere have been concerns that the offshore monopiles withtheir large lateral load compared to dead weight may behavedifferently Moreover the available 119901-119910 curves are validatedfor small piles with diameter of 1-2m under small numberof cycles (119873 lt 100) only while offshore monopiles have5ndash7m diameter and are subject to much more cycles Theexisting 119901-119910 curves may not be applicable to the offshoremonopiles without appropriate validation Achmus et al [1]instead developed design charts for offshore wind turbinefoundation installed in sandy soils based on a stiffness degra-dation model Cyclic triaxial tests and series of numericalanalyses were carried out in order to investigate the long-term behavior under cyclic loading Wichtmann et al [5 6]suggested a High Cycle Accumulation (HCA) model whichcaptures the effect of cyclic loading and performed seriesof finite element analyses The proposed HCA model wasbased on series of drained cyclic triaxial compression andextension tests [7 8] Instead of using complicated FEMpractical software analysing the lateral behavior of a pileby means of the Strain Wedge Model (SWM) was appliedin [9] to quantify the deformation of an offshore wind

turbine illustrating an increase in the lateral displacementwith increasing number of cycles Recently nonlinear 119901-119910curves in cohesive soils were proposed accounting for soil-pile interaction degradation in soil stiffness and strengthdue to cyclic loading soil-pile gap forming and radiationdamping [10]

The numerical analyses were often supported by smallor full scale test results Roesen et al [11 12] investigatedthe soil-pile interaction of a monopile embedded in sandsusing small scale tests The monopile was subject to a highnumber of one way cyclic loads (up to 60000 cycles) withvarying amplitude mean load and loading period It wasreported that the increase of the accumulated rotation withgrowing number of cycles could be approximated by a powerfunction Moller and Christiansen [13] conducted small scaletests to evaluate the existing 119901-119910 methods for static andcyclic loading and compared them with numerical analysisresults

This paper investigates the lateral behavior of a monopilesubjected to a lateral cyclic loading focusing particularly onthe influences of the number of cycles and magnitude ofcyclic lateral load The effect of cyclic loading was assessedin terms of accumulated strains which were later modifiedinto the input strains used in the SWM [14] The load-displacement curves deflection curves and cyclic 119901-119910 curveswere obtained from SWM and the lateral behavior of themonopile was analyzed

Mathematical Problems in Engineering 3

F

B

M

C

L

P 120591120591

Δ120590h

F120601m 120601m

120601m

120601m

120601m

120573m

(h minus x)tan 120573m tan120601m

D

D

h

x

hminusx

C

B

Figure 2 Basic strain wedge in uniform soil (redrawn from [17])

2 Material Properties and Loading Conditions

Figure 1 illustrates the offshore monopile embedded inquartz sands used in this study along with the pile and soilproperties The monopile which has to resist relatively largelateral load was a 60m long and 90mm thick steel tubethe diameter of which was 75m The pile was assumed tobe linear elastic with Youngrsquos modulus of 210GPa the soilswere assumed to be homogeneous through the layer Thematerial properties of the soil were adopted from a previousresearch which investigated the effect of cyclic loading on theaccumulated displacement of pile embedded into quartz sand[1] The investigated sands have a peak internal friction angleof 375∘ effective unit weight of 11 kNm3 and a cohesion of0 kPa The used soil was dense thus the relative density ofthe soil was assumed to be 75 throughout the layer and themaximum andminimum void ratio were assumed to be 0874and 0577 as used in the literature [7]

The information of cyclic lateral load is difficult to quan-tify because the load is site specific and variable in directionand magnitude Therefore the static lateral capacity (119876

119906)

of the monopile was calculated using the load-displacementcurve at seabed from the SWM analysis and the cycliclateral loads were chosen as 25 50 and 75 of the staticcapacity The static lateral capacity was determined to be2204MN at 381mm of lateral displacement at seabed [15]Consequently the used cyclic lateral loads 119876cyc were 5511102 and 1653MN respectively In the simulations afterapplying a certain number of cyclic loads the pile head was

loaded with the lateral load of 2204MN to obtain the load-displacement relationship at pile head and the 119901-119910 curveswere obtained at 5 15 25 and 35m depth along the pile belowseabed

3 Input Parameter Calculation

31 SWM Analysis The Strain Wedge Model was initiallysuggested by Norris [16] and has been used to predict thebehavior of flexible piles under lateral loading Figure 2presents the configuration of the Strain Wedge Model inuniform soil The soil resistance against lateral loading isdeveloped by a 3-dimensional passive wedge of soil at thefront of the pile and the configuration of the passive wedge isdetermined by themobilized friction angle and pile diameter120573119898= 45∘+ 1206011198982 119863 is the pile diameter ℎ is the height of

the passive wedge Δ120590ℎis the variation in horizontal stress at

wedge face and 120591 is the shear stress at pile sideThe advantageof the SWM is the transfer of the stress-strain-strengthbehavior of soil observed in element tests to the 1-dimensionalBeam on Elastic Foundation (BEF) parameters While theconventional 119901-119910 curves were experimentally derived fromstrains developed along the laterally loaded pile the SWMprovides a theoretical link between soil properties and lateralbehavior of pile The horizontal strain (120576

ℎ) in the passive

wedge is used to determine the deflection of the pile (119910) andthe variation of horizontal stress at the passive wedge is usedto determine the soil resistance (119901) associated with the BEF

4 Mathematical Problems in Engineering

120576accN=cyc

fp

f120587

fe

fY

12057650static12057650N=cyc

Pile information(D L E t)

Soil properties(120574998400 120601998400p 12057650N=cyc)

SWM analysis

Lateral behavior of monopile

fampl(25 50 and 75 of Qu)

fN

(N = 1 minus 105)

Figure 3 Flow chart of SWM analysis accounting for cyclic loading

Details of the theory and the assumptions used in the SWMare provided for example in [17]

Figure 3 shows a flow chart illustrating the numericalprocess proposed in this paper to analyze the monopileunder cyclic lateral loading The key input parameters intothe SWM include the static strain at 50 peak strength(12057650static) dimension and Youngrsquos modulus of the steel pile

and the material properties of the surrounding soils TheSWM requires the strain at 50 peak strength (120576

50) in order

to assess the lateral behavior of the pile under the staticloading condition On the other hand the strain under thecyclic loading condition accumulates as the number of cyclesincreases (120576acc119873=cyc) which means the strain will not recoverto the original state of the pile upon unloading [7]Thereforethe strain under cyclic loading needs to be modified into aninput strain (120576

50119873=cyc) considering key factors such as themagnitude of cyclic lateral loads average mean stress voidratio and number of cyclic loadsThe proposedmethodologyto assess the key factors will be presented in later sections

32 Input Strain under Cyclic Loading (12057650119873=cyc) Figure 4

shows the concept of calculating the accumulated straindeveloped by the applied number of cycles in the 119902-120576 planeAt the end of 119873 cycles the plastic strain has accumulated

Peak strength① ②

120576accN=cyc 12057650static

12057650N=cyc 12057650N=cyc

Dev

iato

ric st

ress

q

Axial strain 120576

120576accN=cyc + 12057650static①

② 12057650N=cyc

2qampl qf

Figure 4 Accumulated strain after119873 cycles and input strain in 119902-120576plane

to some extent which is denoted by 120576acc119873=cyc Duringa subsequent monotonic loading the deviatoric stress ofthe soil would start from the accumulated strain at zerodeviatoric stress to peak strength (as shown by line A)

Mathematical Problems in Engineering 5

03 04 05 06 07 08 09 10 11 1200

02

04

06

08

10

12

14

1000

600

200

118

120576 50

()

Void ratio (e)Cu

Figure 5 Relation between static strain (12057650static) and void ratio (119890)

for different 119862119906(redrawn from [16])

The probable strain at failure after the considered numberof cycles was calculated by summing twice the accumulatedstrain (120576acc119873=cyc) and twice the static strain at 50 peakstrength (120576

50static) Then the strain at failure under cyclicloading was modified into input strain at 50 peak strength(12057650119873=cyc) which means the corresponding line for a mono-

tonic loading incorporating the cumulative effects followslineB Even though this approach does not perfectly capturethe real behavior of soils and the real accumulated strain thecalculation procedure seems to be simple and reasonableTheinput strains used in this study were calculated by

