Research ArticleCorrespondence Analysis of Soil around MicropileComposite Structures under Horizontal Load

Hai Shi1 Mingzhou Bai12 Chao Li13 Yunlong Zhang1 and Gang Tian1

1Beijing Jiaotong University No 3 Shangyuancun Haidian District Beijing 100044 China2Beijing Key Laboratory of Track Engineering Beijing China3Beijing Engineering and Technology Research Center of Rail Transit Line Safety and Disaster Prevention Beijing China

Correspondence should be addressed to Hai Shi 516519566qqcom

Received 27 June 2015 Accepted 1 September 2015

Academic Editor Fazal M Mahomed

Copyright copy 2015 Hai Shi et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The current approach which is based on conformal transformation is to map micropile holes in comparison with unit circledomain The stress field of soil around a pile plane as well as the plane strain solution to displacement field distribution canbe obtained by adopting complex variable functions of elastic mechanics This paper proposes an approach based on WinklerFoundation Beam Model with the assumption that the soil around the micropiles stemmed from a series of independent springsThe rigidity coefficient of the springs is to be obtained from the planar solution Based on the deflection curve differential equation ofEuler-Bernoulli beams one can derive the pile deformation and internal force calculationmethod ofmicropile composite structuresunder horizontal load In the end we propose reinforcing highway landslides with micropile composite structure and conductingon-site pile pushing tests The obtained results from the experiment were then compared with the theoretical approach It has beenindicated through validation analysis that the results obtained from the established theoretical approach display a reasonable degreeof accuracy and reliability

1 Introduction

Generally the diameter of a micropile is about 70ndash300mmin a small diameter filling pile [1] The slenderness ratio isrelatively big Its preliminary application and exploitationwere explored by Fondedile in Italy [2] Micropile compositestructures refer to antislide structures that are composed ofseveral miniature and single piles with a cap lid at the piletip which jointly bears the horizontal load [3] The structureadeptly adapts to shifting terrain during construction withsmall vibration and noise caused by the construction It ischaracterized by a small pile diameter rapid constructionand flexible piles Thus it has been widely used in build-ing reinforcements shake-proof foundation underpinningfoundation excavation support landslide control and othertypes of engineering found in buildings [4ndash6]

Previous research on micropile structure mainly dis-cussed ground stabilization building and rectification and soforth at vertical load bearing which was specific to internal

force deformation calculations an analysis of micropiles andthe internal force calculation of a combination of micropilegroups [7ndash9]The initial research yielded some achievementsCantoni et al [10] proposed a design and calculation methodbased on reticular micropiles working under the assumptionthat the retaining structure would need to be complex whenMacklin designed the anchorage retaining wall he simplifiedit as gravity retaining walls in order to analyze the internalforce of the micropile [11] Feng et al [12] proposed aninteraction analysis model for the pile-rock soil and mass-piles found in flat micropile systems they also establisheda mechanical model to calculate the internal force anddeformation of a micropile system using a finite elementmethod Juran et al [1] proposed a design approach usinga mesh micropile reinforced slope by assuming that thedense composite strengthening body formed by reticularmicropiles the internal soil mass and the internal systemwere not subjected to tensile stress furthermore Brownand Shie [13] calculated a pilersquos internal force by applying

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 382427 12 pageshttpdxdoiorg1011552015382427

2 Mathematical Problems in Engineering

a nonlinear elastic-plastic subgrade reaction method (calledthe 119901-119910 curve method) However there was less research onmicropile structures under horizontal load At present mostof the previously discussed engineering designs of micropilesadopt a calculation approach of specific to normal antislidepiles However the forcemodel and design calculation theoryof micropile composite structure have not been perfectedyet Establishing a calculation mode suitable for micropilecomposite structure and proposing a reasonable calculationmethod are urgently neededThus this paper further exploresresearch that capitalizes on this missed opportunity

This paper discusses an analytical solution to stress anddisplacement distribution under horizontal load based onthe mechanics theory of two-dimensional elastic complexfunctions Using the Winkler Foundation Beam Model thispaper assumes that the soil around a micropile stems froma series of independent springs The rigidity coefficient of aspring can be obtained using the planar solution After thatbased on the deflection curve of the differential equation ofan Euler-Bernoulli beam the pile deformation and internalforce calculation methods of a micropile composite structureunder horizontal load can be derived using two modesnamely by fixing one end with the other end sliding aswell as fixing both ends In the end the paper suggestsreinforcing highway landslides using micropile compositestructures and conducting on-site pile pushing tests Theresults obtained from the experiment have been compared tothe theoretical approach to verify the accuracy and reliabilityof the theoretical approach

2 Establishment of the Plane Strain SolutionModel of Micropiles

21 Description of the Problem Then researching the effectsof horizontal load on the micropiles it is important toconsider the internal force and deformation analysis of rigiddisc hole structure around the semi-infinite space surface atthe 119885 plane (rectangular coordinate system) As shown inFigure 1 the119877 region is a region of semi-infinite space exceptfor the disc-structure and 119886 is the distance between the soilaround the pile and the center of the pile 119903 is the radius ofthe disc-structure and 119865 is the horizontal load applied inthe center of the pile (119865

119909is 119909 component of direction 119865

119910

is 119910 component of direction) It is assumed that 1199060is the

displacement under horizontal load applied in the center ofthe pile

22 Fundamental Assumption of the Mechanical ModelWhile obtaining the stress field of the soil around the pile andthe plane strain solution of the displacement field distributionby adopting the complex variable functions of plane elasticmechanics the following assumptions are made

(1) While establishing a strainmodel of two-dimensionalcomplex elastic mechanics since the rigidity of themicrosteel pile is bigger than that around the pileassume thatminimicropiles and cross section aroundthe piles are rigid disc hole structures

x

y

F

Surrounding soil

Figure 1 Mechanical model of micropile-soil

(2) Assume that the deformation of the soil around thepile presents a tendency of elastic variation underhorizontal load

(3) Compare the deformation of the micropile and thesoil around the pile under horizontal load

23 Basic Control Equation According to the complex vari-able function of plane elasticmechanics the complex analyticfunction of stress and displacement components at 119877 regionare to be expressed in 120593

1(119909) and 120595

1(119909) then [14]

120590119909+ 120590119910= 4Re [1205931015840

1(119911)]

120590119910minus 120590119909+ 2119894120591119909119910

= 2 [11991112059310158401015840

1(119911) + 120595

1015840

1(119911)]

2119866 (119906 + 119894V) = 120581120593 (120585) minus 12059310158401(120585)

120596 (120585)

1205961015840 (120585)minus 120595 (120585)

(1)

where 119911 is any point on the micropile hole 120590119909 120590119910 and 120591

119909119910are

the stress and strain component of any point Re separatesthe real part and imaginary part in 120581 = 3 minus 4120583 Planestrain (3minus120583)(1+120583) Plane stress and 119866 = 1198642(1+120583) where119864 is modulus of elasticity and 120583 is the Poisson ratio

The analytic function and expression 1205931(119909) and 120595

1(119909)

under stress at infinity bounded conditions are as follows

1205931(119911) = minus

1

2120587 (1 + 120581)(119865119909+ 119894119865119910) ln 119911 + (119861 + 119894119862) 119911

+ 1205930

1(119911)

1205951(119911) =

1

2120587 (1 + 120581)(119865119909minus 119894119865119910) ln 119911 + (119861

1015840

+ 1198941198621015840

) 119911

+ 1205950

1(119911)

(2)

where 119861 1198611015840 and 119862

1015840 respectively represent the depth ofthe tunnel the bulk density of the surrounding rock andthe relationship of the lateral pressure coefficient and thesurrounding rock stress In this equation 120581 is positive integer

Mathematical Problems in Engineering 3

1205930

1(119911) and 120595

0

1(119911) are the analytic function of the points in the

neighborhood located at infinity Where 119861 = (120590infin

119909+ 120590infin

119910)4

1198611015840

= (120590infin

119910minus 120590infin

119909)2 1198621015840 = 120591

infin

119909119910 120590infin119909 120590infin119910 and 120591

infin

119909119910is the stress

located at infinityAccording to the analytic function under complex vari-

ables functions at bounded conditions the displacement andstress boundary conditions are as follows

The condition of stress boundary 119894 int119861

119860

(119865119909+ 119894119865119910) 119889119904

= 1205931(119911) + 1199111205931015840

1(119911) + 120595

1(119911)

The condition of displacement boundary 1205811205931(119911)

minus 11991112059310158401(119911) minus 120595

1(119911) = 2119866 (119906 + 119894V)

(3)

24 ConformalMapping Onemustmap themicropile orificeto unit annulus The semi-infinite space orifice within 119885

plane can be converted to within the plane 120585 after conformalmapping as shown in Figure 2Then themapping function ofzone 119877 after conformal mapping is shown in [15] Therefore

119911 = 120596 (120585) = minus1198941198861 minus 1205722

1 + 1205722

1 minus 120585

1 + 120585 (4)

where any point 120585 located in the 120585 plane is expressed as polarcoordinates that is 120585 = 120588119890

119894120579 and 120572 is a parameter determinedby 119886 and 119903

119903

119886=

2119886

1 + 1198862 (5)

After transforming the function at plane 119911 to the functionof 120585 the stress and displacement component will change to[16]

120590120588+ 120590120579= 120590119909+ 120590119910= 4Re [Φ (120585)]

120590119909minus 120590119910+ 2119894120591119909119910

=21205852

12058821205961015840 (120585)[120596 (120585)Φ

1015840

(120585) + 1205961015840

(120585) Ψ (119911)]

2119866 (119906 + 119894V) = 120581120593 (120585) minus 12059310158401(120585)

120596 (120585)

1205961015840 (120585)minus 120595 (120585)

(6)

where 119911 = 120596(120585) Φ(120585) = 1205931015840

1(119911) and Ψ(120585) = 120595

1015840

1(119911) From

formula (6) it is detected that in order to get the stressand displacement values through using the fundamentalequation tone has to solve the complex function and obtainthe solution to 120593

0(120585) and 120595

0(120585)

25 Solution to Displacement 1199060under Horizontal Load (1)

According to the plane displacement boundary conditions ofmicropile-soil structure the following can be seen from theassumed conditions

The displacement boundary conditions can be dividedas follows infinity to pile core position namely 119911 = 119911 so

120578

1205790

120588 =1

120585

R-area

Figure 2 Plane of conformal transformation

the soilmass will not cause direct impact on pile deformationwhereas displacement exists at the contact surface of the pilehole boundary and soil mass namely |119911 + 119894119886| = 119903 The strainof themicropiles soil structure to plane 120581 = 3minus4120583 can be seenin formula (3) In other words the displacement boundary isas follows

[(3 minus 4120583) 120593 (120585) minus 120596 (120585)1205931015840 (120585)

1205961015840 (120585)minus 120595 (120585)]

1199040

=119864

1 + 120583(119906 + 119894V)

[(3 minus 4120583) 120593 (120585) minus 120596 (120585)1205931015840 (120585)

1205961015840 (120585)minus 120595 (120585)]

1199041

= 0

(7)

where 1199041is distance between the soil around the pile and the

center of the pile and 1199040is the boundary curve of the pile hole

When the function of the 119911 plane is transformed into afunction of the 120585 plane 119911 = 120596(120585) Φ(120585) = 120593

1015840

1(119911) and Ψ(120585) =

1205951015840

1(119911) are substituted into formula (2) to get the following

120593 (120585) = minus1

8120587 (1 minus 120583)(sum119865119909+ 119894sum119865

119910) ln120596 (120585)

+ (119861 + 119894119862) 120596 (120585) + 1205930(120585)

120595 (120585) =1

8120587 (1 minus 120583)(sum119865119909minus 119894sum119865

119910) ln120596 (120585)

+ (1198611015840

+ 1198941198621015840

) 120596 (120585) + 1205950(120585)

(8)

where 1205930(120585) = sum

119899

119896=0119886119896120585minus119896 and 120595

0(120585) = sum

119899

119896=0119887119896120585minus119896 From the

given situation one can obtain sum119865119909

= 119865 sum119865119910

= 0 and 119861 =

119862 = 1198621015840

= 1198611015840

= 0 When the initial conditions are substitutedinto formula (8) one can get the following

120593 (120585) = minus1

8120587 (1 minus 120583)119865 ln120596 (120585) + 120593

0(120585)

120595 (120585) =1

8120587 (1 minus 120583)119865 ln120596 (120585) + 120595

0(120585)

(9)

4 Mathematical Problems in Engineering

By substituting formula (9) into the displacement bound-ary condition formula (7) it can be derived that

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

1

(10)

where

119891 =119864

1 + 1205831199060minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

1198911= minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

(11)

Through the simultaneous application of the boundaryconditions of the two equations found in formula (10) theexpression of 120593

0(120585) and 120595

0(120585) can be derived (including the

unknown displacement of 1199060)

(2)Considering the micropile hole boundary namely forthe boundary conditions 119904

0 120590 = 119890

119894120579 the stress boundaryconditions are to be expressed with components under theorthogonal curvilinear coordinate system at the 119911 planeNamely

[(3 minus 4120583) 1205930(120590) minus

120596 (120590)

1205961015840 (120590)12059310158400(120590) minus 120595

0(120590)] = 119891

0 (12)

The Cauchy integral operator (12120587119894) ∮(119889120590(120590 minus 120585)) atboth ends of the formula above can be obtained

(3 minus 4120583)

2120587119894∮

1205930(120590) 119889120590

120590 minus 120585minus

1

2120587119894∮

120596 (120590)

1205961015840 (120590)

12059310158400(120590)

120590 minus 120585119889120590

minus1

2120587119894∮

1205950(120590)

