research article correspondence analysis of soil around...

Research ArticleCorrespondence Analysis of Soil around MicropileComposite Structures under Horizontal Load

Hai Shi,1 Mingzhou Bai,1,2 Chao Li,1,3 Yunlong Zhang,1 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; [email protected]

Received 27 June 2015; Accepted 1 September 2015

Academic Editor: Fazal M. Mahomed

Copyright © 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 springs.The 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 70–300mmin 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 construction,and flexible piles. Thus, it has been widely used in build-ing reinforcements, shake-proof, foundation underpinning,foundation excavation support, landslide control, and othertypes of engineering found in buildings [4–6].

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 [7–9].The initial research yielded some achievements.Cantoni 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 pile’s internal force by applying

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 382427, 12 pageshttp://dx.doi.org/10.1155/2015/382427

2 Mathematical Problems in Engineering

a nonlinear, elastic-plastic subgrade reaction method (calledthe 𝑝-𝑦 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 needed.Thus, 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 that,based 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 modes,namely, 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

2.1. 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 𝑍 plane (rectangular coordinate system). As shown inFigure 1, the𝑅 region is a region of semi-infinite space, exceptfor the disc-structure, and 𝑎 is the distance between the soilaround the pile and the center of the pile; 𝑟 is the radius ofthe disc-structure; and 𝐹 is the horizontal load applied inthe center of the pile (𝐹

𝑥is 𝑥 component of direction; 𝐹

𝑦

is 𝑦 component of direction). It is assumed that 𝑢0is the

displacement under horizontal load applied in the center ofthe pile.

2.2. Fundamental Assumption of the Mechanical Model.While 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 pile,assume 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.

2.3. Basic Control Equation. According to the complex vari-able function of plane elasticmechanics, the complex analyticfunction of stress and displacement components at 𝑅 regionare to be expressed in 𝜑

1(𝑥) and 𝜓

1(𝑥); then [14],

𝜎𝑥+ 𝜎𝑦= 4Re [𝜑

1(𝑧)] ,

𝜎𝑦− 𝜎𝑥+ 2𝑖𝜏𝑥𝑦

= 2 [𝑧𝜑

1(𝑧) + 𝜓

1(𝑧)] ,

2𝐺 (𝑢 + 𝑖V) = 𝜅𝜑 (𝜉) − 𝜑1(𝜉)

𝜔 (𝜉)

𝜔 (𝜉)− 𝜓 (𝜉),

(1)

where 𝑧 is any point on the micropile hole, 𝜎𝑥, 𝜎𝑦, and 𝜏

𝑥𝑦are

the stress and strain component of any point, Re separatesthe real part and imaginary part in 𝜅 = 3 − 4𝜇, Planestrain; (3−𝜇)/(1+𝜇), Plane stress, and 𝐺 = 𝐸/2(1+𝜇), where𝐸 is modulus of elasticity and 𝜇 is the Poisson ratio.

The analytic function and expression 𝜑1(𝑥) and 𝜓

1(𝑥)

under stress at infinity bounded conditions are as follows:

𝜑1(𝑧) = −

1

2𝜋 (1 + 𝜅)(𝐹𝑥+ 𝑖𝐹𝑦) ln 𝑧 + (𝐵 + 𝑖𝐶) 𝑧

+ 𝜑0

1(𝑧) ,

𝜓1(𝑧) =

1

2𝜋 (1 + 𝜅)(𝐹𝑥− 𝑖𝐹𝑦) ln 𝑧 + (𝐵 + 𝑖𝐶) 𝑧

+ 𝜓0

1(𝑧) ,

(2)

where 𝐵, 𝐵, and 𝐶, 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, 𝜅 is positive integer.

Mathematical Problems in Engineering 3

𝜑0

1(𝑧) and 𝜓0

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

neighborhood located at infinity. Where 𝐵 = (𝜎∞𝑥

+ 𝜎∞

𝑦)/4,

𝐵

= (𝜎∞

𝑦− 𝜎∞

𝑥)/2; 𝐶 = 𝜏∞

𝑥𝑦, 𝜎∞𝑥, 𝜎∞𝑦, and 𝜏∞

𝑥𝑦is the stress

located at infinity.According 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: 𝑖 ∫𝐵

𝐴

(𝐹𝑥+ 𝑖𝐹𝑦) 𝑑𝑠

= 𝜑1(𝑧) + 𝑧𝜑

1(𝑧) + 𝜓

1(𝑧),

The condition of displacement boundary: 𝜅𝜑1(𝑧)

− 𝑧𝜑1(𝑧) − 𝜓

1(𝑧) = 2𝐺 (𝑢 + 𝑖V) .