12057650119873=cyc = 12057650static + 120576acc119873=cyc (1)

The static strain (12057650static) varies with the void ratio (ie

relative density) and coefficient of uniformity of the sand asshown in Figure 5 With the chosen relative density (75)and coefficient of uniformity (18) from previous research [7]the static strain (120576

50static) of the studied soil was calculatedto be 042 at the reference confining pressure of 425 kPaThe static strain needs modification to account for the effectof confining pressure which varies along the pile lengthThus the static strain under the reference confining pressure(425 kPa) is modified into an input value for the effectiveoverburden pressure at desired depth using (2) proposed in[16] Consider the following

(12057650)119894= (12057650)425

((120590V0)119894

425)

02

(2)

33 Accumulated Strain (120576acc119873=cyc) The accumulated strainsas a result of a large number of cycles were calculated basedon test data from cyclic drained triaxial tests in compressionand could be assessed using the following proposed equation[7]

120576acc = 119891ampl119891119901119891119884119891119890119891119873119891120587 (3)

Peak

stren

gth

Critica

l state

line

q

qav

Qcycle

p

120578av =

qav p

av

pav

qampl

Δ1205901

1205901

1205903

Δ1205901Cyclic axial

stress+

This graph is not scaledΔy

Δy puD (= 2qampl)

Figure 6 Stress path in 119901-119902 plane from a cyclic triaxial test incompression

Table 1 Summary of the partial functions f i and a list of thematerial constants Ci for the tested sand [7]

Function Material constants

119891ampl = (120576ampl

120576amplref

)

2

120576amplref 10minus4

119891119873= 1198621198731[ln(1 + 119862

1198732119873) + 119862

1198733119873]

1198621198731

34 times 10minus4

1198621198732

0551198621198733

60 times 10minus5

119891119901= exp [minus119862

119901(119901av

119901refminus 1)]

119862119901

043119901ref 100 kPa

119891119910= exp(119862

119884119884av) 119862

11988420

119891119890=(119862119890minus 119890)2

1 + 119890

1 + 119890ref

(119862119890minus 119890ref)

2

119862119890

054119890ref 0874

119891120587= 1 if polarization = constant

where 119891ampl is a coefficient for strain amplitude 119891119901is a

coefficient for average mean stress (119901av) 119891119884is a coefficient

for average stress ratio (119884av) 119891119890is a coefficient for void

ratio 119891119873

is a coefficient for number of cycles and 119891120587is a

factor associated with polarization change Equation (3) wasoriginally developed for the norm of the strain tensor butwithin this paper the assumption is made for simplificationthat the total strain predicted by (3) equals the horizontalstrain in the soil at the front of the pile Table 1 summarizesthe partial functions for the coefficients and the suggestedmaterial constants used in this study

Figure 6 illustrates the stress path in the 119901-119902 planeapplied in a cyclic triaxial test in compression When cycliclateral load is applied to the pile head the soils at the pilefront are compressed and released repeatedly which can be

6 Mathematical Problems in Engineering

1 10 100 1000 10000 1000000

1

2

3

4

5

9485746454

44342412

Stra

in am

plitu

de

(104)

Number of cycles Nqampl (kPa)

pav = 200 kPa120578av = 075

120576ampl

Figure 7 Development of strain amplitude with number of cyclesfor different deviatoric stress amplitudes (redrawn from [7])

simulated by cyclic triaxial test in compression In fact theSWM recommends using triaxial compression test results[17] thus the results from cyclic triaxial test in compressionwere adopted in this study Increase in axial (horizontal)stress (Δ120590

1) causes an increasing deviatoric stress from the

initial in situ stress on the 120578avndash line to the desired deviatoricstress amplitude (119902ampl) above the 120578avndash line and then theaxial stress decreased causing a decrease in deviatoric stressto the desired amplitude under the 120578avndash line In one wayloading condition the cyclic lateral load (119876cyc) induces thedeviatoric stress amplitude (119902ampl) above and under the 120578avndashline indicating that the increase in the axial stress (Δ120590

1) is

twice the deviatoric stress amplitude It should be noted thatAshour et al [17] recommend using conventional triaxialcompression test where the confining pressure is isotropicHowever the cyclic triaxial test has anisotropic confiningpressure before applying cyclic axial stress The anisotropicconfining pressure as shown in the 119901-119902 plane is appropriatefor in situ stress condition and the cyclic lateral load inducesaxial stress always greater than zero which occurs in one waycyclic loading condition

The coefficient for the amplitude (119891ampl) could be cal-culated from strain amplitude (120576ampl) which decreased withthe number of cycles in the cyclic triaxial tests with constantstress amplitude Figure 7 shows the development of strainamplitude with the number of cycles for various deviatoricstress amplitudes at an average stress with 119901av = 200 kPa and120578av= 075 In order to calculate deviatoric stress amplitude

along the pile resulting fromcyclic lateral load SWManalyseswere conducted using static strains From the static analysesthe 119901-119910 curves and pile deflection curves were obtained The119901-119910 curves provide the soil resistance per unit length (119901) at aspecific lateral displacement (119910) and the lateral displacement

Table 2 Strain amplitudes along the pile derived from Figure 7for different lateral cyclic loads of the pile and different numbers ofcycles (unit 10minus4)

Depth (m)120576ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 208 205 416 367 549 45515 187 163 349 297 511 42725 084 083 200 177 268 22835 030 031 097 085 201 167

Table 3 Calculated coefficients 119891ampl for strain amplitude

Depth (m)119891ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 432 421 1729 1350 3013 206815 349 265 1218 881 2616 182525 071 069 400 312 720 52135 009 010 094 072 406 278

1 10 100 1000 10000

Little and Briaud [18]Hettler [19]

Achmus et al [1]SWM

Number of cycles N

00

05

10

15

20

25

30

35

40

Accu

mul

atio

n ra

te o

f lat

eral

disp

lace

men

t at s

eabe

dysNy

s1

Figure 8 Comparison of accumulation rates of monopile displace-ment at seabed

can be obtained from pile deflection curves The estimatedsoil resistance per unit lengthwas divided by the pile diameter(75m) resulting in the increase in cyclic axial stress (Δ120590

1)

The deviatoric stress amplitude is half the cyclic axial stressIn the same way the deviatoric stress amplitudes along thepile at 5 15 25 and 35m depth were obtained

Using the obtained deviatoric stress amplitudes along thepile the strain amplitudes were estimated using Figure 7Linear interpolation technique was used when exact devia-toric stress amplitude is unavailable As the strain amplitudewas measured with an average mean pressure of 200 kPa thestrain amplitude needs correction to account for the effectof pressure dependency of the secant stiffness With simple

Mathematical Problems in Engineering 7

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

(c) 75 of119876119906

Figure 9 Load-displacement curves at pile head for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

Table 4 Calculated coefficients for number of cycles average stress ratio void ratio polarization changes and average mean stress

CoefficientDepth(m) 119901

av (kPa) 119902av (kPa) 120578

av119891119873(10minus4)

119891119884

119891119890

119891120587

119891119901

119873 = 1 119873 = 102

119873 = 103

119873 = 104

119873 = 105

5 3667 2750 1401525

1100018333

825013750 075 149 637 1371 2166 5751 153 013 1 096

07035 25667 19250 051

Independent of depth Constant valueswere used

Independentof119873

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

F

B

M

C

L

P 120591120591

Δ120590h

F120601m 120601m

120601m

120601m

120601m

120573m

(h minus x)tan 120573m tan120601m

D

D

h

x

hminusx

C

B

Figure 2 Basic strain wedge in uniform soil (redrawn from [17])

2 Material Properties and Loading Conditions

Figure 1 illustrates the offshore monopile embedded inquartz sands used in this study along with the pile and soilproperties The monopile which has to resist relatively largelateral load was a 60m long and 90mm thick steel tubethe diameter of which was 75m The pile was assumed tobe linear elastic with Youngrsquos modulus of 210GPa the soilswere assumed to be homogeneous through the layer Thematerial properties of the soil were adopted from a previousresearch which investigated the effect of cyclic loading on theaccumulated displacement of pile embedded into quartz sand[1] The investigated sands have a peak internal friction angleof 375∘ effective unit weight of 11 kNm3 and a cohesion of0 kPa The used soil was dense thus the relative density ofthe soil was assumed to be 75 throughout the layer and themaximum andminimum void ratio were assumed to be 0874and 0577 as used in the literature [7]