120590 minus 120585119889120590 =

1

2120587119894∮

1198910

120590 minus 120585119889120590

(13)

In formula (13) ∮(1205930(120590)119889120590(120590 minus 120585)) = minus120593

0(120585)

(12120587119894) ∮(1205950(120590)(120590 minus 120585))119889120590 = 0 so by substituting into

formula (13) one can find 1205930(120585)

Similarly according to formula (12) the value of 1205950(120585)

can be obtained by taking conjugation at both sides and thenapplying the Cauchy integral operator Thus the informationcan be simultaneously obtained by combining formula (10)to get the value of 119906

0

26 Solution to 1205930(120585) and 120595

0(120585) To obtain the boundary

conditions of a miniature pile make 120585 = 120590120588 and unfold 120593(120585)

and 120595(120585) in a Laurent series form namely [17]

120593 (120585) = 1198860+

119899

sum

119896=1

119886119896120585119896

+

119899

sum

119896=1

119887119896120585minus119896

120595 (120585) = 1198880+

119899

sum

119896=1

119888119896120585119896

+

119899

sum

119896=1

119889119896120585minus119896

(14)

Depending on formula (4) one can get the following

120596 (120590)

1205961015840 (120590)= minus

1

2

(1 + 120588120590) (120590 minus 120588)2

1205902 (1 minus 120588120590) (15)

Using formulas (14) and (15) substituted into boundaryconditions 119904

1of formula (7) one can get the following

1198880= minus1198860minus

1

21198861minus

1

21198871

119888119896= minus119887119896+

1

2(119896 minus 1) 119886

119896minus1minus

1

2(119896 + 1) 119886

119896+1

119889119896= minus119886119896+

1

2(119896 minus 1) 119887

119896minus1minus

1

2(119896 + 1) 119887

119896+1

(16)

Make 119891(120585) = 119891(120572120590) = 2119866(119906 + 119894V) and 119891lowast

(120572120590) = (1 minus

120572120590)119891(120572120590) = suminfin

119896119860119896120590119896 Formulas (14) (15) and (17)ndash(19) can

be substituted into the boundary conditions 1199041of formula (7)

by eliminating 119888119896and 119889

119896 one finds 119886

119896and 119887119896

(1 minus 1205722

) (119896 + 1) 119886119896+1

minus (1205722

+ 120581120572minus2119896

) 119887119896+1

= (1 minus 1205722

) 119896119886119896minus (1 + 120581120572

minus2119896

) 119887119896+ 119860minus119896

120572119896

(1 + 1205811205722119896+2

) 119886119896+1

+ (1 minus 1205722

) (119896 + 1) 119887119896+1

= 1205722

(1205722

+ 1205811205722119896

) 119886119896+ (1 minus 120572

2

) 119896119887119896+ 1198601+119896

120572119896+1

(1 minus 1205722

) 1198861minus (120581 + 120572

2

) 1198871= 1198600minus (120581 + 1) 119886

0

(1 + 1205811205722

) 1198861+ (1 minus 120572

2

) 1198871= 1198601120572 + 1205722

(120581 + 1) 1198860

(17)

We simultaneously solved the four formulas in (17) toget all of the coefficients except for 119886

0 since 119886

0represents

rigid body displacement thus no stress will be generatedand it can be deemed as 0 By 119896 times of iteration one canobtain 119886

119896 thus to get 120593

0(120585) and 120595

0(120585) one must substitute

1205930(120585) and 120595

0(120585) in the complex function of the fundamental

equation to ascertain the stress and displacement field ofthe soil around a pile of micropiles under horizontal loadAccording to the assumed conditions specified in Sections21 and 22 the relationship between horizontal load and thehorizontal displacement can be obtained from 119870 = 119865119906

0

3 Establishment of Pile-Soil Mechanics ModelBased on Winkler Foundation Beam

In most cases while ministeel tub piles under horizontalload are applied in landslide control and slope reinforcement

Mathematical Problems in Engineering 5

Micropile

Sloping surface

Roof beam

Landslid

e surfa

ce

Figure 3 Micropile composite structure set on step of slope

a pile groupsrsquo layout will be adopted Especially for multistageslopes platforms are available for each grade Miniature pilescan be set at the platform for reinforcement The multipleminiature piles that are exposed on the platform can be fixedwith a top beam to enhance their sliding resistance forceThemicropile layout is shown in Figure 3

31 Model Assumption While miniature piles are appliedfor landslide reinforcement the mechanism can be used toaddress bigger shearing resistance in miniature pile land-slides Pile-soil structures will resist landslide thrust formedbymicropiles as well as the soil mass For internal calculationtheWinkler low econometricmodel is adoptedThis researchassumes that the soil around the micropile has a series ofdiscrete springs and a rigidity coefficient of 119870 = 119865119906

0 As

shown in Figure 3 since the micropile is fixed at the bed rockbelow the sliding surface it is assumed that the sliding surfaceis as fixed constraint Due to the fact that themicropile is fixedand connected through a top beam as compared tominiaturepile the top beam can be deemed as a rigid member Underthe effects of horizontal load the top beam only experienceshorizontal displacement so the displacement at each pile capwill be the same [18] Thus the internal force calculation ofthe model is shown in Figure 4

32 Internal Force Calculation According to the fundamen-tal theory of elastic mechanics the stress and calculationmodel of a single micropile can be obtained as shown inFigure 5 It is composed of two calculation models namelywith one fixed and one sliding end under a concentrated load(Figure 5(a)) and with both ends fixed under a uniformlydistributed load (Figure 5(b)) Specific to the calculationmodel found in Figure 4 for three-row micropile structurethe internal force of the micropile AD is obtained throughsuperposition of Figures 5(a) and 5(b) The internal forces ofmicropile BE and CF are derived from Figure 5(a)

321 Solution to Calculation Model with One Fixed andOne Sliding End under a Concentrated Load According tothe internal force calculation model specified in Figure 5(a)when considering micropile structures with one fixed endand one sliding end under a concentrated load the boundary

q

F

f f

A B C

D E F

K K K H

Figure 4 Mechanical calculation model of micropile structure

z

x

H

F

(a)

q

z

x

H

(b)

Figure 5 Mechanical calculation model of single micropile

condition on the top of the micropile is 119911 = 0 the bendingmoment is 119872 = 0 and 119865 is horizontal shear The boundarycondition on the bottom of the micropile is 119911 = 119867 the angleis 120593 = 0 and the horizontal displacement is 119906 = 0 Theflexural differential equation of an Euler-Bernoulli beamwithone fixed and one sliding end is [19]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 0 (18)

In the formula above 119864 is the elasticity modulus of theminiature pile 119868 is the inertia moment of a micropile crosssection119870 is the rigidity coefficient of an assumed spring and119906(119911) is the horizontal displacement of the soil around the pilealong the pile body

6 Mathematical Problems in Engineering

Roof beam

GroundSteel plate

Jack

Counterforce deviceRetaining wall

Earth pressure boxSteel bar

Micropile structure

Reac

tion

wal

l

Earth pressure cells

Reinforced stress meter

Inclinometer tube

Thrust 1 2 34 5 6

meter

05m

Figure 6 The schematic drawing of a test model of micropiles composite structure

By substituting the boundary conditions in the flexuraldifferential equation (18) the analytical solution can beobtained

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)]

119872 = 21205822

119864119868119890120582119911

[1198882cos (120582119911) minus 119888

1sin (120582119911)]

+ 21205822

119864119868119890minus120582119911

[1198884cos (120582119911) minus 119888

3sin (120582119911)]

(19)

where 120582 = (1198704119864119868)14 and 119888

1 1198882 1198883 and 119888

4are the integral

constant

322 Solution to Calculation Model with Both Ends Fixedunder a Uniformly Distributed Load According to the inter-nal force calculation model specified in Figure 5(b) formicropile structures with both ends fixed under a uniformlydistributed load the flexural differential equation of an Euler-Bernoulli beam is [18]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 119902 (119911) = 0 (20)

where 119902(119911) is the uniform load of the soil around the pile Tosolve this equation 120582 = (1198704119864119868)

14 can be substituted intoformula (19) to get the following

1198894

119906

1198891199114+ 4120582119906 (119911) =

119902

119864119868 (21)

By substituting the boundary conditions into the flex-ural differential equation (18) the analytical solution can

Steel bar welded to

The type of bracket

2

1

120∘

3Φ28

Φ8 3

empty150

steel pipe

Diameter 50mm-pipe

B4000

Figure 7 Micropiles sectional drawing

Figure 8 The field tests of micropile

be obtained The general solution 119902(119911) = 0 and specialconnection is 119902(119911) that is

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)] +

119902

119870

(22)

Mathematical Problems in Engineering 7

Figure 9 The horizontal static load test of micropiles

323 The Determination of the Elastic Modulus (119864) In theprevious calculation 119864 is a crucial parameter usually deter-mined by laboratory experiment and experimental relation-ship of previous practice However as laboratory experimentit is very difficult to reproduce the soil pile in live loadeffect on the stress path and experiential relationship cannotdetermine the value of 119864 either as it can be influencedobviously by human interference Due to the above reason inthis paper the field test of single pile at the scene of the elasticdeformation range was appliedThe elastic model value is theinversion analysis through the stress-strain curve with thebasic process described as follows

Under the load themicropile is loaded step by step usingthe same load increment Record each time loaddisplacementdeformation therefore according to the stress-strain curveobtained from test to calculate the value of elastic modulus

4 Engineering Practice Analysis andVerification of Pile-Soil Mechanics Model

41 Engineering Examples In this example the landslide ofa highway is medium type about 80ndash100m wild and 160mlong The front is about 5m thick whereas the central partis 10ndash15m thick and the back part is 35ndash80m thick Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160000m3 The landslidedemonstrates typical factors with the slope facing the emptyand the steep slope The features of landslide appearance areobvious the back has tensile cracks and the two sides havepinnate cracks The front is flanked by a shear seam with aballooning extrusion crack forming and radioactive cracksSpecific to the characteristics of this landslide it is proposedto adopt a light retaining structure with a ministeel tub pilescomposite structure for reinforcement

42 Establishment of Experiment Process andNumerical Simu-lation According to the design requirements we performedon-site horizontal static load tests within the landslide rein-forcement range We then simulated an action mechanismand stress distribution in the micropile composite structureby adopting a jack to provide load Through earth pressurecells that were installed before and after the micropile wewere able to monitor the earth pressure variation and thelandslide thrust while checking the status of each row of

piles The stress of the miniature piles is measured througha reinforcement meter that was welded on a miniature steelpipe The pile body deformation was indirectly measuredthrough an inclinometer that was installed on the pile sidesFigure 6 shows the model demonstration diagram

421 Experiment Process The in situ test uses the micropileto determine the rate of reinforcement The grouting coagu-lation of the soil strength grade is C25 The micropile lengthindicates the landslide segment which is 8m long The pilediameter is 150mmand the tube diameter is 50mmThemainreinforcement pile contains 3 roots made up of 28 reinforcedsteel pipesThemicrocap sets the C30 concrete capping beamwith a beam that is 05m high and 15m wide In accordancewith the requirements for the load test the test is conductedwith a grade 11 effective load using two gauges to record thedata The average is taken as the final result On the 12thlevel (96 t) load the counterforce device becomes damagedindicating the end of the test The load of the destruction isthe horizontal limit load At themoment when themaximumamount of bending occurs the steel of the yielding tensilezone is the corresponding loadThemicropile section and testprocess are shown in Figures 7 8 and 9

422 Result Analysis The rate of pile displacement of eachgrade of load was obtained through on-site experimentsand analysis as shown in Figures 10 and 11 In Figures 10and 11 it can be detected that under horizontal load thedisplacement of the micropile composite structure above thesliding surface (3m) is more obvious than that below thesliding surface which indicates that the micropile compositestructure presents a tendency to lean forward Since the pileonly leaned 2m forward at the base below the sliding surfacethe horizontal displacement is basically 0 which indicatesthat the anchorage effects at the anchorage section are com-paratively betterWhen consistently increasing the horizontalload the variable quantities of the displacement of pile top ofeach row of piles are the sameThis consistency in the variablequantities is caused by the lid cap contracting to make themicropile composite structure act as a whole for antislidingby adopting a jack to provide step loadThe bending momentand the displacement distribution laws of the three-row pilesare similar A bendingmoment above 025m is 0 No bendingdeflection of the micropile is generated due to the constraint

8 Mathematical Problems in Engineering

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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Mathematical Problems in Engineering

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Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

a nonlinear elastic-plastic subgrade reaction method (calledthe 119901-119910 curve method) However there was less research onmicropile structures under horizontal load At present mostof the previously discussed engineering designs of micropilesadopt a calculation approach of specific to normal antislidepiles However the forcemodel and design calculation theoryof micropile composite structure have not been perfectedyet Establishing a calculation mode suitable for micropilecomposite structure and proposing a reasonable calculationmethod are urgently neededThus this paper further exploresresearch that capitalizes on this missed opportunity

This paper discusses an analytical solution to stress anddisplacement distribution under horizontal load based onthe mechanics theory of two-dimensional elastic complexfunctions Using the Winkler Foundation Beam Model thispaper assumes that the soil around a micropile stems froma series of independent springs The rigidity coefficient of aspring can be obtained using the planar solution After thatbased on the deflection curve of the differential equation ofan Euler-Bernoulli beam the pile deformation and internalforce calculation methods of a micropile composite structureunder horizontal load can be derived using two modesnamely by fixing one end with the other end sliding aswell as fixing both ends In the end the paper suggestsreinforcing highway landslides using micropile compositestructures and conducting on-site pile pushing tests Theresults obtained from the experiment have been compared tothe theoretical approach to verify the accuracy and reliabilityof the theoretical approach