(3)

2.4. ConformalMapping. Onemustmap themicropile orificeto unit annulus. The semi-infinite space orifice within 𝑍plane can be converted to within the plane 𝜉 after conformalmapping as shown in Figure 2.Then, themapping function ofzone 𝑅 after conformal mapping is shown in [15]. Therefore,

𝑧 = 𝜔 (𝜉) = −𝑖𝑎1 − 𝛼2

1 + 𝛼2

1 − 𝜉

1 + 𝜉, (4)

where any point 𝜉 located in the 𝜉 plane is expressed as polarcoordinates; that is, 𝜉 = 𝜌𝑒𝑖𝜃 and 𝛼 is a parameter determinedby 𝑎 and 𝑟:

𝑟

𝑎=

2𝑎

1 + 𝑎2. (5)

After transforming the function at plane 𝑧 to the functionof 𝜉, the stress and displacement component will change to[16]

𝜎𝜌+ 𝜎𝜃= 𝜎𝑥+ 𝜎𝑦= 4Re [Φ (𝜉)] ,

𝜎𝑥− 𝜎𝑦+ 2𝑖𝜏𝑥𝑦

=2𝜉2

𝜌2𝜔 (𝜉)[𝜔 (𝜉)Φ

(𝜉) + 𝜔

(𝜉) Ψ (𝑧)] ,

2𝐺 (𝑢 + 𝑖V) = 𝜅𝜑 (𝜉) − 𝜑1(𝜉)

𝜔 (𝜉)

𝜔 (𝜉)− 𝜓 (𝜉),

(6)

where 𝑧 = 𝜔(𝜉), Φ(𝜉) = 𝜑1(𝑧), and Ψ(𝜉) = 𝜓

1(𝑧). 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 𝜑

0(𝜉) and 𝜓

0(𝜉).

2.5. Solution to Displacement 𝑢0under 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, 𝑧 = 𝑧; so

𝜂

𝜃0

𝜌 =1

𝜉

R-area

Figure 2: Plane of conformal transformation.

the soilmass will not cause direct impact on pile deformation,whereas displacement exists at the contact surface of the pilehole boundary and soil mass, namely, |𝑧 + 𝑖𝑎| = 𝑟. The strainof themicropiles soil structure to plane 𝜅 = 3−4𝜇 can be seenin formula (3). In other words, the displacement boundary isas follows:

[(3 − 4𝜇) 𝜑 (𝜉) − 𝜔 (𝜉)𝜑 (𝜉)

𝜔 (𝜉)− 𝜓 (𝜉)]

𝑠0

=𝐸

1 + 𝜇(𝑢 + 𝑖V) ,

[(3 − 4𝜇) 𝜑 (𝜉) − 𝜔 (𝜉)𝜑 (𝜉)

𝜔 (𝜉)− 𝜓 (𝜉)]

𝑠1

= 0,

(7)

where 𝑠1is distance between the soil around the pile and the

center of the pile and 𝑠0is the boundary curve of the pile hole.

When the function of the 𝑧 plane is transformed into afunction of the 𝜉 plane, 𝑧 = 𝜔(𝜉), Φ(𝜉) = 𝜑

1(𝑧), and Ψ(𝜉) =

𝜓

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

𝜑 (𝜉) = −1

8𝜋 (1 − 𝜇)(∑𝐹𝑥+ 𝑖∑𝐹

𝑦) ln𝜔 (𝜉)

+ (𝐵 + 𝑖𝐶) 𝜔 (𝜉) + 𝜑0(𝜉) ,

𝜓 (𝜉) =1

8𝜋 (1 − 𝜇)(∑𝐹𝑥− 𝑖∑𝐹

𝑦) ln𝜔 (𝜉)

+ (𝐵

+ 𝑖𝐶

) 𝜔 (𝜉) + 𝜓0(𝜉) ,

(8)

where 𝜑0(𝜉) = ∑

𝑛

𝑘=0𝑎𝑘𝜉−𝑘 and 𝜓

0(𝜉) = ∑

𝑛

𝑘=0𝑏𝑘𝜉−𝑘. From the

given situation, one can obtain ∑𝐹𝑥

= 𝐹, ∑𝐹𝑦

= 0, and 𝐵 =𝐶 = 𝐶

= 𝐵

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

𝜑 (𝜉) = −1

8𝜋 (1 − 𝜇)𝐹 ln𝜔 (𝜉) + 𝜑

0(𝜉) ,

𝜓 (𝜉) =1

8𝜋 (1 − 𝜇)𝐹 ln𝜔 (𝜉) + 𝜓

0(𝜉) .

(9)


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

[(3 − 4𝜇) 𝜑0(𝜉) −

𝜔 (𝜉)

𝜔 (𝜉)𝜑0(𝜉) − 𝜓

0(𝜉)] = 𝑓,

[(3 − 4𝜇) 𝜑0(𝜉) −

𝜔 (𝜉)

𝜔 (𝜉)𝜑0(𝜉) − 𝜓

0(𝜉)] = 𝑓

1,

(10)

where

𝑓 =𝐸

1 + 𝜇𝑢0−

𝐹

8𝜋 (1 − 𝜇)[ln𝜔 (𝜉) − 𝜔 (𝜉)

𝜔 (𝜉)

+ (3 − 4𝜇) ln𝜔 (𝜉)] ,

𝑓1= −

𝐹

8𝜋 (1 − 𝜇)[ln𝜔 (𝜉) − 𝜔 (𝜉)

𝜔 (𝜉)

+ (3 − 4𝜇) ln𝜔 (𝜉)] .