The information of cyclic lateral load is difficult to quan-tify because the load is site specific and variable in directionand magnitude Therefore the static lateral capacity (119876

119906)

of the monopile was calculated using the load-displacementcurve at seabed from the SWM analysis and the cycliclateral loads were chosen as 25 50 and 75 of the staticcapacity The static lateral capacity was determined to be2204MN at 381mm of lateral displacement at seabed [15]Consequently the used cyclic lateral loads 119876cyc were 5511102 and 1653MN respectively In the simulations afterapplying a certain number of cyclic loads the pile head was

loaded with the lateral load of 2204MN to obtain the load-displacement relationship at pile head and the 119901-119910 curveswere obtained at 5 15 25 and 35m depth along the pile belowseabed

3 Input Parameter Calculation

31 SWM Analysis The Strain Wedge Model was initiallysuggested by Norris [16] and has been used to predict thebehavior of flexible piles under lateral loading Figure 2presents the configuration of the Strain Wedge Model inuniform soil The soil resistance against lateral loading isdeveloped by a 3-dimensional passive wedge of soil at thefront of the pile and the configuration of the passive wedge isdetermined by themobilized friction angle and pile diameter120573119898= 45∘+ 1206011198982 119863 is the pile diameter ℎ is the height of

the passive wedge Δ120590ℎis the variation in horizontal stress at

wedge face and 120591 is the shear stress at pile sideThe advantageof the SWM is the transfer of the stress-strain-strengthbehavior of soil observed in element tests to the 1-dimensionalBeam on Elastic Foundation (BEF) parameters While theconventional 119901-119910 curves were experimentally derived fromstrains developed along the laterally loaded pile the SWMprovides a theoretical link between soil properties and lateralbehavior of pile The horizontal strain (120576

ℎ) in the passive

wedge is used to determine the deflection of the pile (119910) andthe variation of horizontal stress at the passive wedge is usedto determine the soil resistance (119901) associated with the BEF

4 Mathematical Problems in Engineering

120576accN=cyc

fp

f120587

fe

fY

12057650static12057650N=cyc

Pile information(D L E t)

Soil properties(120574998400 120601998400p 12057650N=cyc)

SWM analysis

Lateral behavior of monopile

fampl(25 50 and 75 of Qu)

fN

(N = 1 minus 105)

Figure 3 Flow chart of SWM analysis accounting for cyclic loading

Details of the theory and the assumptions used in the SWMare provided for example in [17]

Figure 3 shows a flow chart illustrating the numericalprocess proposed in this paper to analyze the monopileunder cyclic lateral loading The key input parameters intothe SWM include the static strain at 50 peak strength(12057650static) dimension and Youngrsquos modulus of the steel pile

and the material properties of the surrounding soils TheSWM requires the strain at 50 peak strength (120576

50) in order

to assess the lateral behavior of the pile under the staticloading condition On the other hand the strain under thecyclic loading condition accumulates as the number of cyclesincreases (120576acc119873=cyc) which means the strain will not recoverto the original state of the pile upon unloading [7]Thereforethe strain under cyclic loading needs to be modified into aninput strain (120576

50119873=cyc) considering key factors such as themagnitude of cyclic lateral loads average mean stress voidratio and number of cyclic loadsThe proposedmethodologyto assess the key factors will be presented in later sections

32 Input Strain under Cyclic Loading (12057650119873=cyc) Figure 4

shows the concept of calculating the accumulated straindeveloped by the applied number of cycles in the 119902-120576 planeAt the end of 119873 cycles the plastic strain has accumulated

Peak strength① ②

120576accN=cyc 12057650static

12057650N=cyc 12057650N=cyc

Dev

iato

ric st

ress

q

Axial strain 120576

120576accN=cyc + 12057650static①

② 12057650N=cyc

2qampl qf

Figure 4 Accumulated strain after119873 cycles and input strain in 119902-120576plane

to some extent which is denoted by 120576acc119873=cyc Duringa subsequent monotonic loading the deviatoric stress ofthe soil would start from the accumulated strain at zerodeviatoric stress to peak strength (as shown by line A)

Mathematical Problems in Engineering 5

03 04 05 06 07 08 09 10 11 1200

02

04

06

08

10

12

14

1000

600

200

118

120576 50

()

Void ratio (e)Cu

Figure 5 Relation between static strain (12057650static) and void ratio (119890)

for different 119862119906(redrawn from [16])

The probable strain at failure after the considered numberof cycles was calculated by summing twice the accumulatedstrain (120576acc119873=cyc) and twice the static strain at 50 peakstrength (120576

50static) Then the strain at failure under cyclicloading was modified into input strain at 50 peak strength(12057650119873=cyc) which means the corresponding line for a mono-

tonic loading incorporating the cumulative effects followslineB Even though this approach does not perfectly capturethe real behavior of soils and the real accumulated strain thecalculation procedure seems to be simple and reasonableTheinput strains used in this study were calculated by

12057650119873=cyc = 12057650static + 120576acc119873=cyc (1)

The static strain (12057650static) varies with the void ratio (ie

relative density) and coefficient of uniformity of the sand asshown in Figure 5 With the chosen relative density (75)and coefficient of uniformity (18) from previous research [7]the static strain (120576

50static) of the studied soil was calculatedto be 042 at the reference confining pressure of 425 kPaThe static strain needs modification to account for the effectof confining pressure which varies along the pile lengthThus the static strain under the reference confining pressure(425 kPa) is modified into an input value for the effectiveoverburden pressure at desired depth using (2) proposed in[16] Consider the following

(12057650)119894= (12057650)425

((120590V0)119894

425)

02

(2)

33 Accumulated Strain (120576acc119873=cyc) The accumulated strainsas a result of a large number of cycles were calculated basedon test data from cyclic drained triaxial tests in compressionand could be assessed using the following proposed equation[7]

120576acc = 119891ampl119891119901119891119884119891119890119891119873119891120587 (3)

Peak

stren

gth

Critica

l state

line

q

qav

Qcycle

p

120578av =

qav p

av

pav

qampl

Δ1205901

1205901

1205903

Δ1205901Cyclic axial

stress+

This graph is not scaledΔy

Δy puD (= 2qampl)

Figure 6 Stress path in 119901-119902 plane from a cyclic triaxial test incompression

Table 1 Summary of the partial functions f i and a list of thematerial constants Ci for the tested sand [7]

Function Material constants

119891ampl = (120576ampl

120576amplref

)

2

120576amplref 10minus4

119891119873= 1198621198731[ln(1 + 119862

1198732119873) + 119862

1198733119873]

1198621198731

34 times 10minus4

1198621198732

0551198621198733

60 times 10minus5

119891119901= exp [minus119862

119901(119901av

119901refminus 1)]

119862119901

043119901ref 100 kPa

119891119910= exp(119862

119884119884av) 119862

11988420

119891119890=(119862119890minus 119890)2

1 + 119890

1 + 119890ref

(119862119890minus 119890ref)

2

119862119890

054119890ref 0874

119891120587= 1 if polarization = constant

where 119891ampl is a coefficient for strain amplitude 119891119901is a

coefficient for average mean stress (119901av) 119891119884is a coefficient

for average stress ratio (119884av) 119891119890is a coefficient for void

ratio 119891119873

is a coefficient for number of cycles and 119891120587is a

factor associated with polarization change Equation (3) wasoriginally developed for the norm of the strain tensor butwithin this paper the assumption is made for simplificationthat the total strain predicted by (3) equals the horizontalstrain in the soil at the front of the pile Table 1 summarizesthe partial functions for the coefficients and the suggestedmaterial constants used in this study

Figure 6 illustrates the stress path in the 119901-119902 planeapplied in a cyclic triaxial test in compression When cycliclateral load is applied to the pile head the soils at the pilefront are compressed and released repeatedly which can be

6 Mathematical Problems in Engineering

1 10 100 1000 10000 1000000

1

2

3

4

5

9485746454

44342412

Stra

in am

plitu

de

(104)

Number of cycles Nqampl (kPa)

pav = 200 kPa120578av = 075

120576ampl

Figure 7 Development of strain amplitude with number of cyclesfor different deviatoric stress amplitudes (redrawn from [7])

simulated by cyclic triaxial test in compression In fact theSWM recommends using triaxial compression test results[17] thus the results from cyclic triaxial test in compressionwere adopted in this study Increase in axial (horizontal)stress (Δ120590