2 Establishment of the Plane Strain SolutionModel of Micropiles

21 Description of the Problem Then researching the effectsof horizontal load on the micropiles it is important toconsider the internal force and deformation analysis of rigiddisc hole structure around the semi-infinite space surface atthe 119885 plane (rectangular coordinate system) As shown inFigure 1 the119877 region is a region of semi-infinite space exceptfor the disc-structure and 119886 is the distance between the soilaround the pile and the center of the pile 119903 is the radius ofthe disc-structure and 119865 is the horizontal load applied inthe center of the pile (119865

119909is 119909 component of direction 119865

119910

is 119910 component of direction) It is assumed that 1199060is the

displacement under horizontal load applied in the center ofthe pile

22 Fundamental Assumption of the Mechanical ModelWhile obtaining the stress field of the soil around the pile andthe plane strain solution of the displacement field distributionby adopting the complex variable functions of plane elasticmechanics the following assumptions are made

(1) While establishing a strainmodel of two-dimensionalcomplex elastic mechanics since the rigidity of themicrosteel pile is bigger than that around the pileassume thatminimicropiles and cross section aroundthe piles are rigid disc hole structures

x

y

F

Surrounding soil

Figure 1 Mechanical model of micropile-soil

(2) Assume that the deformation of the soil around thepile presents a tendency of elastic variation underhorizontal load

(3) Compare the deformation of the micropile and thesoil around the pile under horizontal load

23 Basic Control Equation According to the complex vari-able function of plane elasticmechanics the complex analyticfunction of stress and displacement components at 119877 regionare to be expressed in 120593

1(119909) and 120595

1(119909) then [14]

120590119909+ 120590119910= 4Re [1205931015840

1(119911)]

120590119910minus 120590119909+ 2119894120591119909119910

= 2 [11991112059310158401015840

1(119911) + 120595

1015840

1(119911)]

2119866 (119906 + 119894V) = 120581120593 (120585) minus 12059310158401(120585)

120596 (120585)

1205961015840 (120585)minus 120595 (120585)

(1)

where 119911 is any point on the micropile hole 120590119909 120590119910 and 120591

119909119910are

the stress and strain component of any point Re separatesthe real part and imaginary part in 120581 = 3 minus 4120583 Planestrain (3minus120583)(1+120583) Plane stress and 119866 = 1198642(1+120583) where119864 is modulus of elasticity and 120583 is the Poisson ratio

The analytic function and expression 1205931(119909) and 120595

1(119909)

under stress at infinity bounded conditions are as follows

1205931(119911) = minus

1

2120587 (1 + 120581)(119865119909+ 119894119865119910) ln 119911 + (119861 + 119894119862) 119911

+ 1205930

1(119911)

1205951(119911) =

1

2120587 (1 + 120581)(119865119909minus 119894119865119910) ln 119911 + (119861

1015840

+ 1198941198621015840

) 119911

+ 1205950

1(119911)

(2)

where 119861 1198611015840 and 119862

1015840 respectively represent the depth ofthe tunnel the bulk density of the surrounding rock andthe relationship of the lateral pressure coefficient and thesurrounding rock stress In this equation 120581 is positive integer

Mathematical Problems in Engineering 3

1205930

1(119911) and 120595

0

1(119911) are the analytic function of the points in the

neighborhood located at infinity Where 119861 = (120590infin

119909+ 120590infin

119910)4

1198611015840

= (120590infin

119910minus 120590infin

119909)2 1198621015840 = 120591

infin

119909119910 120590infin119909 120590infin119910 and 120591

infin

119909119910is the stress

located at infinityAccording to the analytic function under complex vari-

ables functions at bounded conditions the displacement andstress boundary conditions are as follows

The condition of stress boundary 119894 int119861

119860

(119865119909+ 119894119865119910) 119889119904

= 1205931(119911) + 1199111205931015840

1(119911) + 120595

1(119911)

The condition of displacement boundary 1205811205931(119911)

minus 11991112059310158401(119911) minus 120595

1(119911) = 2119866 (119906 + 119894V)

(3)

24 ConformalMapping Onemustmap themicropile orificeto unit annulus The semi-infinite space orifice within 119885

plane can be converted to within the plane 120585 after conformalmapping as shown in Figure 2Then themapping function ofzone 119877 after conformal mapping is shown in [15] Therefore

119911 = 120596 (120585) = minus1198941198861 minus 1205722

1 + 1205722

1 minus 120585

1 + 120585 (4)

where any point 120585 located in the 120585 plane is expressed as polarcoordinates that is 120585 = 120588119890

119894120579 and 120572 is a parameter determinedby 119886 and 119903

119903

119886=

2119886

1 + 1198862 (5)

After transforming the function at plane 119911 to the functionof 120585 the stress and displacement component will change to[16]

120590120588+ 120590120579= 120590119909+ 120590119910= 4Re [Φ (120585)]

120590119909minus 120590119910+ 2119894120591119909119910

=21205852

12058821205961015840 (120585)[120596 (120585)Φ

1015840

(120585) + 1205961015840

(120585) Ψ (119911)]

2119866 (119906 + 119894V) = 120581120593 (120585) minus 12059310158401(120585)

120596 (120585)

1205961015840 (120585)minus 120595 (120585)

(6)

where 119911 = 120596(120585) Φ(120585) = 1205931015840

1(119911) and Ψ(120585) = 120595

1015840

1(119911) From

formula (6) it is detected that in order to get the stressand displacement values through using the fundamentalequation tone has to solve the complex function and obtainthe solution to 120593

0(120585) and 120595

0(120585)

25 Solution to Displacement 1199060under Horizontal Load (1)

According to the plane displacement boundary conditions ofmicropile-soil structure the following can be seen from theassumed conditions

The displacement boundary conditions can be dividedas follows infinity to pile core position namely 119911 = 119911 so

120578

1205790

120588 =1

120585

R-area

Figure 2 Plane of conformal transformation

the soilmass will not cause direct impact on pile deformationwhereas displacement exists at the contact surface of the pilehole boundary and soil mass namely |119911 + 119894119886| = 119903 The strainof themicropiles soil structure to plane 120581 = 3minus4120583 can be seenin formula (3) In other words the displacement boundary isas follows

[(3 minus 4120583) 120593 (120585) minus 120596 (120585)1205931015840 (120585)

1205961015840 (120585)minus 120595 (120585)]

1199040

=119864

1 + 120583(119906 + 119894V)

[(3 minus 4120583) 120593 (120585) minus 120596 (120585)1205931015840 (120585)

1205961015840 (120585)minus 120595 (120585)]

1199041

= 0

(7)

where 1199041is distance between the soil around the pile and the

center of the pile and 1199040is the boundary curve of the pile hole

When the function of the 119911 plane is transformed into afunction of the 120585 plane 119911 = 120596(120585) Φ(120585) = 120593

1015840

1(119911) and Ψ(120585) =

1205951015840

1(119911) are substituted into formula (2) to get the following

120593 (120585) = minus1

8120587 (1 minus 120583)(sum119865119909+ 119894sum119865

119910) ln120596 (120585)

+ (119861 + 119894119862) 120596 (120585) + 1205930(120585)

120595 (120585) =1

8120587 (1 minus 120583)(sum119865119909minus 119894sum119865

119910) ln120596 (120585)

+ (1198611015840

+ 1198941198621015840

) 120596 (120585) + 1205950(120585)

(8)

where 1205930(120585) = sum

119899

119896=0119886119896120585minus119896 and 120595

0(120585) = sum

119899

119896=0119887119896120585minus119896 From the

given situation one can obtain sum119865119909

= 119865 sum119865119910

= 0 and 119861 =

119862 = 1198621015840

= 1198611015840

= 0 When the initial conditions are substitutedinto formula (8) one can get the following

120593 (120585) = minus1

8120587 (1 minus 120583)119865 ln120596 (120585) + 120593

0(120585)

120595 (120585) =1

8120587 (1 minus 120583)119865 ln120596 (120585) + 120595

0(120585)

(9)

4 Mathematical Problems in Engineering

By substituting formula (9) into the displacement bound-ary condition formula (7) it can be derived that

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

1

(10)

where

119891 =119864

1 + 1205831199060minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

1198911= minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

(11)

Through the simultaneous application of the boundaryconditions of the two equations found in formula (10) theexpression of 120593

0(120585) and 120595

0(120585) can be derived (including the

unknown displacement of 1199060)

(2)Considering the micropile hole boundary namely forthe boundary conditions 119904

0 120590 = 119890

119894120579 the stress boundaryconditions are to be expressed with components under theorthogonal curvilinear coordinate system at the 119911 planeNamely

[(3 minus 4120583) 1205930(120590) minus

120596 (120590)

1205961015840 (120590)12059310158400(120590) minus 120595

0(120590)] = 119891

0 (12)

The Cauchy integral operator (12120587119894) ∮(119889120590(120590 minus 120585)) atboth ends of the formula above can be obtained

(3 minus 4120583)

2120587119894∮

1205930(120590) 119889120590

120590 minus 120585minus

1

2120587119894∮

120596 (120590)

1205961015840 (120590)

12059310158400(120590)

120590 minus 120585119889120590

minus1

2120587119894∮

1205950(120590)

120590 minus 120585119889120590 =

1

2120587119894∮

1198910

120590 minus 120585119889120590

(13)

In formula (13) ∮(1205930(120590)119889120590(120590 minus 120585)) = minus120593

0(120585)

(12120587119894) ∮(1205950(120590)(120590 minus 120585))119889120590 = 0 so by substituting into

formula (13) one can find 1205930(120585)

Similarly according to formula (12) the value of 1205950(120585)

can be obtained by taking conjugation at both sides and thenapplying the Cauchy integral operator Thus the informationcan be simultaneously obtained by combining formula (10)to get the value of 119906

0

26 Solution to 1205930(120585) and 120595

0(120585) To obtain the boundary

conditions of a miniature pile make 120585 = 120590120588 and unfold 120593(120585)

and 120595(120585) in a Laurent series form namely [17]

120593 (120585) = 1198860+

119899

sum

119896=1

119886119896120585119896

+

119899

sum

119896=1

119887119896120585minus119896

120595 (120585) = 1198880+

119899

sum

119896=1

119888119896120585119896

+

119899

sum

119896=1

119889119896120585minus119896

(14)

Depending on formula (4) one can get the following

120596 (120590)

1205961015840 (120590)= minus

1

2

(1 + 120588120590) (120590 minus 120588)2

1205902 (1 minus 120588120590) (15)

Using formulas (14) and (15) substituted into boundaryconditions 119904

1of formula (7) one can get the following

1198880= minus1198860minus

1

21198861minus

1

21198871

119888119896= minus119887119896+

1

2(119896 minus 1) 119886

119896minus1minus

1

2(119896 + 1) 119886

119896+1

119889119896= minus119886119896+

1

2(119896 minus 1) 119887

119896minus1minus

1

2(119896 + 1) 119887

119896+1

(16)

Make 119891(120585) = 119891(120572120590) = 2119866(119906 + 119894V) and 119891lowast

(120572120590) = (1 minus

120572120590)119891(120572120590) = suminfin

119896119860119896120590119896 Formulas (14) (15) and (17)ndash(19) can

be substituted into the boundary conditions 1199041of formula (7)

by eliminating 119888119896and 119889

119896 one finds 119886

119896and 119887119896

(1 minus 1205722

) (119896 + 1) 119886119896+1

minus (1205722

+ 120581120572minus2119896

) 119887119896+1

= (1 minus 1205722

) 119896119886119896minus (1 + 120581120572

minus2119896

) 119887119896+ 119860minus119896

120572119896

(1 + 1205811205722119896+2

) 119886119896+1

+ (1 minus 1205722

) (119896 + 1) 119887119896+1

= 1205722

(1205722

+ 1205811205722119896

) 119886119896+ (1 minus 120572

2

) 119896119887119896+ 1198601+119896

120572119896+1

(1 minus 1205722

) 1198861minus (120581 + 120572

2

) 1198871= 1198600minus (120581 + 1) 119886

0

(1 + 1205811205722

) 1198861+ (1 minus 120572

2

) 1198871= 1198601120572 + 1205722

(120581 + 1) 1198860

(17)

We simultaneously solved the four formulas in (17) toget all of the coefficients except for 119886

0 since 119886

0represents

rigid body displacement thus no stress will be generatedand it can be deemed as 0 By 119896 times of iteration one canobtain 119886

119896 thus to get 120593

0(120585) and 120595

0(120585) one must substitute

1205930(120585) and 120595

0(120585) in the complex function of the fundamental

equation to ascertain the stress and displacement field ofthe soil around a pile of micropiles under horizontal loadAccording to the assumed conditions specified in Sections21 and 22 the relationship between horizontal load and thehorizontal displacement can be obtained from 119870 = 119865119906

0

3 Establishment of Pile-Soil Mechanics ModelBased on Winkler Foundation Beam

In most cases while ministeel tub piles under horizontalload are applied in landslide control and slope reinforcement

Mathematical Problems in Engineering 5

Micropile

Sloping surface

Roof beam

Landslid

e surfa

ce

Figure 3 Micropile composite structure set on step of slope

a pile groupsrsquo layout will be adopted Especially for multistageslopes platforms are available for each grade Miniature pilescan be set at the platform for reinforcement The multipleminiature piles that are exposed on the platform can be fixedwith a top beam to enhance their sliding resistance forceThemicropile layout is shown in Figure 3