1) causes an increasing deviatoric stress from the

initial in situ stress on the 120578avndash line to the desired deviatoricstress amplitude (119902ampl) above the 120578avndash line and then theaxial stress decreased causing a decrease in deviatoric stressto the desired amplitude under the 120578avndash line In one wayloading condition the cyclic lateral load (119876cyc) induces thedeviatoric stress amplitude (119902ampl) above and under the 120578avndashline indicating that the increase in the axial stress (Δ120590

1) is

twice the deviatoric stress amplitude It should be noted thatAshour et al [17] recommend using conventional triaxialcompression test where the confining pressure is isotropicHowever the cyclic triaxial test has anisotropic confiningpressure before applying cyclic axial stress The anisotropicconfining pressure as shown in the 119901-119902 plane is appropriatefor in situ stress condition and the cyclic lateral load inducesaxial stress always greater than zero which occurs in one waycyclic loading condition

The coefficient for the amplitude (119891ampl) could be cal-culated from strain amplitude (120576ampl) which decreased withthe number of cycles in the cyclic triaxial tests with constantstress amplitude Figure 7 shows the development of strainamplitude with the number of cycles for various deviatoricstress amplitudes at an average stress with 119901av = 200 kPa and120578av= 075 In order to calculate deviatoric stress amplitude

along the pile resulting fromcyclic lateral load SWManalyseswere conducted using static strains From the static analysesthe 119901-119910 curves and pile deflection curves were obtained The119901-119910 curves provide the soil resistance per unit length (119901) at aspecific lateral displacement (119910) and the lateral displacement

Table 2 Strain amplitudes along the pile derived from Figure 7for different lateral cyclic loads of the pile and different numbers ofcycles (unit 10minus4)

Depth (m)120576ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 208 205 416 367 549 45515 187 163 349 297 511 42725 084 083 200 177 268 22835 030 031 097 085 201 167

Table 3 Calculated coefficients 119891ampl for strain amplitude

Depth (m)119891ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 432 421 1729 1350 3013 206815 349 265 1218 881 2616 182525 071 069 400 312 720 52135 009 010 094 072 406 278

1 10 100 1000 10000

Little and Briaud [18]Hettler [19]

Achmus et al [1]SWM

Number of cycles N

00

05

10

15

20

25

30

35

40

Accu

mul

atio

n ra

te o

f lat

eral

disp

lace

men

t at s

eabe

dysNy

s1

Figure 8 Comparison of accumulation rates of monopile displace-ment at seabed

can be obtained from pile deflection curves The estimatedsoil resistance per unit lengthwas divided by the pile diameter(75m) resulting in the increase in cyclic axial stress (Δ120590

1)

The deviatoric stress amplitude is half the cyclic axial stressIn the same way the deviatoric stress amplitudes along thepile at 5 15 25 and 35m depth were obtained

Using the obtained deviatoric stress amplitudes along thepile the strain amplitudes were estimated using Figure 7Linear interpolation technique was used when exact devia-toric stress amplitude is unavailable As the strain amplitudewas measured with an average mean pressure of 200 kPa thestrain amplitude needs correction to account for the effectof pressure dependency of the secant stiffness With simple

Mathematical Problems in Engineering 7

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

(c) 75 of119876119906

Figure 9 Load-displacement curves at pile head for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

Table 4 Calculated coefficients for number of cycles average stress ratio void ratio polarization changes and average mean stress

CoefficientDepth(m) 119901

av (kPa) 119902av (kPa) 120578

av119891119873(10minus4)

119891119884

119891119890

119891120587

119891119901

119873 = 1 119873 = 102

119873 = 103

119873 = 104

119873 = 105

5 3667 2750 1401525

1100018333

825013750 075 149 637 1371 2166 5751 153 013 1 096

07035 25667 19250 051

Independent of depth Constant valueswere used

Independentof119873

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

120576accN=cyc

fp

f120587

fe

fY

12057650static12057650N=cyc

Pile information(D L E t)

Soil properties(120574998400 120601998400p 12057650N=cyc)

SWM analysis

Lateral behavior of monopile

fampl(25 50 and 75 of Qu)

fN

(N = 1 minus 105)

Figure 3 Flow chart of SWM analysis accounting for cyclic loading

Details of the theory and the assumptions used in the SWMare provided for example in [17]

Figure 3 shows a flow chart illustrating the numericalprocess proposed in this paper to analyze the monopileunder cyclic lateral loading The key input parameters intothe SWM include the static strain at 50 peak strength(12057650static) dimension and Youngrsquos modulus of the steel pile

and the material properties of the surrounding soils TheSWM requires the strain at 50 peak strength (120576

50) in order

to assess the lateral behavior of the pile under the staticloading condition On the other hand the strain under thecyclic loading condition accumulates as the number of cyclesincreases (120576acc119873=cyc) which means the strain will not recoverto the original state of the pile upon unloading [7]Thereforethe strain under cyclic loading needs to be modified into aninput strain (120576

50119873=cyc) considering key factors such as themagnitude of cyclic lateral loads average mean stress voidratio and number of cyclic loadsThe proposedmethodologyto assess the key factors will be presented in later sections

32 Input Strain under Cyclic Loading (12057650119873=cyc) Figure 4

shows the concept of calculating the accumulated straindeveloped by the applied number of cycles in the 119902-120576 planeAt the end of 119873 cycles the plastic strain has accumulated

Peak strength① ②

120576accN=cyc 12057650static

12057650N=cyc 12057650N=cyc

Dev

iato

ric st

ress

q

Axial strain 120576

120576accN=cyc + 12057650static①

② 12057650N=cyc

2qampl qf

Figure 4 Accumulated strain after119873 cycles and input strain in 119902-120576plane

to some extent which is denoted by 120576acc119873=cyc Duringa subsequent monotonic loading the deviatoric stress ofthe soil would start from the accumulated strain at zerodeviatoric stress to peak strength (as shown by line A)

Mathematical Problems in Engineering 5

03 04 05 06 07 08 09 10 11 1200

02

04

06

08

10

12

14

1000

600

200

118

120576 50

()

Void ratio (e)Cu

Figure 5 Relation between static strain (12057650static) and void ratio (119890)

for different 119862119906(redrawn from [16])

The probable strain at failure after the considered numberof cycles was calculated by summing twice the accumulatedstrain (120576acc119873=cyc) and twice the static strain at 50 peakstrength (120576

50static) Then the strain at failure under cyclicloading was modified into input strain at 50 peak strength(12057650119873=cyc) which means the corresponding line for a mono-

tonic loading incorporating the cumulative effects followslineB Even though this approach does not perfectly capturethe real behavior of soils and the real accumulated strain thecalculation procedure seems to be simple and reasonableTheinput strains used in this study were calculated by

12057650119873=cyc = 12057650static + 120576acc119873=cyc (1)

The static strain (12057650static) varies with the void ratio (ie

relative density) and coefficient of uniformity of the sand asshown in Figure 5 With the chosen relative density (75)and coefficient of uniformity (18) from previous research [7]the static strain (120576

50static) of the studied soil was calculatedto be 042 at the reference confining pressure of 425 kPaThe static strain needs modification to account for the effectof confining pressure which varies along the pile lengthThus the static strain under the reference confining pressure(425 kPa) is modified into an input value for the effectiveoverburden pressure at desired depth using (2) proposed in[16] Consider the following

(12057650)119894= (12057650)425

((120590V0)119894

425)

02

(2)

33 Accumulated Strain (120576acc119873=cyc) The accumulated strainsas a result of a large number of cycles were calculated basedon test data from cyclic drained triaxial tests in compressionand could be assessed using the following proposed equation[7]

120576acc = 119891ampl119891119901119891119884119891119890119891119873119891120587 (3)

Peak

stren

gth

Critica

l state

line

q

qav

Qcycle

p

120578av =

qav p

av

pav

qampl

Δ1205901

1205901

1205903

Δ1205901Cyclic axial

stress+

This graph is not scaledΔy

Δy puD (= 2qampl)

Figure 6 Stress path in 119901-119902 plane from a cyclic triaxial test incompression