31 Model Assumption While miniature piles are appliedfor landslide reinforcement the mechanism can be used toaddress bigger shearing resistance in miniature pile land-slides Pile-soil structures will resist landslide thrust formedbymicropiles as well as the soil mass For internal calculationtheWinkler low econometricmodel is adoptedThis researchassumes that the soil around the micropile has a series ofdiscrete springs and a rigidity coefficient of 119870 = 119865119906

0 As

shown in Figure 3 since the micropile is fixed at the bed rockbelow the sliding surface it is assumed that the sliding surfaceis as fixed constraint Due to the fact that themicropile is fixedand connected through a top beam as compared tominiaturepile the top beam can be deemed as a rigid member Underthe effects of horizontal load the top beam only experienceshorizontal displacement so the displacement at each pile capwill be the same [18] Thus the internal force calculation ofthe model is shown in Figure 4

32 Internal Force Calculation According to the fundamen-tal theory of elastic mechanics the stress and calculationmodel of a single micropile can be obtained as shown inFigure 5 It is composed of two calculation models namelywith one fixed and one sliding end under a concentrated load(Figure 5(a)) and with both ends fixed under a uniformlydistributed load (Figure 5(b)) Specific to the calculationmodel found in Figure 4 for three-row micropile structurethe internal force of the micropile AD is obtained throughsuperposition of Figures 5(a) and 5(b) The internal forces ofmicropile BE and CF are derived from Figure 5(a)

321 Solution to Calculation Model with One Fixed andOne Sliding End under a Concentrated Load According tothe internal force calculation model specified in Figure 5(a)when considering micropile structures with one fixed endand one sliding end under a concentrated load the boundary

q

F

f f

A B C

D E F

K K K H

Figure 4 Mechanical calculation model of micropile structure

z

x

H

F

(a)

q

z

x

H

(b)

Figure 5 Mechanical calculation model of single micropile

condition on the top of the micropile is 119911 = 0 the bendingmoment is 119872 = 0 and 119865 is horizontal shear The boundarycondition on the bottom of the micropile is 119911 = 119867 the angleis 120593 = 0 and the horizontal displacement is 119906 = 0 Theflexural differential equation of an Euler-Bernoulli beamwithone fixed and one sliding end is [19]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 0 (18)

In the formula above 119864 is the elasticity modulus of theminiature pile 119868 is the inertia moment of a micropile crosssection119870 is the rigidity coefficient of an assumed spring and119906(119911) is the horizontal displacement of the soil around the pilealong the pile body

6 Mathematical Problems in Engineering

Roof beam

GroundSteel plate

Jack

Counterforce deviceRetaining wall

Earth pressure boxSteel bar

Micropile structure

Reac

tion

wal

l

Earth pressure cells

Reinforced stress meter

Inclinometer tube

Thrust 1 2 34 5 6

meter

05m

Figure 6 The schematic drawing of a test model of micropiles composite structure

By substituting the boundary conditions in the flexuraldifferential equation (18) the analytical solution can beobtained

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)]

119872 = 21205822

119864119868119890120582119911

[1198882cos (120582119911) minus 119888

1sin (120582119911)]

+ 21205822

119864119868119890minus120582119911

[1198884cos (120582119911) minus 119888

3sin (120582119911)]

(19)

where 120582 = (1198704119864119868)14 and 119888

1 1198882 1198883 and 119888

4are the integral

constant

322 Solution to Calculation Model with Both Ends Fixedunder a Uniformly Distributed Load According to the inter-nal force calculation model specified in Figure 5(b) formicropile structures with both ends fixed under a uniformlydistributed load the flexural differential equation of an Euler-Bernoulli beam is [18]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 119902 (119911) = 0 (20)

where 119902(119911) is the uniform load of the soil around the pile Tosolve this equation 120582 = (1198704119864119868)

14 can be substituted intoformula (19) to get the following

1198894

119906

1198891199114+ 4120582119906 (119911) =

119902

119864119868 (21)

By substituting the boundary conditions into the flex-ural differential equation (18) the analytical solution can

Steel bar welded to

The type of bracket

2

1

120∘

3Φ28

Φ8 3

empty150

steel pipe

Diameter 50mm-pipe

B4000

Figure 7 Micropiles sectional drawing

Figure 8 The field tests of micropile

be obtained The general solution 119902(119911) = 0 and specialconnection is 119902(119911) that is

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)] +

119902

119870

(22)

Mathematical Problems in Engineering 7

Figure 9 The horizontal static load test of micropiles

323 The Determination of the Elastic Modulus (119864) In theprevious calculation 119864 is a crucial parameter usually deter-mined by laboratory experiment and experimental relation-ship of previous practice However as laboratory experimentit is very difficult to reproduce the soil pile in live loadeffect on the stress path and experiential relationship cannotdetermine the value of 119864 either as it can be influencedobviously by human interference Due to the above reason inthis paper the field test of single pile at the scene of the elasticdeformation range was appliedThe elastic model value is theinversion analysis through the stress-strain curve with thebasic process described as follows

Under the load themicropile is loaded step by step usingthe same load increment Record each time loaddisplacementdeformation therefore according to the stress-strain curveobtained from test to calculate the value of elastic modulus

4 Engineering Practice Analysis andVerification of Pile-Soil Mechanics Model

41 Engineering Examples In this example the landslide ofa highway is medium type about 80ndash100m wild and 160mlong The front is about 5m thick whereas the central partis 10ndash15m thick and the back part is 35ndash80m thick Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160000m3 The landslidedemonstrates typical factors with the slope facing the emptyand the steep slope The features of landslide appearance areobvious the back has tensile cracks and the two sides havepinnate cracks The front is flanked by a shear seam with aballooning extrusion crack forming and radioactive cracksSpecific to the characteristics of this landslide it is proposedto adopt a light retaining structure with a ministeel tub pilescomposite structure for reinforcement

42 Establishment of Experiment Process andNumerical Simu-lation According to the design requirements we performedon-site horizontal static load tests within the landslide rein-forcement range We then simulated an action mechanismand stress distribution in the micropile composite structureby adopting a jack to provide load Through earth pressurecells that were installed before and after the micropile wewere able to monitor the earth pressure variation and thelandslide thrust while checking the status of each row of

piles The stress of the miniature piles is measured througha reinforcement meter that was welded on a miniature steelpipe The pile body deformation was indirectly measuredthrough an inclinometer that was installed on the pile sidesFigure 6 shows the model demonstration diagram

421 Experiment Process The in situ test uses the micropileto determine the rate of reinforcement The grouting coagu-lation of the soil strength grade is C25 The micropile lengthindicates the landslide segment which is 8m long The pilediameter is 150mmand the tube diameter is 50mmThemainreinforcement pile contains 3 roots made up of 28 reinforcedsteel pipesThemicrocap sets the C30 concrete capping beamwith a beam that is 05m high and 15m wide In accordancewith the requirements for the load test the test is conductedwith a grade 11 effective load using two gauges to record thedata The average is taken as the final result On the 12thlevel (96 t) load the counterforce device becomes damagedindicating the end of the test The load of the destruction isthe horizontal limit load At themoment when themaximumamount of bending occurs the steel of the yielding tensilezone is the corresponding loadThemicropile section and testprocess are shown in Figures 7 8 and 9

422 Result Analysis The rate of pile displacement of eachgrade of load was obtained through on-site experimentsand analysis as shown in Figures 10 and 11 In Figures 10and 11 it can be detected that under horizontal load thedisplacement of the micropile composite structure above thesliding surface (3m) is more obvious than that below thesliding surface which indicates that the micropile compositestructure presents a tendency to lean forward Since the pileonly leaned 2m forward at the base below the sliding surfacethe horizontal displacement is basically 0 which indicatesthat the anchorage effects at the anchorage section are com-paratively betterWhen consistently increasing the horizontalload the variable quantities of the displacement of pile top ofeach row of piles are the sameThis consistency in the variablequantities is caused by the lid cap contracting to make themicropile composite structure act as a whole for antislidingby adopting a jack to provide step loadThe bending momentand the displacement distribution laws of the three-row pilesare similar A bendingmoment above 025m is 0 No bendingdeflection of the micropile is generated due to the constraint

8 Mathematical Problems in Engineering

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

1205930

1(119911) and 120595

0

1(119911) are the analytic function of the points in the

neighborhood located at infinity Where 119861 = (120590infin

119909+ 120590infin

119910)4

1198611015840

= (120590infin

119910minus 120590infin

119909)2 1198621015840 = 120591

infin

119909119910 120590infin119909 120590infin119910 and 120591

infin

119909119910is the stress

located at infinityAccording to the analytic function under complex vari-

ables functions at bounded conditions the displacement andstress boundary conditions are as follows

The condition of stress boundary 119894 int119861

119860

(119865119909+ 119894119865119910) 119889119904

= 1205931(119911) + 1199111205931015840

1(119911) + 120595

1(119911)

The condition of displacement boundary 1205811205931(119911)

minus 11991112059310158401(119911) minus 120595

1(119911) = 2119866 (119906 + 119894V)

(3)

24 ConformalMapping Onemustmap themicropile orificeto unit annulus The semi-infinite space orifice within 119885

plane can be converted to within the plane 120585 after conformalmapping as shown in Figure 2Then themapping function ofzone 119877 after conformal mapping is shown in [15] Therefore

119911 = 120596 (120585) = minus1198941198861 minus 1205722

1 + 1205722

1 minus 120585

1 + 120585 (4)

where any point 120585 located in the 120585 plane is expressed as polarcoordinates that is 120585 = 120588119890

119894120579 and 120572 is a parameter determinedby 119886 and 119903

119903

119886=

2119886

1 + 1198862 (5)

After transforming the function at plane 119911 to the functionof 120585 the stress and displacement component will change to[16]

120590120588+ 120590120579= 120590119909+ 120590119910= 4Re [Φ (120585)]

120590119909minus 120590119910+ 2119894120591119909119910

=21205852

12058821205961015840 (120585)[120596 (120585)Φ

1015840

(120585) + 1205961015840

(120585) Ψ (119911)]

2119866 (119906 + 119894V) = 120581120593 (120585) minus 12059310158401(120585)

120596 (120585)

1205961015840 (120585)minus 120595 (120585)

(6)

where 119911 = 120596(120585) Φ(120585) = 1205931015840

1(119911) and Ψ(120585) = 120595

1015840

1(119911) From

formula (6) it is detected that in order to get the stressand displacement values through using the fundamentalequation tone has to solve the complex function and obtainthe solution to 120593

0(120585) and 120595

0(120585)

25 Solution to Displacement 1199060under Horizontal Load (1)

According to the plane displacement boundary conditions ofmicropile-soil structure the following can be seen from theassumed conditions

The displacement boundary conditions can be dividedas follows infinity to pile core position namely 119911 = 119911 so

120578

1205790

120588 =1

120585

R-area

Figure 2 Plane of conformal transformation

the soilmass will not cause direct impact on pile deformationwhereas displacement exists at the contact surface of the pilehole boundary and soil mass namely |119911 + 119894119886| = 119903 The strainof themicropiles soil structure to plane 120581 = 3minus4120583 can be seenin formula (3) In other words the displacement boundary isas follows

[(3 minus 4120583) 120593 (120585) minus 120596 (120585)1205931015840 (120585)

1205961015840 (120585)minus 120595 (120585)]

1199040

=119864

1 + 120583(119906 + 119894V)

[(3 minus 4120583) 120593 (120585) minus 120596 (120585)1205931015840 (120585)

1205961015840 (120585)minus 120595 (120585)]

1199041

= 0

(7)

where 1199041is distance between the soil around the pile and the

center of the pile and 1199040is the boundary curve of the pile hole

When the function of the 119911 plane is transformed into afunction of the 120585 plane 119911 = 120596(120585) Φ(120585) = 120593

1015840

1(119911) and Ψ(120585) =

1205951015840

1(119911) are substituted into formula (2) to get the following

120593 (120585) = minus1

8120587 (1 minus 120583)(sum119865119909+ 119894sum119865

119910) ln120596 (120585)

+ (119861 + 119894119862) 120596 (120585) + 1205930(120585)

120595 (120585) =1

8120587 (1 minus 120583)(sum119865119909minus 119894sum119865

119910) ln120596 (120585)

+ (1198611015840

+ 1198941198621015840

) 120596 (120585) + 1205950(120585)

(8)

where 1205930(120585) = sum

119899

119896=0119886119896120585minus119896 and 120595

0(120585) = sum

119899

119896=0119887119896120585minus119896 From the

given situation one can obtain sum119865119909

= 119865 sum119865119910

= 0 and 119861 =

119862 = 1198621015840

= 1198611015840

= 0 When the initial conditions are substitutedinto formula (8) one can get the following

120593 (120585) = minus1

8120587 (1 minus 120583)119865 ln120596 (120585) + 120593

0(120585)

120595 (120585) =1

8120587 (1 minus 120583)119865 ln120596 (120585) + 120595

0(120585)

(9)

4 Mathematical Problems in Engineering

By substituting formula (9) into the displacement bound-ary condition formula (7) it can be derived that

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

1

(10)

where

119891 =119864

1 + 1205831199060minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

1198911= minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

(11)

Through the simultaneous application of the boundaryconditions of the two equations found in formula (10) theexpression of 120593

0(120585) and 120595

0(120585) can be derived (including the

unknown displacement of 1199060)

(2)Considering the micropile hole boundary namely forthe boundary conditions 119904

0 120590 = 119890

119894120579 the stress boundaryconditions are to be expressed with components under theorthogonal curvilinear coordinate system at the 119911 planeNamely

[(3 minus 4120583) 1205930(120590) minus

120596 (120590)

1205961015840 (120590)12059310158400(120590) minus 120595

0(120590)] = 119891

0 (12)