Table 1 Summary of the partial functions f i and a list of thematerial constants Ci for the tested sand [7]

Function Material constants

119891ampl = (120576ampl

120576amplref

)

2

120576amplref 10minus4

119891119873= 1198621198731[ln(1 + 119862

1198732119873) + 119862

1198733119873]

1198621198731

34 times 10minus4

1198621198732

0551198621198733

60 times 10minus5

119891119901= exp [minus119862

119901(119901av

119901refminus 1)]

119862119901

043119901ref 100 kPa

119891119910= exp(119862

119884119884av) 119862

11988420

119891119890=(119862119890minus 119890)2

1 + 119890

1 + 119890ref

(119862119890minus 119890ref)

2

119862119890

054119890ref 0874

119891120587= 1 if polarization = constant

where 119891ampl is a coefficient for strain amplitude 119891119901is a

coefficient for average mean stress (119901av) 119891119884is a coefficient

for average stress ratio (119884av) 119891119890is a coefficient for void

ratio 119891119873

is a coefficient for number of cycles and 119891120587is a

factor associated with polarization change Equation (3) wasoriginally developed for the norm of the strain tensor butwithin this paper the assumption is made for simplificationthat the total strain predicted by (3) equals the horizontalstrain in the soil at the front of the pile Table 1 summarizesthe partial functions for the coefficients and the suggestedmaterial constants used in this study

Figure 6 illustrates the stress path in the 119901-119902 planeapplied in a cyclic triaxial test in compression When cycliclateral load is applied to the pile head the soils at the pilefront are compressed and released repeatedly which can be

6 Mathematical Problems in Engineering

1 10 100 1000 10000 1000000

1

2

3

4

5

9485746454

44342412

Stra

in am

plitu

de

(104)

Number of cycles Nqampl (kPa)

pav = 200 kPa120578av = 075

120576ampl

Figure 7 Development of strain amplitude with number of cyclesfor different deviatoric stress amplitudes (redrawn from [7])

simulated by cyclic triaxial test in compression In fact theSWM recommends using triaxial compression test results[17] thus the results from cyclic triaxial test in compressionwere adopted in this study Increase in axial (horizontal)stress (Δ120590

1) causes an increasing deviatoric stress from the

initial in situ stress on the 120578avndash line to the desired deviatoricstress amplitude (119902ampl) above the 120578avndash line and then theaxial stress decreased causing a decrease in deviatoric stressto the desired amplitude under the 120578avndash line In one wayloading condition the cyclic lateral load (119876cyc) induces thedeviatoric stress amplitude (119902ampl) above and under the 120578avndashline indicating that the increase in the axial stress (Δ120590

1) is

twice the deviatoric stress amplitude It should be noted thatAshour et al [17] recommend using conventional triaxialcompression test where the confining pressure is isotropicHowever the cyclic triaxial test has anisotropic confiningpressure before applying cyclic axial stress The anisotropicconfining pressure as shown in the 119901-119902 plane is appropriatefor in situ stress condition and the cyclic lateral load inducesaxial stress always greater than zero which occurs in one waycyclic loading condition

The coefficient for the amplitude (119891ampl) could be cal-culated from strain amplitude (120576ampl) which decreased withthe number of cycles in the cyclic triaxial tests with constantstress amplitude Figure 7 shows the development of strainamplitude with the number of cycles for various deviatoricstress amplitudes at an average stress with 119901av = 200 kPa and120578av= 075 In order to calculate deviatoric stress amplitude

along the pile resulting fromcyclic lateral load SWManalyseswere conducted using static strains From the static analysesthe 119901-119910 curves and pile deflection curves were obtained The119901-119910 curves provide the soil resistance per unit length (119901) at aspecific lateral displacement (119910) and the lateral displacement

Table 2 Strain amplitudes along the pile derived from Figure 7for different lateral cyclic loads of the pile and different numbers ofcycles (unit 10minus4)

Depth (m)120576ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 208 205 416 367 549 45515 187 163 349 297 511 42725 084 083 200 177 268 22835 030 031 097 085 201 167

Table 3 Calculated coefficients 119891ampl for strain amplitude

Depth (m)119891ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 432 421 1729 1350 3013 206815 349 265 1218 881 2616 182525 071 069 400 312 720 52135 009 010 094 072 406 278

1 10 100 1000 10000

Little and Briaud [18]Hettler [19]

Achmus et al [1]SWM

Number of cycles N

00

05

10

15

20

25

30

35

40

Accu

mul

atio

n ra

te o

f lat

eral

disp

lace

men

t at s

eabe

dysNy

s1

Figure 8 Comparison of accumulation rates of monopile displace-ment at seabed

can be obtained from pile deflection curves The estimatedsoil resistance per unit lengthwas divided by the pile diameter(75m) resulting in the increase in cyclic axial stress (Δ120590

1)

The deviatoric stress amplitude is half the cyclic axial stressIn the same way the deviatoric stress amplitudes along thepile at 5 15 25 and 35m depth were obtained

Using the obtained deviatoric stress amplitudes along thepile the strain amplitudes were estimated using Figure 7Linear interpolation technique was used when exact devia-toric stress amplitude is unavailable As the strain amplitudewas measured with an average mean pressure of 200 kPa thestrain amplitude needs correction to account for the effectof pressure dependency of the secant stiffness With simple

Mathematical Problems in Engineering 7

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

(c) 75 of119876119906

Figure 9 Load-displacement curves at pile head for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

Table 4 Calculated coefficients for number of cycles average stress ratio void ratio polarization changes and average mean stress

CoefficientDepth(m) 119901

av (kPa) 119902av (kPa) 120578

av119891119873(10minus4)

119891119884

119891119890

119891120587

119891119901

119873 = 1 119873 = 102

119873 = 103

119873 = 104

119873 = 105

5 3667 2750 1401525

1100018333

825013750 075 149 637 1371 2166 5751 153 013 1 096

07035 25667 19250 051

Independent of depth Constant valueswere used

Independentof119873

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

03 04 05 06 07 08 09 10 11 1200

02

04

06

08

10

12

14

1000

600

200

118

120576 50

()

Void ratio (e)Cu

Figure 5 Relation between static strain (12057650static) and void ratio (119890)

for different 119862119906(redrawn from [16])

The probable strain at failure after the considered numberof cycles was calculated by summing twice the accumulatedstrain (120576acc119873=cyc) and twice the static strain at 50 peakstrength (120576

50static) Then the strain at failure under cyclicloading was modified into input strain at 50 peak strength(12057650119873=cyc) which means the corresponding line for a mono-

tonic loading incorporating the cumulative effects followslineB Even though this approach does not perfectly capturethe real behavior of soils and the real accumulated strain thecalculation procedure seems to be simple and reasonableTheinput strains used in this study were calculated by

12057650119873=cyc = 12057650static + 120576acc119873=cyc (1)

The static strain (12057650static) varies with the void ratio (ie

relative density) and coefficient of uniformity of the sand asshown in Figure 5 With the chosen relative density (75)and coefficient of uniformity (18) from previous research [7]the static strain (120576

50static) of the studied soil was calculatedto be 042 at the reference confining pressure of 425 kPaThe static strain needs modification to account for the effectof confining pressure which varies along the pile lengthThus the static strain under the reference confining pressure(425 kPa) is modified into an input value for the effectiveoverburden pressure at desired depth using (2) proposed in[16] Consider the following

(12057650)119894= (12057650)425

((120590V0)119894

425)

02

(2)

33 Accumulated Strain (120576acc119873=cyc) The accumulated strainsas a result of a large number of cycles were calculated basedon test data from cyclic drained triaxial tests in compressionand could be assessed using the following proposed equation[7]

120576acc = 119891ampl119891119901119891119884119891119890119891119873119891120587 (3)

Peak

stren

gth

Critica

l state

line

q

qav

Qcycle

p

120578av =

qav p

av

pav

qampl

Δ1205901

1205901

1205903

Δ1205901Cyclic axial

stress+

This graph is not scaledΔy

Δy puD (= 2qampl)

Figure 6 Stress path in 119901-119902 plane from a cyclic triaxial test incompression

Table 1 Summary of the partial functions f i and a list of thematerial constants Ci for the tested sand [7]

Function Material constants

119891ampl = (120576ampl

120576amplref

)