The Cauchy integral operator (12120587119894) ∮(119889120590(120590 minus 120585)) atboth ends of the formula above can be obtained

(3 minus 4120583)

2120587119894∮

1205930(120590) 119889120590

120590 minus 120585minus

1

2120587119894∮

120596 (120590)

1205961015840 (120590)

12059310158400(120590)

120590 minus 120585119889120590

minus1

2120587119894∮

1205950(120590)

120590 minus 120585119889120590 =

1

2120587119894∮

1198910

120590 minus 120585119889120590

(13)

In formula (13) ∮(1205930(120590)119889120590(120590 minus 120585)) = minus120593

0(120585)

(12120587119894) ∮(1205950(120590)(120590 minus 120585))119889120590 = 0 so by substituting into

formula (13) one can find 1205930(120585)

Similarly according to formula (12) the value of 1205950(120585)

can be obtained by taking conjugation at both sides and thenapplying the Cauchy integral operator Thus the informationcan be simultaneously obtained by combining formula (10)to get the value of 119906

0

26 Solution to 1205930(120585) and 120595

0(120585) To obtain the boundary

conditions of a miniature pile make 120585 = 120590120588 and unfold 120593(120585)

and 120595(120585) in a Laurent series form namely [17]

120593 (120585) = 1198860+

119899

sum

119896=1

119886119896120585119896

+

119899

sum

119896=1

119887119896120585minus119896

120595 (120585) = 1198880+

119899

sum

119896=1

119888119896120585119896

+

119899

sum

119896=1

119889119896120585minus119896

(14)

Depending on formula (4) one can get the following

120596 (120590)

1205961015840 (120590)= minus

1

2

(1 + 120588120590) (120590 minus 120588)2

1205902 (1 minus 120588120590) (15)

Using formulas (14) and (15) substituted into boundaryconditions 119904

1of formula (7) one can get the following

1198880= minus1198860minus

1

21198861minus

1

21198871

119888119896= minus119887119896+

1

2(119896 minus 1) 119886

119896minus1minus

1

2(119896 + 1) 119886

119896+1

119889119896= minus119886119896+

1

2(119896 minus 1) 119887

119896minus1minus

1

2(119896 + 1) 119887

119896+1

(16)

Make 119891(120585) = 119891(120572120590) = 2119866(119906 + 119894V) and 119891lowast

(120572120590) = (1 minus

120572120590)119891(120572120590) = suminfin

119896119860119896120590119896 Formulas (14) (15) and (17)ndash(19) can

be substituted into the boundary conditions 1199041of formula (7)

by eliminating 119888119896and 119889

119896 one finds 119886

119896and 119887119896

(1 minus 1205722

) (119896 + 1) 119886119896+1

minus (1205722

+ 120581120572minus2119896

) 119887119896+1

= (1 minus 1205722

) 119896119886119896minus (1 + 120581120572

minus2119896

) 119887119896+ 119860minus119896

120572119896

(1 + 1205811205722119896+2

) 119886119896+1

+ (1 minus 1205722

) (119896 + 1) 119887119896+1

= 1205722

(1205722

+ 1205811205722119896

) 119886119896+ (1 minus 120572

2

) 119896119887119896+ 1198601+119896

120572119896+1

(1 minus 1205722

) 1198861minus (120581 + 120572

2

) 1198871= 1198600minus (120581 + 1) 119886

0

(1 + 1205811205722

) 1198861+ (1 minus 120572

2

) 1198871= 1198601120572 + 1205722

(120581 + 1) 1198860

(17)

We simultaneously solved the four formulas in (17) toget all of the coefficients except for 119886

0 since 119886

0represents

rigid body displacement thus no stress will be generatedand it can be deemed as 0 By 119896 times of iteration one canobtain 119886

119896 thus to get 120593

0(120585) and 120595

0(120585) one must substitute

1205930(120585) and 120595

0(120585) in the complex function of the fundamental

equation to ascertain the stress and displacement field ofthe soil around a pile of micropiles under horizontal loadAccording to the assumed conditions specified in Sections21 and 22 the relationship between horizontal load and thehorizontal displacement can be obtained from 119870 = 119865119906

0

3 Establishment of Pile-Soil Mechanics ModelBased on Winkler Foundation Beam

In most cases while ministeel tub piles under horizontalload are applied in landslide control and slope reinforcement

Mathematical Problems in Engineering 5

Micropile

Sloping surface

Roof beam

Landslid

e surfa

ce

Figure 3 Micropile composite structure set on step of slope

a pile groupsrsquo layout will be adopted Especially for multistageslopes platforms are available for each grade Miniature pilescan be set at the platform for reinforcement The multipleminiature piles that are exposed on the platform can be fixedwith a top beam to enhance their sliding resistance forceThemicropile layout is shown in Figure 3

31 Model Assumption While miniature piles are appliedfor landslide reinforcement the mechanism can be used toaddress bigger shearing resistance in miniature pile land-slides Pile-soil structures will resist landslide thrust formedbymicropiles as well as the soil mass For internal calculationtheWinkler low econometricmodel is adoptedThis researchassumes that the soil around the micropile has a series ofdiscrete springs and a rigidity coefficient of 119870 = 119865119906

0 As

shown in Figure 3 since the micropile is fixed at the bed rockbelow the sliding surface it is assumed that the sliding surfaceis as fixed constraint Due to the fact that themicropile is fixedand connected through a top beam as compared tominiaturepile the top beam can be deemed as a rigid member Underthe effects of horizontal load the top beam only experienceshorizontal displacement so the displacement at each pile capwill be the same [18] Thus the internal force calculation ofthe model is shown in Figure 4

32 Internal Force Calculation According to the fundamen-tal theory of elastic mechanics the stress and calculationmodel of a single micropile can be obtained as shown inFigure 5 It is composed of two calculation models namelywith one fixed and one sliding end under a concentrated load(Figure 5(a)) and with both ends fixed under a uniformlydistributed load (Figure 5(b)) Specific to the calculationmodel found in Figure 4 for three-row micropile structurethe internal force of the micropile AD is obtained throughsuperposition of Figures 5(a) and 5(b) The internal forces ofmicropile BE and CF are derived from Figure 5(a)

321 Solution to Calculation Model with One Fixed andOne Sliding End under a Concentrated Load According tothe internal force calculation model specified in Figure 5(a)when considering micropile structures with one fixed endand one sliding end under a concentrated load the boundary

q

F

f f

A B C

D E F

K K K H

Figure 4 Mechanical calculation model of micropile structure

z

x

H

F

(a)

q

z

x

H

(b)

Figure 5 Mechanical calculation model of single micropile

condition on the top of the micropile is 119911 = 0 the bendingmoment is 119872 = 0 and 119865 is horizontal shear The boundarycondition on the bottom of the micropile is 119911 = 119867 the angleis 120593 = 0 and the horizontal displacement is 119906 = 0 Theflexural differential equation of an Euler-Bernoulli beamwithone fixed and one sliding end is [19]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 0 (18)

In the formula above 119864 is the elasticity modulus of theminiature pile 119868 is the inertia moment of a micropile crosssection119870 is the rigidity coefficient of an assumed spring and119906(119911) is the horizontal displacement of the soil around the pilealong the pile body

6 Mathematical Problems in Engineering

Roof beam

GroundSteel plate

Jack

Counterforce deviceRetaining wall

Earth pressure boxSteel bar

Micropile structure

Reac

tion

wal

l

Earth pressure cells

Reinforced stress meter

Inclinometer tube

Thrust 1 2 34 5 6

meter

05m

Figure 6 The schematic drawing of a test model of micropiles composite structure

By substituting the boundary conditions in the flexuraldifferential equation (18) the analytical solution can beobtained

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)]

119872 = 21205822

119864119868119890120582119911

[1198882cos (120582119911) minus 119888

1sin (120582119911)]

+ 21205822

119864119868119890minus120582119911

[1198884cos (120582119911) minus 119888

3sin (120582119911)]

(19)

where 120582 = (1198704119864119868)14 and 119888

1 1198882 1198883 and 119888

4are the integral

constant

322 Solution to Calculation Model with Both Ends Fixedunder a Uniformly Distributed Load According to the inter-nal force calculation model specified in Figure 5(b) formicropile structures with both ends fixed under a uniformlydistributed load the flexural differential equation of an Euler-Bernoulli beam is [18]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 119902 (119911) = 0 (20)

where 119902(119911) is the uniform load of the soil around the pile Tosolve this equation 120582 = (1198704119864119868)

14 can be substituted intoformula (19) to get the following

1198894

119906

1198891199114+ 4120582119906 (119911) =

119902

119864119868 (21)

By substituting the boundary conditions into the flex-ural differential equation (18) the analytical solution can

Steel bar welded to

The type of bracket

2

1

120∘

3Φ28

Φ8 3

empty150

steel pipe

Diameter 50mm-pipe

B4000

Figure 7 Micropiles sectional drawing

Figure 8 The field tests of micropile

be obtained The general solution 119902(119911) = 0 and specialconnection is 119902(119911) that is

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)] +

119902

119870

(22)

Mathematical Problems in Engineering 7

Figure 9 The horizontal static load test of micropiles

323 The Determination of the Elastic Modulus (119864) In theprevious calculation 119864 is a crucial parameter usually deter-mined by laboratory experiment and experimental relation-ship of previous practice However as laboratory experimentit is very difficult to reproduce the soil pile in live loadeffect on the stress path and experiential relationship cannotdetermine the value of 119864 either as it can be influencedobviously by human interference Due to the above reason inthis paper the field test of single pile at the scene of the elasticdeformation range was appliedThe elastic model value is theinversion analysis through the stress-strain curve with thebasic process described as follows

Under the load themicropile is loaded step by step usingthe same load increment Record each time loaddisplacementdeformation therefore according to the stress-strain curveobtained from test to calculate the value of elastic modulus

4 Engineering Practice Analysis andVerification of Pile-Soil Mechanics Model

41 Engineering Examples In this example the landslide ofa highway is medium type about 80ndash100m wild and 160mlong The front is about 5m thick whereas the central partis 10ndash15m thick and the back part is 35ndash80m thick Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160000m3 The landslidedemonstrates typical factors with the slope facing the emptyand the steep slope The features of landslide appearance areobvious the back has tensile cracks and the two sides havepinnate cracks The front is flanked by a shear seam with aballooning extrusion crack forming and radioactive cracksSpecific to the characteristics of this landslide it is proposedto adopt a light retaining structure with a ministeel tub pilescomposite structure for reinforcement

42 Establishment of Experiment Process andNumerical Simu-lation According to the design requirements we performedon-site horizontal static load tests within the landslide rein-forcement range We then simulated an action mechanismand stress distribution in the micropile composite structureby adopting a jack to provide load Through earth pressurecells that were installed before and after the micropile wewere able to monitor the earth pressure variation and thelandslide thrust while checking the status of each row of

piles The stress of the miniature piles is measured througha reinforcement meter that was welded on a miniature steelpipe The pile body deformation was indirectly measuredthrough an inclinometer that was installed on the pile sidesFigure 6 shows the model demonstration diagram

421 Experiment Process The in situ test uses the micropileto determine the rate of reinforcement The grouting coagu-lation of the soil strength grade is C25 The micropile lengthindicates the landslide segment which is 8m long The pilediameter is 150mmand the tube diameter is 50mmThemainreinforcement pile contains 3 roots made up of 28 reinforcedsteel pipesThemicrocap sets the C30 concrete capping beamwith a beam that is 05m high and 15m wide In accordancewith the requirements for the load test the test is conductedwith a grade 11 effective load using two gauges to record thedata The average is taken as the final result On the 12thlevel (96 t) load the counterforce device becomes damagedindicating the end of the test The load of the destruction isthe horizontal limit load At themoment when themaximumamount of bending occurs the steel of the yielding tensilezone is the corresponding loadThemicropile section and testprocess are shown in Figures 7 8 and 9

422 Result Analysis The rate of pile displacement of eachgrade of load was obtained through on-site experimentsand analysis as shown in Figures 10 and 11 In Figures 10and 11 it can be detected that under horizontal load thedisplacement of the micropile composite structure above thesliding surface (3m) is more obvious than that below thesliding surface which indicates that the micropile compositestructure presents a tendency to lean forward Since the pileonly leaned 2m forward at the base below the sliding surfacethe horizontal displacement is basically 0 which indicatesthat the anchorage effects at the anchorage section are com-paratively betterWhen consistently increasing the horizontalload the variable quantities of the displacement of pile top ofeach row of piles are the sameThis consistency in the variablequantities is caused by the lid cap contracting to make themicropile composite structure act as a whole for antislidingby adopting a jack to provide step loadThe bending momentand the displacement distribution laws of the three-row pilesare similar A bendingmoment above 025m is 0 No bendingdeflection of the micropile is generated due to the constraint

8 Mathematical Problems in Engineering

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

4 Mathematical Problems in Engineering

By substituting formula (9) into the displacement bound-ary condition formula (7) it can be derived that

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

[(3 minus 4120583) 1205930(120585) minus

120596 (120585)

1205961015840 (120585)12059310158400(120585) minus 120595

0(120585)] = 119891

1

(10)

where

119891 =119864

1 + 1205831199060minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

1198911= minus

119865

8120587 (1 minus 120583)[ln120596 (120585) minus

120596 (120585)

120596 (120585)

+ (3 minus 4120583) ln120596 (120585)]

(11)

Through the simultaneous application of the boundaryconditions of the two equations found in formula (10) theexpression of 120593

0(120585) and 120595

0(120585) can be derived (including the

unknown displacement of 1199060)