2

120576amplref 10minus4

119891119873= 1198621198731[ln(1 + 119862

1198732119873) + 119862

1198733119873]

1198621198731

34 times 10minus4

1198621198732

0551198621198733

60 times 10minus5

119891119901= exp [minus119862

119901(119901av

119901refminus 1)]

119862119901

043119901ref 100 kPa

119891119910= exp(119862

119884119884av) 119862

11988420

119891119890=(119862119890minus 119890)2

1 + 119890

1 + 119890ref

(119862119890minus 119890ref)

2

119862119890

054119890ref 0874

119891120587= 1 if polarization = constant

where 119891ampl is a coefficient for strain amplitude 119891119901is a

coefficient for average mean stress (119901av) 119891119884is a coefficient

for average stress ratio (119884av) 119891119890is a coefficient for void

ratio 119891119873

is a coefficient for number of cycles and 119891120587is a

factor associated with polarization change Equation (3) wasoriginally developed for the norm of the strain tensor butwithin this paper the assumption is made for simplificationthat the total strain predicted by (3) equals the horizontalstrain in the soil at the front of the pile Table 1 summarizesthe partial functions for the coefficients and the suggestedmaterial constants used in this study

Figure 6 illustrates the stress path in the 119901-119902 planeapplied in a cyclic triaxial test in compression When cycliclateral load is applied to the pile head the soils at the pilefront are compressed and released repeatedly which can be

6 Mathematical Problems in Engineering

1 10 100 1000 10000 1000000

1

2

3

4

5

9485746454

44342412

Stra

in am

plitu

de

(104)

Number of cycles Nqampl (kPa)

pav = 200 kPa120578av = 075

120576ampl

Figure 7 Development of strain amplitude with number of cyclesfor different deviatoric stress amplitudes (redrawn from [7])

simulated by cyclic triaxial test in compression In fact theSWM recommends using triaxial compression test results[17] thus the results from cyclic triaxial test in compressionwere adopted in this study Increase in axial (horizontal)stress (Δ120590

1) causes an increasing deviatoric stress from the

initial in situ stress on the 120578avndash line to the desired deviatoricstress amplitude (119902ampl) above the 120578avndash line and then theaxial stress decreased causing a decrease in deviatoric stressto the desired amplitude under the 120578avndash line In one wayloading condition the cyclic lateral load (119876cyc) induces thedeviatoric stress amplitude (119902ampl) above and under the 120578avndashline indicating that the increase in the axial stress (Δ120590

1) is

twice the deviatoric stress amplitude It should be noted thatAshour et al [17] recommend using conventional triaxialcompression test where the confining pressure is isotropicHowever the cyclic triaxial test has anisotropic confiningpressure before applying cyclic axial stress The anisotropicconfining pressure as shown in the 119901-119902 plane is appropriatefor in situ stress condition and the cyclic lateral load inducesaxial stress always greater than zero which occurs in one waycyclic loading condition

The coefficient for the amplitude (119891ampl) could be cal-culated from strain amplitude (120576ampl) which decreased withthe number of cycles in the cyclic triaxial tests with constantstress amplitude Figure 7 shows the development of strainamplitude with the number of cycles for various deviatoricstress amplitudes at an average stress with 119901av = 200 kPa and120578av= 075 In order to calculate deviatoric stress amplitude

along the pile resulting fromcyclic lateral load SWManalyseswere conducted using static strains From the static analysesthe 119901-119910 curves and pile deflection curves were obtained The119901-119910 curves provide the soil resistance per unit length (119901) at aspecific lateral displacement (119910) and the lateral displacement

Table 2 Strain amplitudes along the pile derived from Figure 7for different lateral cyclic loads of the pile and different numbers ofcycles (unit 10minus4)

Depth (m)120576ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 208 205 416 367 549 45515 187 163 349 297 511 42725 084 083 200 177 268 22835 030 031 097 085 201 167

Table 3 Calculated coefficients 119891ampl for strain amplitude

Depth (m)119891ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 432 421 1729 1350 3013 206815 349 265 1218 881 2616 182525 071 069 400 312 720 52135 009 010 094 072 406 278

1 10 100 1000 10000

Little and Briaud [18]Hettler [19]

Achmus et al [1]SWM

Number of cycles N

00

05

10

15

20

25

30

35

40

Accu

mul

atio

n ra

te o

f lat

eral

disp

lace

men

t at s

eabe

dysNy

s1

Figure 8 Comparison of accumulation rates of monopile displace-ment at seabed

can be obtained from pile deflection curves The estimatedsoil resistance per unit lengthwas divided by the pile diameter(75m) resulting in the increase in cyclic axial stress (Δ120590

1)

The deviatoric stress amplitude is half the cyclic axial stressIn the same way the deviatoric stress amplitudes along thepile at 5 15 25 and 35m depth were obtained

Using the obtained deviatoric stress amplitudes along thepile the strain amplitudes were estimated using Figure 7Linear interpolation technique was used when exact devia-toric stress amplitude is unavailable As the strain amplitudewas measured with an average mean pressure of 200 kPa thestrain amplitude needs correction to account for the effectof pressure dependency of the secant stiffness With simple

Mathematical Problems in Engineering 7

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

(c) 75 of119876119906

Figure 9 Load-displacement curves at pile head for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

Table 4 Calculated coefficients for number of cycles average stress ratio void ratio polarization changes and average mean stress

CoefficientDepth(m) 119901

av (kPa) 119902av (kPa) 120578

av119891119873(10minus4)

119891119884

119891119890

119891120587

119891119901

119873 = 1 119873 = 102

119873 = 103

119873 = 104

119873 = 105

5 3667 2750 1401525

1100018333

825013750 075 149 637 1371 2166 5751 153 013 1 096

07035 25667 19250 051

Independent of depth Constant valueswere used

Independentof119873

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

1 10 100 1000 10000 1000000

1

2

3

4

5

9485746454

44342412

Stra

in am

plitu

de

(104)

Number of cycles Nqampl (kPa)

pav = 200 kPa120578av = 075

120576ampl

Figure 7 Development of strain amplitude with number of cyclesfor different deviatoric stress amplitudes (redrawn from [7])

simulated by cyclic triaxial test in compression In fact theSWM recommends using triaxial compression test results[17] thus the results from cyclic triaxial test in compressionwere adopted in this study Increase in axial (horizontal)stress (Δ120590

1) causes an increasing deviatoric stress from the

initial in situ stress on the 120578avndash line to the desired deviatoricstress amplitude (119902ampl) above the 120578avndash line and then theaxial stress decreased causing a decrease in deviatoric stressto the desired amplitude under the 120578avndash line In one wayloading condition the cyclic lateral load (119876cyc) induces thedeviatoric stress amplitude (119902ampl) above and under the 120578avndashline indicating that the increase in the axial stress (Δ120590

1) is

twice the deviatoric stress amplitude It should be noted thatAshour et al [17] recommend using conventional triaxialcompression test where the confining pressure is isotropicHowever the cyclic triaxial test has anisotropic confiningpressure before applying cyclic axial stress The anisotropicconfining pressure as shown in the 119901-119902 plane is appropriatefor in situ stress condition and the cyclic lateral load inducesaxial stress always greater than zero which occurs in one waycyclic loading condition

The coefficient for the amplitude (119891ampl) could be cal-culated from strain amplitude (120576ampl) which decreased withthe number of cycles in the cyclic triaxial tests with constantstress amplitude Figure 7 shows the development of strainamplitude with the number of cycles for various deviatoricstress amplitudes at an average stress with 119901av = 200 kPa and120578av= 075 In order to calculate deviatoric stress amplitude

along the pile resulting fromcyclic lateral load SWManalyseswere conducted using static strains From the static analysesthe 119901-119910 curves and pile deflection curves were obtained The119901-119910 curves provide the soil resistance per unit length (119901) at aspecific lateral displacement (119910) and the lateral displacement

Table 2 Strain amplitudes along the pile derived from Figure 7for different lateral cyclic loads of the pile and different numbers ofcycles (unit 10minus4)

Depth (m)120576ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 208 205 416 367 549 45515 187 163 349 297 511 42725 084 083 200 177 268 22835 030 031 097 085 201 167

Table 3 Calculated coefficients 119891ampl for strain amplitude

Depth (m)119891ampl

25 119876119906

50 119876119906

75 119876119906

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 432 421 1729 1350 3013 206815 349 265 1218 881 2616 182525 071 069 400 312 720 52135 009 010 094 072 406 278