(2)Considering the micropile hole boundary namely forthe boundary conditions 119904

0 120590 = 119890

119894120579 the stress boundaryconditions are to be expressed with components under theorthogonal curvilinear coordinate system at the 119911 planeNamely

[(3 minus 4120583) 1205930(120590) minus

120596 (120590)

1205961015840 (120590)12059310158400(120590) minus 120595

0(120590)] = 119891

0 (12)

The Cauchy integral operator (12120587119894) ∮(119889120590(120590 minus 120585)) atboth ends of the formula above can be obtained

(3 minus 4120583)

2120587119894∮

1205930(120590) 119889120590

120590 minus 120585minus

1

2120587119894∮

120596 (120590)

1205961015840 (120590)

12059310158400(120590)

120590 minus 120585119889120590

minus1

2120587119894∮

1205950(120590)

120590 minus 120585119889120590 =

1

2120587119894∮

1198910

120590 minus 120585119889120590

(13)

In formula (13) ∮(1205930(120590)119889120590(120590 minus 120585)) = minus120593

0(120585)

(12120587119894) ∮(1205950(120590)(120590 minus 120585))119889120590 = 0 so by substituting into

formula (13) one can find 1205930(120585)

Similarly according to formula (12) the value of 1205950(120585)

can be obtained by taking conjugation at both sides and thenapplying the Cauchy integral operator Thus the informationcan be simultaneously obtained by combining formula (10)to get the value of 119906

0

26 Solution to 1205930(120585) and 120595

0(120585) To obtain the boundary

conditions of a miniature pile make 120585 = 120590120588 and unfold 120593(120585)

and 120595(120585) in a Laurent series form namely [17]

120593 (120585) = 1198860+

119899

sum

119896=1

119886119896120585119896

+

119899

sum

119896=1

119887119896120585minus119896

120595 (120585) = 1198880+

119899

sum

119896=1

119888119896120585119896

+

119899

sum

119896=1

119889119896120585minus119896

(14)

Depending on formula (4) one can get the following

120596 (120590)

1205961015840 (120590)= minus

1

2

(1 + 120588120590) (120590 minus 120588)2

1205902 (1 minus 120588120590) (15)

Using formulas (14) and (15) substituted into boundaryconditions 119904

1of formula (7) one can get the following

1198880= minus1198860minus

1

21198861minus

1

21198871

119888119896= minus119887119896+

1

2(119896 minus 1) 119886

119896minus1minus

1

2(119896 + 1) 119886

119896+1

119889119896= minus119886119896+

1

2(119896 minus 1) 119887

119896minus1minus

1

2(119896 + 1) 119887

119896+1

(16)

Make 119891(120585) = 119891(120572120590) = 2119866(119906 + 119894V) and 119891lowast

(120572120590) = (1 minus

120572120590)119891(120572120590) = suminfin

119896119860119896120590119896 Formulas (14) (15) and (17)ndash(19) can

be substituted into the boundary conditions 1199041of formula (7)

by eliminating 119888119896and 119889

119896 one finds 119886

119896and 119887119896

(1 minus 1205722

) (119896 + 1) 119886119896+1

minus (1205722

+ 120581120572minus2119896

) 119887119896+1

= (1 minus 1205722

) 119896119886119896minus (1 + 120581120572

minus2119896

) 119887119896+ 119860minus119896

120572119896

(1 + 1205811205722119896+2

) 119886119896+1

+ (1 minus 1205722

) (119896 + 1) 119887119896+1

= 1205722

(1205722

+ 1205811205722119896

) 119886119896+ (1 minus 120572

2

) 119896119887119896+ 1198601+119896

120572119896+1

(1 minus 1205722

) 1198861minus (120581 + 120572

2

) 1198871= 1198600minus (120581 + 1) 119886

0

(1 + 1205811205722

) 1198861+ (1 minus 120572

2

) 1198871= 1198601120572 + 1205722

(120581 + 1) 1198860

(17)

We simultaneously solved the four formulas in (17) toget all of the coefficients except for 119886

0 since 119886

0represents

rigid body displacement thus no stress will be generatedand it can be deemed as 0 By 119896 times of iteration one canobtain 119886

119896 thus to get 120593

0(120585) and 120595

0(120585) one must substitute

1205930(120585) and 120595

0(120585) in the complex function of the fundamental

equation to ascertain the stress and displacement field ofthe soil around a pile of micropiles under horizontal loadAccording to the assumed conditions specified in Sections21 and 22 the relationship between horizontal load and thehorizontal displacement can be obtained from 119870 = 119865119906

0

3 Establishment of Pile-Soil Mechanics ModelBased on Winkler Foundation Beam

In most cases while ministeel tub piles under horizontalload are applied in landslide control and slope reinforcement

Mathematical Problems in Engineering 5

Micropile

Sloping surface

Roof beam

Landslid

e surfa

ce

Figure 3 Micropile composite structure set on step of slope

a pile groupsrsquo layout will be adopted Especially for multistageslopes platforms are available for each grade Miniature pilescan be set at the platform for reinforcement The multipleminiature piles that are exposed on the platform can be fixedwith a top beam to enhance their sliding resistance forceThemicropile layout is shown in Figure 3

31 Model Assumption While miniature piles are appliedfor landslide reinforcement the mechanism can be used toaddress bigger shearing resistance in miniature pile land-slides Pile-soil structures will resist landslide thrust formedbymicropiles as well as the soil mass For internal calculationtheWinkler low econometricmodel is adoptedThis researchassumes that the soil around the micropile has a series ofdiscrete springs and a rigidity coefficient of 119870 = 119865119906

0 As

shown in Figure 3 since the micropile is fixed at the bed rockbelow the sliding surface it is assumed that the sliding surfaceis as fixed constraint Due to the fact that themicropile is fixedand connected through a top beam as compared tominiaturepile the top beam can be deemed as a rigid member Underthe effects of horizontal load the top beam only experienceshorizontal displacement so the displacement at each pile capwill be the same [18] Thus the internal force calculation ofthe model is shown in Figure 4

32 Internal Force Calculation According to the fundamen-tal theory of elastic mechanics the stress and calculationmodel of a single micropile can be obtained as shown inFigure 5 It is composed of two calculation models namelywith one fixed and one sliding end under a concentrated load(Figure 5(a)) and with both ends fixed under a uniformlydistributed load (Figure 5(b)) Specific to the calculationmodel found in Figure 4 for three-row micropile structurethe internal force of the micropile AD is obtained throughsuperposition of Figures 5(a) and 5(b) The internal forces ofmicropile BE and CF are derived from Figure 5(a)

321 Solution to Calculation Model with One Fixed andOne Sliding End under a Concentrated Load According tothe internal force calculation model specified in Figure 5(a)when considering micropile structures with one fixed endand one sliding end under a concentrated load the boundary

q

F

f f

A B C

D E F

K K K H

Figure 4 Mechanical calculation model of micropile structure

z

x

H

F

(a)

q

z

x

H

(b)

Figure 5 Mechanical calculation model of single micropile

condition on the top of the micropile is 119911 = 0 the bendingmoment is 119872 = 0 and 119865 is horizontal shear The boundarycondition on the bottom of the micropile is 119911 = 119867 the angleis 120593 = 0 and the horizontal displacement is 119906 = 0 Theflexural differential equation of an Euler-Bernoulli beamwithone fixed and one sliding end is [19]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 0 (18)

In the formula above 119864 is the elasticity modulus of theminiature pile 119868 is the inertia moment of a micropile crosssection119870 is the rigidity coefficient of an assumed spring and119906(119911) is the horizontal displacement of the soil around the pilealong the pile body

6 Mathematical Problems in Engineering

Roof beam

GroundSteel plate

Jack

Counterforce deviceRetaining wall

Earth pressure boxSteel bar

Micropile structure

Reac

tion

wal

l

Earth pressure cells

Reinforced stress meter

Inclinometer tube

Thrust 1 2 34 5 6

meter

05m

Figure 6 The schematic drawing of a test model of micropiles composite structure

By substituting the boundary conditions in the flexuraldifferential equation (18) the analytical solution can beobtained

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)]

119872 = 21205822

119864119868119890120582119911

[1198882cos (120582119911) minus 119888

1sin (120582119911)]

+ 21205822

119864119868119890minus120582119911

[1198884cos (120582119911) minus 119888

3sin (120582119911)]

(19)

where 120582 = (1198704119864119868)14 and 119888

1 1198882 1198883 and 119888

4are the integral

constant

322 Solution to Calculation Model with Both Ends Fixedunder a Uniformly Distributed Load According to the inter-nal force calculation model specified in Figure 5(b) formicropile structures with both ends fixed under a uniformlydistributed load the flexural differential equation of an Euler-Bernoulli beam is [18]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 119902 (119911) = 0 (20)

where 119902(119911) is the uniform load of the soil around the pile Tosolve this equation 120582 = (1198704119864119868)

14 can be substituted intoformula (19) to get the following

1198894

119906

1198891199114+ 4120582119906 (119911) =

119902

119864119868 (21)

By substituting the boundary conditions into the flex-ural differential equation (18) the analytical solution can

Steel bar welded to

The type of bracket

2

1

120∘

3Φ28

Φ8 3

empty150

steel pipe

Diameter 50mm-pipe

B4000

Figure 7 Micropiles sectional drawing

Figure 8 The field tests of micropile

be obtained The general solution 119902(119911) = 0 and specialconnection is 119902(119911) that is

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)] +

119902

119870

(22)

Mathematical Problems in Engineering 7

Figure 9 The horizontal static load test of micropiles

323 The Determination of the Elastic Modulus (119864) In theprevious calculation 119864 is a crucial parameter usually deter-mined by laboratory experiment and experimental relation-ship of previous practice However as laboratory experimentit is very difficult to reproduce the soil pile in live loadeffect on the stress path and experiential relationship cannotdetermine the value of 119864 either as it can be influencedobviously by human interference Due to the above reason inthis paper the field test of single pile at the scene of the elasticdeformation range was appliedThe elastic model value is theinversion analysis through the stress-strain curve with thebasic process described as follows

Under the load themicropile is loaded step by step usingthe same load increment Record each time loaddisplacementdeformation therefore according to the stress-strain curveobtained from test to calculate the value of elastic modulus

4 Engineering Practice Analysis andVerification of Pile-Soil Mechanics Model

41 Engineering Examples In this example the landslide ofa highway is medium type about 80ndash100m wild and 160mlong The front is about 5m thick whereas the central partis 10ndash15m thick and the back part is 35ndash80m thick Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160000m3 The landslidedemonstrates typical factors with the slope facing the emptyand the steep slope The features of landslide appearance areobvious the back has tensile cracks and the two sides havepinnate cracks The front is flanked by a shear seam with aballooning extrusion crack forming and radioactive cracksSpecific to the characteristics of this landslide it is proposedto adopt a light retaining structure with a ministeel tub pilescomposite structure for reinforcement

42 Establishment of Experiment Process andNumerical Simu-lation According to the design requirements we performedon-site horizontal static load tests within the landslide rein-forcement range We then simulated an action mechanismand stress distribution in the micropile composite structureby adopting a jack to provide load Through earth pressurecells that were installed before and after the micropile wewere able to monitor the earth pressure variation and thelandslide thrust while checking the status of each row of

piles The stress of the miniature piles is measured througha reinforcement meter that was welded on a miniature steelpipe The pile body deformation was indirectly measuredthrough an inclinometer that was installed on the pile sidesFigure 6 shows the model demonstration diagram

421 Experiment Process The in situ test uses the micropileto determine the rate of reinforcement The grouting coagu-lation of the soil strength grade is C25 The micropile lengthindicates the landslide segment which is 8m long The pilediameter is 150mmand the tube diameter is 50mmThemainreinforcement pile contains 3 roots made up of 28 reinforcedsteel pipesThemicrocap sets the C30 concrete capping beamwith a beam that is 05m high and 15m wide In accordancewith the requirements for the load test the test is conductedwith a grade 11 effective load using two gauges to record thedata The average is taken as the final result On the 12thlevel (96 t) load the counterforce device becomes damagedindicating the end of the test The load of the destruction isthe horizontal limit load At themoment when themaximumamount of bending occurs the steel of the yielding tensilezone is the corresponding loadThemicropile section and testprocess are shown in Figures 7 8 and 9

422 Result Analysis The rate of pile displacement of eachgrade of load was obtained through on-site experimentsand analysis as shown in Figures 10 and 11 In Figures 10and 11 it can be detected that under horizontal load thedisplacement of the micropile composite structure above thesliding surface (3m) is more obvious than that below thesliding surface which indicates that the micropile compositestructure presents a tendency to lean forward Since the pileonly leaned 2m forward at the base below the sliding surfacethe horizontal displacement is basically 0 which indicatesthat the anchorage effects at the anchorage section are com-paratively betterWhen consistently increasing the horizontalload the variable quantities of the displacement of pile top ofeach row of piles are the sameThis consistency in the variablequantities is caused by the lid cap contracting to make themicropile composite structure act as a whole for antislidingby adopting a jack to provide step loadThe bending momentand the displacement distribution laws of the three-row pilesare similar A bendingmoment above 025m is 0 No bendingdeflection of the micropile is generated due to the constraint

8 Mathematical Problems in Engineering

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

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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 5

Micropile

Sloping surface

Roof beam

Landslid

e surfa

ce

Figure 3 Micropile composite structure set on step of slope

a pile groupsrsquo layout will be adopted Especially for multistageslopes platforms are available for each grade Miniature pilescan be set at the platform for reinforcement The multipleminiature piles that are exposed on the platform can be fixedwith a top beam to enhance their sliding resistance forceThemicropile layout is shown in Figure 3