1 10 100 1000 10000

Little and Briaud [18]Hettler [19]

Achmus et al [1]SWM

Number of cycles N

00

05

10

15

20

25

30

35

40

Accu

mul

atio

n ra

te o

f lat

eral

disp

lace

men

t at s

eabe

dysNy

s1

Figure 8 Comparison of accumulation rates of monopile displace-ment at seabed

can be obtained from pile deflection curves The estimatedsoil resistance per unit lengthwas divided by the pile diameter(75m) resulting in the increase in cyclic axial stress (Δ120590

1)

The deviatoric stress amplitude is half the cyclic axial stressIn the same way the deviatoric stress amplitudes along thepile at 5 15 25 and 35m depth were obtained

Using the obtained deviatoric stress amplitudes along thepile the strain amplitudes were estimated using Figure 7Linear interpolation technique was used when exact devia-toric stress amplitude is unavailable As the strain amplitudewas measured with an average mean pressure of 200 kPa thestrain amplitude needs correction to account for the effectof pressure dependency of the secant stiffness With simple

Mathematical Problems in Engineering 7

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

(c) 75 of119876119906

Figure 9 Load-displacement curves at pile head for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

Table 4 Calculated coefficients for number of cycles average stress ratio void ratio polarization changes and average mean stress

CoefficientDepth(m) 119901

av (kPa) 119902av (kPa) 120578

av119891119873(10minus4)

119891119884

119891119890

119891120587

119891119901

119873 = 1 119873 = 102

119873 = 103

119873 = 104

119873 = 105

5 3667 2750 1401525

1100018333

825013750 075 149 637 1371 2166 5751 153 013 1 096

07035 25667 19250 051

Independent of depth Constant valueswere used

Independentof119873

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 50 100 150 200 250 3000

5000

10000

15000

20000

25000

Late

ral l

oad

(kN

)

Displacement at pile head (mm)

(c) 75 of119876119906

Figure 9 Load-displacement curves at pile head for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

Table 4 Calculated coefficients for number of cycles average stress ratio void ratio polarization changes and average mean stress

CoefficientDepth(m) 119901

av (kPa) 119902av (kPa) 120578

av119891119873(10minus4)

119891119884

119891119890

119891120587

119891119901

119873 = 1 119873 = 102

119873 = 103

119873 = 104

119873 = 105

5 3667 2750 1401525

1100018333

825013750 075 149 637 1371 2166 5751 153 013 1 096

07035 25667 19250 051

Independent of depth Constant valueswere used

Independentof119873

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 5 Accumulated strains and input strains at119873 = 1 and119873 = 105 (unit 10minus4)

Depth (m)25 119876

11990650 119876

11990675 119876

119906

120576acc119873=cyc 12057650119873=cyc 120576acc119873=cyc 120576

50119873=cyc 120576acc119873=cyc 12057650119873=cyc

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

119873 = 1 119873 = 105

5 017 652 459 1093 069 2090 511 2531 121 3202 562 364315 010 281 559 831 033 935 583 1484 072 1936 621 248525 001 054 610 662 008 242 617 850 014 403 623 101235 000 005 651 657 001 040 652 691 006 157 657 808

1 10 100 1000 10000 1000000

50

100

150

200

250

300

Number of cycles N

Disp

lace

men

t at p

ile h

ead

(mm

)

75 of Qu

50 of Qu

25 of Qu

Figure 10 Effect of number of cycles on the displacement at pilehead

assumption that Youngrsquos modulus is proportional to thesquare root of average mean stress (4) the strain amplitudesat different mean stresses (different depth) were calculatedUsing the correction factor 119888

119901 provided in (5) the strain

amplitudes for every mean stress were calculated as shown in(6) Applying the correction factors to the strain amplitudesthe pressure dependency of strain amplitudes was consideredalong the pile

Table 2 provides the strain amplitudes for different depthsand number of cycles (1 and 105) under 3 cyclic lateralloads The cyclic lateral loads were selected at 25 50 and75 of the static capacity As shown in the table the strainamplitudes were found to be larger for higher cyclic lateralloads implying that the largest cyclic lateral load (75 of thestatic capacity) would lead to the largest accumulated strainsand consequently to the largest lateral displacement

119864 prop (119901av)05

(4)

119888119901= radic

119901avref (= 200 kPa)

119901av (5)

120576ampl119901av = 119888

119901120576ampl119901avref (6)

The coefficient for the number of cycles (119891119873) was simply

calculated by inserting the number of cycles under consider-ation together with the suggested material constants into thepartial function provided in Table 1 It was reported that theaccumulated strain increased with decreasing average meanstress and the coefficient for average mean stress (119891

119901) was

slightly affected by the number of cycles [7] However therecommended material constants neglected the effect of thenumber of cycles and were adopted as such in this study Theaverage stress ratio (the ratio of the average deviatoric stress tothe averagemean stress) as well as the void ratio was assumedto be constant accordingly the coefficients 119891

119884and 119891

119890 were

calculated to be 153 and 013 The coefficient 119891120587was 10 since

polarization changes were not considered in this studyTables 3 and 4 present the calculated coefficients for

the influences of the strain amplitude number of cyclesaverage stress ratio void ratio polarization and averagemean stress The homogeneous subground was divided into10m thick layers and the coefficients were calculated foreach layer It is shown that the coefficients 119891ampl for strainamplitude significantly varywith depths but only slightlywiththe number of cycles and that the coefficients 119891

119873for the

number of cycles vary from 149 (10minus4) to 5751 (10minus4) for 1and 105 cycles respectively Table 5 tabulates the calculatedaccumulated strains and input strains at 119873 = 1 and 105accounting for all the coefficients provided in Tables 3 and4 It is clearly shown that most accumulated strains developat a depth shallower than 25m regardless of the number ofcyclesThis agrees well with the fact that the strain amplitudesare very small at the greater depths for the given cyclic lateralloadsThemagnitude of input strains is found to be larger forlarger cyclic lateral loads and larger numbers of cycles

4 Numerical Results and Discussion

41 SWM Analysis versus FEM Achmus et al [1] performedthe Finite Element Method (FEM) with the stiffness degra-dation model to investigate the accumulated displacementdue to cyclic loading In order to check the applicabilityof the SWM analysis for cyclic lateral loading the lateraldisplacement of pile at seabed level was compared with theFEM results (Figure 8) The used soil and pile propertiesare identical to those in Figure 1 and the used cyclic lateralload was 150MN The number of cycles ranges from 1 to104 It is shown that the displacement at seabed is similar forboth analyses when 119873 = 1 but the SWM analysis tends tounderestimate the displacement when the number of cycles

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

60

50

40

30

20

10

0 0 50 100 150 200 250 300Deflection (mm)

Dep

th fr

om p

ile h

ead

(m)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 11 Pile deflection curves for 3 cyclic lateral loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

gets higher For 119873 = 104 the SWM analysis predicts 46increase in lateral displacement whereas the FEM predicts80 increase Both FEM and SWM analyses predict theincrease in accumulated lateral displacement with increas-ing number of cycles but underestimate the displacementscompared to the predictions proposed by other researchers[18 19]

42 Load-Displacement Curve Figure 9 shows the load-displacement curves at the pile head for different cyclic lateralloads and numbers of cycles For the given lateral load of2204MN the pile was found to be more displaced as thenumber of cycles increased When cyclic lateral load wasassumed to be 25 of the static capacity the displacementfor 1 cycle of lateral load was 18311mm but increased up

to 20193mm for 105 cycles The 105 cycles are likely toinduce approximately 111 additional displacement for thesame magnitude of lateral load (2204MN) The discrepancybetween the displacements at 119873 = 1 and 105 appearedto significantly grow when the largest magnitude in cycliclateral load was considered (75 of the static capacity)With the largest cyclic load the lateral displacement for105 cycles was calculated to be approximately 140 of thedisplacement under static loading It was found that anincrease both in themagnitude of the cyclic lateral load and inthe number of cycles significantly increases the accumulatedlateral displacement

Figure 10 summarizes the effect of the number of cyclesand the cyclic lateral load on the lateral displacement atpile head The lateral displacement appeared to increase

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(a) 25 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(b) 50 of119876119906