31 Model Assumption While miniature piles are appliedfor landslide reinforcement the mechanism can be used toaddress bigger shearing resistance in miniature pile land-slides Pile-soil structures will resist landslide thrust formedbymicropiles as well as the soil mass For internal calculationtheWinkler low econometricmodel is adoptedThis researchassumes that the soil around the micropile has a series ofdiscrete springs and a rigidity coefficient of 119870 = 119865119906

0 As

shown in Figure 3 since the micropile is fixed at the bed rockbelow the sliding surface it is assumed that the sliding surfaceis as fixed constraint Due to the fact that themicropile is fixedand connected through a top beam as compared tominiaturepile the top beam can be deemed as a rigid member Underthe effects of horizontal load the top beam only experienceshorizontal displacement so the displacement at each pile capwill be the same [18] Thus the internal force calculation ofthe model is shown in Figure 4

32 Internal Force Calculation According to the fundamen-tal theory of elastic mechanics the stress and calculationmodel of a single micropile can be obtained as shown inFigure 5 It is composed of two calculation models namelywith one fixed and one sliding end under a concentrated load(Figure 5(a)) and with both ends fixed under a uniformlydistributed load (Figure 5(b)) Specific to the calculationmodel found in Figure 4 for three-row micropile structurethe internal force of the micropile AD is obtained throughsuperposition of Figures 5(a) and 5(b) The internal forces ofmicropile BE and CF are derived from Figure 5(a)

321 Solution to Calculation Model with One Fixed andOne Sliding End under a Concentrated Load According tothe internal force calculation model specified in Figure 5(a)when considering micropile structures with one fixed endand one sliding end under a concentrated load the boundary

q

F

f f

A B C

D E F

K K K H

Figure 4 Mechanical calculation model of micropile structure

z

x

H

F

(a)

q

z

x

H

(b)

Figure 5 Mechanical calculation model of single micropile

condition on the top of the micropile is 119911 = 0 the bendingmoment is 119872 = 0 and 119865 is horizontal shear The boundarycondition on the bottom of the micropile is 119911 = 119867 the angleis 120593 = 0 and the horizontal displacement is 119906 = 0 Theflexural differential equation of an Euler-Bernoulli beamwithone fixed and one sliding end is [19]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 0 (18)

In the formula above 119864 is the elasticity modulus of theminiature pile 119868 is the inertia moment of a micropile crosssection119870 is the rigidity coefficient of an assumed spring and119906(119911) is the horizontal displacement of the soil around the pilealong the pile body

6 Mathematical Problems in Engineering

Roof beam

GroundSteel plate

Jack

Counterforce deviceRetaining wall

Earth pressure boxSteel bar

Micropile structure

Reac

tion

wal

l

Earth pressure cells

Reinforced stress meter

Inclinometer tube

Thrust 1 2 34 5 6

meter

05m

Figure 6 The schematic drawing of a test model of micropiles composite structure

By substituting the boundary conditions in the flexuraldifferential equation (18) the analytical solution can beobtained

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)]

119872 = 21205822

119864119868119890120582119911

[1198882cos (120582119911) minus 119888

1sin (120582119911)]

+ 21205822

119864119868119890minus120582119911

[1198884cos (120582119911) minus 119888

3sin (120582119911)]

(19)

where 120582 = (1198704119864119868)14 and 119888

1 1198882 1198883 and 119888

4are the integral

constant

322 Solution to Calculation Model with Both Ends Fixedunder a Uniformly Distributed Load According to the inter-nal force calculation model specified in Figure 5(b) formicropile structures with both ends fixed under a uniformlydistributed load the flexural differential equation of an Euler-Bernoulli beam is [18]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 119902 (119911) = 0 (20)

where 119902(119911) is the uniform load of the soil around the pile Tosolve this equation 120582 = (1198704119864119868)

14 can be substituted intoformula (19) to get the following

1198894

119906

1198891199114+ 4120582119906 (119911) =

119902

119864119868 (21)

By substituting the boundary conditions into the flex-ural differential equation (18) the analytical solution can

Steel bar welded to

The type of bracket

2

1

120∘

3Φ28

Φ8 3

empty150

steel pipe

Diameter 50mm-pipe

B4000

Figure 7 Micropiles sectional drawing

Figure 8 The field tests of micropile

be obtained The general solution 119902(119911) = 0 and specialconnection is 119902(119911) that is

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)] +

119902

119870

(22)

Mathematical Problems in Engineering 7

Figure 9 The horizontal static load test of micropiles

323 The Determination of the Elastic Modulus (119864) In theprevious calculation 119864 is a crucial parameter usually deter-mined by laboratory experiment and experimental relation-ship of previous practice However as laboratory experimentit is very difficult to reproduce the soil pile in live loadeffect on the stress path and experiential relationship cannotdetermine the value of 119864 either as it can be influencedobviously by human interference Due to the above reason inthis paper the field test of single pile at the scene of the elasticdeformation range was appliedThe elastic model value is theinversion analysis through the stress-strain curve with thebasic process described as follows

Under the load themicropile is loaded step by step usingthe same load increment Record each time loaddisplacementdeformation therefore according to the stress-strain curveobtained from test to calculate the value of elastic modulus

4 Engineering Practice Analysis andVerification of Pile-Soil Mechanics Model

41 Engineering Examples In this example the landslide ofa highway is medium type about 80ndash100m wild and 160mlong The front is about 5m thick whereas the central partis 10ndash15m thick and the back part is 35ndash80m thick Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160000m3 The landslidedemonstrates typical factors with the slope facing the emptyand the steep slope The features of landslide appearance areobvious the back has tensile cracks and the two sides havepinnate cracks The front is flanked by a shear seam with aballooning extrusion crack forming and radioactive cracksSpecific to the characteristics of this landslide it is proposedto adopt a light retaining structure with a ministeel tub pilescomposite structure for reinforcement

42 Establishment of Experiment Process andNumerical Simu-lation According to the design requirements we performedon-site horizontal static load tests within the landslide rein-forcement range We then simulated an action mechanismand stress distribution in the micropile composite structureby adopting a jack to provide load Through earth pressurecells that were installed before and after the micropile wewere able to monitor the earth pressure variation and thelandslide thrust while checking the status of each row of

piles The stress of the miniature piles is measured througha reinforcement meter that was welded on a miniature steelpipe The pile body deformation was indirectly measuredthrough an inclinometer that was installed on the pile sidesFigure 6 shows the model demonstration diagram

421 Experiment Process The in situ test uses the micropileto determine the rate of reinforcement The grouting coagu-lation of the soil strength grade is C25 The micropile lengthindicates the landslide segment which is 8m long The pilediameter is 150mmand the tube diameter is 50mmThemainreinforcement pile contains 3 roots made up of 28 reinforcedsteel pipesThemicrocap sets the C30 concrete capping beamwith a beam that is 05m high and 15m wide In accordancewith the requirements for the load test the test is conductedwith a grade 11 effective load using two gauges to record thedata The average is taken as the final result On the 12thlevel (96 t) load the counterforce device becomes damagedindicating the end of the test The load of the destruction isthe horizontal limit load At themoment when themaximumamount of bending occurs the steel of the yielding tensilezone is the corresponding loadThemicropile section and testprocess are shown in Figures 7 8 and 9

422 Result Analysis The rate of pile displacement of eachgrade of load was obtained through on-site experimentsand analysis as shown in Figures 10 and 11 In Figures 10and 11 it can be detected that under horizontal load thedisplacement of the micropile composite structure above thesliding surface (3m) is more obvious than that below thesliding surface which indicates that the micropile compositestructure presents a tendency to lean forward Since the pileonly leaned 2m forward at the base below the sliding surfacethe horizontal displacement is basically 0 which indicatesthat the anchorage effects at the anchorage section are com-paratively betterWhen consistently increasing the horizontalload the variable quantities of the displacement of pile top ofeach row of piles are the sameThis consistency in the variablequantities is caused by the lid cap contracting to make themicropile composite structure act as a whole for antislidingby adopting a jack to provide step loadThe bending momentand the displacement distribution laws of the three-row pilesare similar A bendingmoment above 025m is 0 No bendingdeflection of the micropile is generated due to the constraint

8 Mathematical Problems in Engineering

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

6 Mathematical Problems in Engineering

Roof beam

GroundSteel plate

Jack

Counterforce deviceRetaining wall

Earth pressure boxSteel bar

Micropile structure

Reac

tion

wal

l

Earth pressure cells

Reinforced stress meter

Inclinometer tube

Thrust 1 2 34 5 6

meter

05m

Figure 6 The schematic drawing of a test model of micropiles composite structure

By substituting the boundary conditions in the flexuraldifferential equation (18) the analytical solution can beobtained

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)]

119872 = 21205822

119864119868119890120582119911

[1198882cos (120582119911) minus 119888

1sin (120582119911)]

+ 21205822

119864119868119890minus120582119911

[1198884cos (120582119911) minus 119888

3sin (120582119911)]

(19)

where 120582 = (1198704119864119868)14 and 119888

1 1198882 1198883 and 119888

4are the integral

constant

322 Solution to Calculation Model with Both Ends Fixedunder a Uniformly Distributed Load According to the inter-nal force calculation model specified in Figure 5(b) formicropile structures with both ends fixed under a uniformlydistributed load the flexural differential equation of an Euler-Bernoulli beam is [18]

1198641198681198894

119906

1198891199114+ 119870119906 (119911) = 119902 (119911) = 0 (20)

where 119902(119911) is the uniform load of the soil around the pile Tosolve this equation 120582 = (1198704119864119868)

14 can be substituted intoformula (19) to get the following

1198894

119906

1198891199114+ 4120582119906 (119911) =

119902

119864119868 (21)

By substituting the boundary conditions into the flex-ural differential equation (18) the analytical solution can

Steel bar welded to

The type of bracket

2

1

120∘

3Φ28

Φ8 3

empty150

steel pipe

Diameter 50mm-pipe

B4000

Figure 7 Micropiles sectional drawing

Figure 8 The field tests of micropile

be obtained The general solution 119902(119911) = 0 and specialconnection is 119902(119911) that is

119906 = 119890120582119911

[1198881cos (120582119911) + 119888

2sin (120582119911)]

+ 119890minus120582119911

[1198883cos (120582119911) + 119888

4sin (120582119911)] +

119902

119870

(22)

Mathematical Problems in Engineering 7

Figure 9 The horizontal static load test of micropiles

323 The Determination of the Elastic Modulus (119864) In theprevious calculation 119864 is a crucial parameter usually deter-mined by laboratory experiment and experimental relation-ship of previous practice However as laboratory experimentit is very difficult to reproduce the soil pile in live loadeffect on the stress path and experiential relationship cannotdetermine the value of 119864 either as it can be influencedobviously by human interference Due to the above reason inthis paper the field test of single pile at the scene of the elasticdeformation range was appliedThe elastic model value is theinversion analysis through the stress-strain curve with thebasic process described as follows

Under the load themicropile is loaded step by step usingthe same load increment Record each time loaddisplacementdeformation therefore according to the stress-strain curveobtained from test to calculate the value of elastic modulus

4 Engineering Practice Analysis andVerification of Pile-Soil Mechanics Model

41 Engineering Examples In this example the landslide ofa highway is medium type about 80ndash100m wild and 160mlong The front is about 5m thick whereas the central partis 10ndash15m thick and the back part is 35ndash80m thick Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160000m3 The landslidedemonstrates typical factors with the slope facing the emptyand the steep slope The features of landslide appearance areobvious the back has tensile cracks and the two sides havepinnate cracks The front is flanked by a shear seam with aballooning extrusion crack forming and radioactive cracksSpecific to the characteristics of this landslide it is proposedto adopt a light retaining structure with a ministeel tub pilescomposite structure for reinforcement

42 Establishment of Experiment Process andNumerical Simu-lation According to the design requirements we performedon-site horizontal static load tests within the landslide rein-forcement range We then simulated an action mechanismand stress distribution in the micropile composite structureby adopting a jack to provide load Through earth pressurecells that were installed before and after the micropile wewere able to monitor the earth pressure variation and thelandslide thrust while checking the status of each row of

piles The stress of the miniature piles is measured througha reinforcement meter that was welded on a miniature steelpipe The pile body deformation was indirectly measuredthrough an inclinometer that was installed on the pile sidesFigure 6 shows the model demonstration diagram

421 Experiment Process The in situ test uses the micropileto determine the rate of reinforcement The grouting coagu-lation of the soil strength grade is C25 The micropile lengthindicates the landslide segment which is 8m long The pilediameter is 150mmand the tube diameter is 50mmThemainreinforcement pile contains 3 roots made up of 28 reinforcedsteel pipesThemicrocap sets the C30 concrete capping beamwith a beam that is 05m high and 15m wide In accordancewith the requirements for the load test the test is conductedwith a grade 11 effective load using two gauges to record thedata The average is taken as the final result On the 12thlevel (96 t) load the counterforce device becomes damagedindicating the end of the test The load of the destruction isthe horizontal limit load At themoment when themaximumamount of bending occurs the steel of the yielding tensilezone is the corresponding loadThemicropile section and testprocess are shown in Figures 7 8 and 9

422 Result Analysis The rate of pile displacement of eachgrade of load was obtained through on-site experimentsand analysis as shown in Figures 10 and 11 In Figures 10and 11 it can be detected that under horizontal load thedisplacement of the micropile composite structure above thesliding surface (3m) is more obvious than that below thesliding surface which indicates that the micropile compositestructure presents a tendency to lean forward Since the pileonly leaned 2m forward at the base below the sliding surfacethe horizontal displacement is basically 0 which indicatesthat the anchorage effects at the anchorage section are com-paratively betterWhen consistently increasing the horizontalload the variable quantities of the displacement of pile top ofeach row of piles are the sameThis consistency in the variablequantities is caused by the lid cap contracting to make themicropile composite structure act as a whole for antislidingby adopting a jack to provide step loadThe bending momentand the displacement distribution laws of the three-row pilesare similar A bendingmoment above 025m is 0 No bendingdeflection of the micropile is generated due to the constraint

8 Mathematical Problems in Engineering

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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 7

Figure 9 The horizontal static load test of micropiles

323 The Determination of the Elastic Modulus (119864) In theprevious calculation 119864 is a crucial parameter usually deter-mined by laboratory experiment and experimental relation-ship of previous practice However as laboratory experimentit is very difficult to reproduce the soil pile in live loadeffect on the stress path and experiential relationship cannotdetermine the value of 119864 either as it can be influencedobviously by human interference Due to the above reason inthis paper the field test of single pile at the scene of the elasticdeformation range was appliedThe elastic model value is theinversion analysis through the stress-strain curve with thebasic process described as follows

Under the load themicropile is loaded step by step usingthe same load increment Record each time loaddisplacementdeformation therefore according to the stress-strain curveobtained from test to calculate the value of elastic modulus

4 Engineering Practice Analysis andVerification of Pile-Soil Mechanics Model

41 Engineering Examples In this example the landslide ofa highway is medium type about 80ndash100m wild and 160mlong The front is about 5m thick whereas the central partis 10ndash15m thick and the back part is 35ndash80m thick Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160000m3 The landslidedemonstrates typical factors with the slope facing the emptyand the steep slope The features of landslide appearance areobvious the back has tensile cracks and the two sides havepinnate cracks The front is flanked by a shear seam with aballooning extrusion crack forming and radioactive cracksSpecific to the characteristics of this landslide it is proposedto adopt a light retaining structure with a ministeel tub pilescomposite structure for reinforcement

42 Establishment of Experiment Process andNumerical Simu-lation According to the design requirements we performedon-site horizontal static load tests within the landslide rein-forcement range We then simulated an action mechanismand stress distribution in the micropile composite structureby adopting a jack to provide load Through earth pressurecells that were installed before and after the micropile wewere able to monitor the earth pressure variation and thelandslide thrust while checking the status of each row of

piles The stress of the miniature piles is measured througha reinforcement meter that was welded on a miniature steelpipe The pile body deformation was indirectly measuredthrough an inclinometer that was installed on the pile sidesFigure 6 shows the model demonstration diagram

421 Experiment Process The in situ test uses the micropileto determine the rate of reinforcement The grouting coagu-lation of the soil strength grade is C25 The micropile lengthindicates the landslide segment which is 8m long The pilediameter is 150mmand the tube diameter is 50mmThemainreinforcement pile contains 3 roots made up of 28 reinforcedsteel pipesThemicrocap sets the C30 concrete capping beamwith a beam that is 05m high and 15m wide In accordancewith the requirements for the load test the test is conductedwith a grade 11 effective load using two gauges to record thedata The average is taken as the final result On the 12thlevel (96 t) load the counterforce device becomes damagedindicating the end of the test The load of the destruction isthe horizontal limit load At themoment when themaximumamount of bending occurs the steel of the yielding tensilezone is the corresponding loadThemicropile section and testprocess are shown in Figures 7 8 and 9

422 Result Analysis The rate of pile displacement of eachgrade of load was obtained through on-site experimentsand analysis as shown in Figures 10 and 11 In Figures 10and 11 it can be detected that under horizontal load thedisplacement of the micropile composite structure above thesliding surface (3m) is more obvious than that below thesliding surface which indicates that the micropile compositestructure presents a tendency to lean forward Since the pileonly leaned 2m forward at the base below the sliding surfacethe horizontal displacement is basically 0 which indicatesthat the anchorage effects at the anchorage section are com-paratively betterWhen consistently increasing the horizontalload the variable quantities of the displacement of pile top ofeach row of piles are the sameThis consistency in the variablequantities is caused by the lid cap contracting to make themicropile composite structure act as a whole for antislidingby adopting a jack to provide step loadThe bending momentand the displacement distribution laws of the three-row pilesare similar A bendingmoment above 025m is 0 No bendingdeflection of the micropile is generated due to the constraint

8 Mathematical Problems in Engineering

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

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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

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

0Displacement (mm)

2 4 6 8

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(b) The second row of piles

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

2 4 6 80Displacement (mm)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

(c) The third row of piles

Figure 10 Comparative curves of distribution of pile deflection

Mathematical Problems in Engineering 9

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0Pi

le le

ngth

(mm

)

0 5 10minus5minus10Moment (KN m)

(a) The first row of piles

Landslide surface

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

0 5 10minus5minus10Moment (KN m)

(b) The second row of piles

Landslide surface

minus8

minus6

minus4

minus2

0

Pile

leng

th (m

)

5minus5 0 10minus15 minus10Moment (KN m)

1467KPa133KPa1197KPa1064KPa931KPa

798KPa665KPa532KPa399KPa266KPa133KPa

(c) The third row of piles

The first row pile +The second row pile +The third row pile +

The first row pile minusThe second row pile minusThe third row pile minus

minus10

minus5

0

5

10

Mom

ent (

KN m

)

2 4 6 8 10 120Load series

(d) The maximum positive (negative) moment

Figure 11 Comparative curves of distribution of pile model

10 Mathematical Problems in Engineering

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

minus12

minus10

minus8

minus6

minus4

minus2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(a) The first row of piles

1 2 3 4 50Displacement (mm)

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(b) The second row of piles

minus12

minus10

minus8

minus6

minus4

minus2

0

Hig

h (m

)

1 2 3 4 50Displacement (mm)

1197KPa (calculation)1197KPa (test)798KPa (calculation)

798KPa (test)266KPa (calculation)266KPa (test)

(c) The third row of piles

Figure 12 Comparative curves of the distribution of pile deflection

of the cap lid The point of contraflexure occurred at 09mabove the sliding surface and 25m below the sliding surfacefor the three rows of piles The maximum sagging moment isat 1m above the sliding surface and the maximum hoggingmoment is at 05m below the sliding surface The value ofbending moment increases along with the horizontal loadAs the horizontal load increased to grades 9ndash120KPa (closeto the upper limit of the horizontal load of an antislidestructure of the micropile combination) the increment of themaximum bending moment (the absolute value) at the thirdrow will be the maximum followed by that of the second rowand then the first row If each row of piles adopts the samebending strength design without considering the impact ofplastic failure on the soil mass between the piles underthe effects of landslide thrust beyond the upper limit of the

horizontal load of the micropile combined mechanism thenthe sequence for each row of piles is the third row followedby the second row and the first row

43 Contrastive Analysis to Theoretical Calculation Theproposed approach incorporated the theoretical calculationmodel of Sections 2 and 3 and the mechanical parametersand conditions of the experiment in order to calculate thedisplacement of each row of micropiles under 266 KPa798 KPa and 1197 KPa as well as the bendingmoment under266 KPa and 1197 KPa as shown in Figures 12 and 13

By comparing Figures 10 and 12 to Figures 11 and 13 itcan be seen that according to the pile-soil response theorycalculation method under horizontal load the pile displace-ment and bending moment are similar to the results found

Mathematical Problems in Engineering 11

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

04

02

00

06

minus0

2

minus0

8minus

06

minus0

4

Moment (KN m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

(a) The first row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

minus8

minus6

minus4

minus2

0H

igh

(m)

minus8

minus6

minus4

minus2

0

Hig

h (m

)minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus6 minus4 minus2 0 2 4 6minus8Moment (KN m)

minus6 minus4 minus2minus8 2 4 60Moment (KN m)0

00

20

4

minus0

4minus

06

minus0

2

minus0

8

Moment (KN m)

minus0

8minus

06

Moment (KN m)

minus0

4minus

02

00

02

04

(b) The second row of piles

1197KPa (test) 1197KPa (calculate)266KPa (test) 266KPa (calculate)

Moment (KN m)

minus10

minus5

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

minus8

minus6

minus4

minus2

0

Hig

h (m

)

20 4 6minus4minus6 minus2minus8Moment (KN m)

20 4 6minus4minus6 minus2minus8Moment (KN m)minus

08

minus0

6

Moment (KN m)

minus0

4minus

02

00

02

04

minus0

8minus

06

minus0

4minus

02

00

02

04

(c) The third row of piles

Figure 13 Comparative curves of the distribution of the pile model

12 Mathematical Problems in Engineering

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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

in simulated field experiments under each grade of loadwhich shows that the theoretical approach demonstrated inthis paper is feasible From the figures it can be detected thatthe results of the experiment are smaller than the bendingmoment and the rate of pile body displacement This resultoccurs because in most cases while searching for solution tothe Winkler Foundation Beam Model the sheer force of thesoil between piles is generally ignored But for engineeringdesign the solution from theoretical approach adopted by thepaper is simply safe thus it can satisfy design accuracy

5 Conclusion

(1) This paper discusses an analytical solution to stressaround a micropile Displacement distribution underhorizontal load is obtained according to the planarcomplex function theory The defects found in theinternal force analysis after applying uniform sec-tions were addressed The stress and displacementdistribution were analyzed by utilizing the proposedanalytical solution in the soil around the micropiles

(2) Based on the Winkler Foundation Beam Model weassumed that the soil around the micropiles stemmedfrom a series of independent springs The rigiditycoefficient of a spring is obtained using a planarsolution After that based on the deflection curvedifferential equation of an Euler-Bernoulli beamthe pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modeswhich have provided theoretical guidance for engi-neering designs On the one hand one end is fixedwith the other end sliding on the other hand bothends are fixed

(3) By comparing the results obtained from on-sitepile pushing tests it is determined that the resultsobtained of the established theoretical approach arereasonably accurate and reliable

Conflict of Interests

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

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (no 2015YJS121) Theauthors would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] H JuranA Benslimane andDA Bruce ldquoSlope stabilization bymicropile reinforcementrdquo in Proceedings of the 7th InternationalSymposium on Landslides pp 1718ndash1726 Trondheim NorwayJune 1996

[2] D A Bruce A F Dimillio and I Juran ldquoIntroduction tomicropiles an international perspectiverdquo in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention pp 1ndash26 San DiegoCalif USA 1995

[3] S-G Xiao F Xian and H-L Wang ldquoAnalytical method ofinternal forces of a combining micropiles structurerdquo Rock andSoil Mechanics vol 31 no 8 pp 2553ndash2259 2010

[4] G Russo ldquoDiscussion full-scale load tests on instrumentedmicropiles technology and behaviorrdquoGeotechnical Engineeringvol 157 pp 127ndash135 2004

[5] R Z Moayed and S A Naeini ldquoImrovement of loose sandy soildeposits using micropilesrdquo KSCE Journal of Civil Engineeringvol 16 no 3 pp 334ndash340 2012

[6] A Ghorbani H Hasanzadehshooiili E Ghamari and JMedzvieckas ldquoComprehensive three dimensional finite elementanalysis parametric study and sensitivity analysis on the seis-mic performance of soil-micropile-superstructure interactionrdquoSoil Dynamics and Earthquake Engineering vol 58 pp 21ndash362014

[7] J M Duncan L T Evans Jr and P S K Ooi ldquoLateral loadanalysis of single piles and drilled shaftsrdquo Journal of GeotechnicalEngineering vol 120 no 6 pp 1018ndash1033 1994

[8] R L Mokwa and J M Duncan ldquoLaterally loaded pile groupeffects and P-Y multipliersrdquo Geotechnical Special Publicationvol 113 pp 728ndash742 2001

[9] M JThompson andD JWhite ldquoDesign of slope reinforcementwith small-diameter pilesrdquo in Proceedings of the Advances inEarth Structures pp 67ndash73 ASCE Shanghai China June 2006

[10] R Cantoni T Collotta and V N Ghionna ldquoA design methodfor reticulated micropiles structure in sliding sloperdquo GroundEngineering vol 22 no 1 pp 41ndash47 1989

[11] P R Macklin D Berger W Zietlow W Herring and J CullenldquoCase history micropile use for temporary excavation supportrdquoin Proceedings of Sessions of the Geosupport Conference Innova-tion and Cooperation in Geo pp 653ndash661 Geotechnical SpecialPublication ASCE Reston Va USA January 2004

[12] J FengD-P ZhouN Jiang andT Yang ldquoModel for calculationof internal force of micropile system to reinforce bedding rocksloperdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 2 pp 284ndash288 2006

[13] D A Brown and C-F Shie ldquoNumerical experiments into groupeffects on the response of piles to lateral loadingrdquo Computersand Geotechnics vol 10 no 3 pp 211ndash230 1990

[14] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 1970

[15] AVerruijt ldquoDeformations of an elastic half planewith a circularcavityrdquo International Journal of Solids and Structures vol 35 no21 pp 2795ndash2804 1998

[16] O E Strack and A Verruijt ldquoA complex variable solutionfor a deforming buoyant tunnel in a heavy elastic half-planerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 12 pp 1235ndash1252 2002

[17] A Z Lv andLQ ZhangUndergroundTunnel ComplexVariableMethod of Mechanical Analysis Science Press Beijing China2007

[18] P D Zhou H L Wang and H W Sun ldquoMicropile compos-ite structure and its design theoryrdquo Chinese Journal of RockMechanics and Engineering vol 28 no 7 pp 1353ndash1361 2009

[19] F Baguelin R Frank and Y H Said ldquoTheoretical study oflateral reaction mechanism of pilesrdquo Geotechnique vol 27 no3 pp 405ndash433 1977

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