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

0 20 40 60 80 100 1200

200

400

600

800

1000

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

(c) 75 of119876119906

Figure 12 119901-119910 curves at 2m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

exponentially with increasing logarithm of the number ofcycles leading to about 40 increase of the displacementwhen comparing119873 = 1with119873 = 105 cyclesThis implies thatthe lateral displacement properly predicted for static lateralload would be inappropriate for cyclic loading conditions Itshould be noted that the partial safety factor for displacementis usually 10 in offshore pile design

43 Pile Deflection Curve Figure 11 presents deflectioncurves along the pile Similar to the load-displacementcurves the pile deflection increased with increasing numberof cycles and growing cyclic lateral load Even though the piledoes not deflect significantly below 30mdepth the deflectionnear the ground surface considerably differs depending on

loading conditions For the smallest cyclic lateral load thepile deflections calculated for the various numbers of cyclesdid not significantly differ however the deflectionwas clearlydependent on the number of cycles for the biggest lateral load

44 Cyclic 119901-119910 Curves Figures 12-13 present 119901-119910 curves at2 and 4m depth considering the effect of cyclic loadingindicating that the consideration of the cumulative deforma-tions due to cyclic loading leads to a considerable decreaseof the initial stiffness of the 119901-119910 curves Even though SWMprovides 119901-119910 curves at any desired locations two shallowdepths were particularly chosen because soil properties nearthe ground surface play a key role in the lateral behavior ofthe monopile As shown in the figures the initial stiffness of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(a) 25 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(b) 50 of119876119906

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

1600

Soil

resis

tanc

ep

(kN

m)

Displacement y (mm)

N = 1

N = 10

N = 102

N = 103

N = 104

N = 105

(c) 75 of119876119906

Figure 13 119901-119910 curves at 4m depth for 3 cyclic loads (a) 25 (b) 50 and (c) 75 of the static capacity (119876119906)

the 119901-119910 curves sharply decreased with increasing number ofcycles and increasing cyclic lateral loadThe stiffness decreasebecomes more prominent with increasing cyclic lateral loadSuch trends in the 119901-119910 curves do not differ with depth

5 Conclusions

In this study the lateral behavior of amonopile for an offshorewind turbine subject to cyclic loading was investigated usingthe SWManalysis in combination with the data from drainedcyclic triaxial compression test results on quartz sand Fromthe SWM analyses the load-displacement curves deflectioncurves and 119901-119910 curves were obtained and compared witheach other The following conclusions were drawn

(1) A new approach was proposed to investigate theeffect of cyclic lateral loads on the lateral behaviorof monopiles The approach employed the inputstrains (120576

50119873=cyc) which were calculated based onthe accumulated strains caused by cyclic lateral loads(120576acc119873=cyc) and the static strains (120576

50static)(2) The displacement at pile head representing the sum

of accumulated displacements due to a cyclic load-ing and a subsequent monotonic loading towardsa maximum load larger than the amplitude of thecycles was found to increase exponentially as thelogarithm of the number of cycles increased For agiven lateral load of 2204MN (lateral static capacityof the pile under consideration) the increase in lateral

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

displacement at pile headwas 111 if 105 cycleswith anamplitude corresponding to 25 of the static capacityhad been previously applied The increase of thelateral displacement due to preceding cycles becomesas much as 140 at an amplitude corresponding to75 of the static capacity

(3) The119901-119910 curves obtained at 2 and 4mdepths indicatedthat the initial stiffness of the 119901-119910 curves considerablydecreased with increasing number of cycles Thedecrease became prominent when the cyclic lateralload was 75 of the static capacity However it shouldbe kept in mind that this decrease of stiffness issome kind of ldquocalculationrdquo in order to consider theaccumulated deformations The real stiffness of thesand surrounding the pile usually increases due tocompaction

(4) The application of existing 119901-119910 curves without con-sidering cyclic loading would lead to an underestima-tion of the lateral displacement at pile head Modified119901-119910 curves accounting for the magnitude of cyclicloading and number of cycles are required to be usedin the BEF analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by National ResearchFoundation of Korea (NRF) funded by Ministry of ScienceICT amp Future Planning (NRF-2013R1A1A1011983) and KoreaInstitute of Ocean Science and Technology (20113010020010-11-3-510)

References

[1] M Achmus Y-S Kuo and K Abdel-Rahman ldquoBehavior ofmonopile foundations under cyclic lateral loadrdquoComputers andGeotechnics vol 36 no 5 pp 725ndash735 2009

[2] E Winkler Theory of Elasticity and Strength DomimicusPrague Czech Republic 1867

[3] L C Reese W R Cox and F D Koop ldquoAnalysis of laterallyloaded piles in sandrdquo in Proceedings of the Offshore Technologyin Civil Engineering sHall of Fame Papers from the Early Yearspp 95ndash105 Houston Tex USA May 1974

[4] D Bogard and H Matlock ldquoSimplified calculation of p-ycurves for laterally loaded piles in sandrdquo Tech Rep The EarthTechnology Houston Tex USA 1980

[5] T Wichtmann A Niemunis and T Triantafyllidis ldquoPredictionof long-term deformations for monopile foundations of off-shore wind power plantsrdquo in Proceedings of the 11th Baltic SeaGeotechnical Conference Geotechnics in Maritime EngineeringGdansk Poland September 2008

[6] T Wichtmann A Niemunis and T Triantafyllidis ldquoTowardsthe FE prediction of permanent deformations of offshorewind power plant foundations using a high-cycle accumulation

modelrdquo in Proceedings of the 2nd International Symposium onFrontiers in Offshore Geotechnics (ISFOG rsquo10) pp 635ndash640November 2010

[7] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained triaxialtestsrdquo Soil Dynamics and Earthquake Engineering vol 25 no12 pp 967ndash979 2005

[8] T Wichtmann A Niemunis and T Triantafyllidis ldquoStrainaccumulation in sand due to cyclic loading drained cyclictests with triaxial extensionrdquo Soil Dynamics and EarthquakeEngineering vol 27 no 1 pp 42ndash48 2007

[9] P Hinz K Lesny and W Richwien ldquoPrediction of monopiledeformation under high cyclic lateral loadingrdquo in Proceedingsof the 8th German Wind Energy Conference DEWEK BremenGermany 2006

[10] M Heidari M Jahanandish H E Naggar and A GhahramanildquoNonlinear cyclic behavior of laterally loaded pile in cohesivesoilrdquo Canadian Geotechnical Journal vol 51 no 2 pp 129ndash1432014

[11] H R Roesen L B Ibsen and L V Andersen ldquoExperimentaltesting of monopiles in sand subjected to one-way long-termcyclic lateral loadingrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical Engineering2013

[12] H R Roesen L B Ibsen M Hansen T K Wolf and K LRasmussen ldquoLaboratory testing of cyclic laterally loaded pilein cohesionless soilrdquo in Proceedings of the 23rd InternationalOffshore and Polar Engineering Conference (ISOPE rsquo13) pp 594ndash601 July 2013

[13] I F Moller and T H Christiansen Laterally loaded monopile indry and saturated sandmdashstatic and cyclic loading [MS thesis]Aalborg University Aalborg Denmark 2011

[14] L C C Geopile ldquoSWM 62 version 62 Laterally and axiallyloaded single pile and pile grouprdquo 2000

[15] AASHTO AASHTO LRFD Bridge Design Specifications Amer-ican Association of Highway and Transportation OfficialsWashington DC USA 2007

[16] G Norris ldquoTheoretically based BEF laterally loaded pile anal-ysisrdquo in Proceedings of the 3rd International Conference onNumerical Methods in Offshore Piling pp 361ndash386 NantesFrance 1986

[17] M Ashour G Norris and P Pilling ldquoLateral loading of apile in layered soil using the strain wedge modelrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 4pp 303ndash314 1998

[18] R L Little and J L Briaud ldquoFull scale cyclic lateral loadtests on six single piles in sandrdquo Miscellaneous Paper GL-88-27 Geotechnical Division Texas AampM Unuiversity CollegeStation Tex USA 1988

[19] A Hettler Verschiebungen Starrer und Elastischer Grund-ungskorper in Sand bei Monotoner und Zyklischer BelastungInstitut fur Bodenmechanik und Felsmechanik der UniversitatFridericiana Karlsruhe Germany 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of