(11)

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

0(𝜉) and 𝜓

0(𝜉) can be derived (including the

unknown displacement of 𝑢0).

(2)Considering the micropile hole boundary, namely, forthe boundary conditions 𝑠

0, 𝜎 = 𝑒𝑖𝜃, the stress boundary

conditions are to be expressed with components under theorthogonal curvilinear coordinate system at the 𝑧 plane.Namely,

[(3 − 4𝜇) 𝜑0(𝜎) −

𝜔 (𝜎)

𝜔 (𝜎)𝜑0(𝜎) − 𝜓

0(𝜎)] = 𝑓

0. (12)

The Cauchy integral operator (1/2𝜋𝑖) ∮(𝑑𝜎/(𝜎 − 𝜉)) atboth ends of the formula above can be obtained:

(3 − 4𝜇)

2𝜋𝑖∮

𝜑0(𝜎) 𝑑𝜎

𝜎 − 𝜉−

1

2𝜋𝑖∮

𝜔 (𝜎)

𝜔 (𝜎)

𝜑0(𝜎)

𝜎 − 𝜉𝑑𝜎

−1

2𝜋𝑖∮

𝜓0(𝜎)

𝜎 − 𝜉𝑑𝜎 =

1

2𝜋𝑖∮

𝑓0

𝜎 − 𝜉𝑑𝜎.

(13)

In formula (13), ∮(𝜑0(𝜎)𝑑𝜎/(𝜎 − 𝜉)) = −𝜑

0(𝜉),

(1/2𝜋𝑖) ∮(𝜓0(𝜎)/(𝜎 − 𝜉))𝑑𝜎 = 0; so by substituting into

formula (13), one can find 𝜑0(𝜉).

Similarly, according to formula (12), the value of 𝜓0(𝜉)

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 𝑢

0.

2.6. Solution to 𝜑0(𝜉) and 𝜓

0(𝜉). To obtain the boundary

conditions of a miniature pile, make 𝜉 = 𝜎𝜌 and unfold 𝜑(𝜉)and 𝜓(𝜉) in a Laurent series form; namely [17],

𝜑 (𝜉) = 𝑎0+

𝑛

∑

𝑘=1

𝑎𝑘𝜉𝑘

+

𝑛

∑

𝑘=1

𝑏𝑘𝜉−𝑘

,

𝜓 (𝜉) = 𝑐0+

𝑛

∑

𝑘=1

𝑐𝑘𝜉𝑘

+

𝑛

∑

𝑘=1

𝑑𝑘𝜉−𝑘

.

(14)

Depending on formula (4), one can get the following:

𝜔 (𝜎)

𝜔 (𝜎)= −

1

2

(1 + 𝜌𝜎) (𝜎 − 𝜌)2

𝜎2 (1 − 𝜌𝜎). (15)

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

1of formula (7), one can get the following:

𝑐0= −𝑎0−

1

2𝑎1−

1

2𝑏1,

𝑐𝑘= −𝑏𝑘+

1

2(𝑘 − 1) 𝑎

𝑘−1−

1

2(𝑘 + 1) 𝑎

𝑘+1,

𝑑𝑘= −𝑎𝑘+

1

2(𝑘 − 1) 𝑏

𝑘−1−

1

2(𝑘 + 1) 𝑏

𝑘+1.

(16)

Make 𝑓(𝜉) = 𝑓(𝛼𝜎) = 2𝐺(𝑢 + 𝑖V) and 𝑓∗(𝛼𝜎) = (1 −𝛼𝜎)𝑓(𝛼𝜎) = ∑

∞

𝑘𝐴𝑘𝜎𝑘. Formulas (14), (15), and (17)–(19) can

be substituted into the boundary conditions 𝑠1of formula (7);

by eliminating 𝑐𝑘and 𝑑

𝑘, one finds 𝑎

𝑘and 𝑏𝑘:

(1 − 𝛼2

) (𝑘 + 1) 𝑎𝑘+1

− (𝛼2

+ 𝜅𝛼−2𝑘

) 𝑏𝑘+1

= (1 − 𝛼2

) 𝑘𝑎𝑘− (1 + 𝜅𝛼

−2𝑘

) 𝑏𝑘+ 𝐴−𝑘

𝛼𝑘

,

(1 + 𝜅𝛼2𝑘+2

) 𝑎𝑘+1

+ (1 − 𝛼2

) (𝑘 + 1) 𝑏𝑘+1

= 𝛼2

(𝛼2

+ 𝜅𝛼2𝑘

) 𝑎𝑘+ (1 − 𝛼

2

) 𝑘𝑏𝑘+ 𝐴1+𝑘

𝛼𝑘+1

,

(1 − 𝛼2

) 𝑎1− (𝜅 + 𝛼

2

) 𝑏1= 𝐴0− (𝜅 + 1) 𝑎

0,

(1 + 𝜅𝛼2

) 𝑎1+ (1 − 𝛼

2

) 𝑏1= 𝐴1𝛼 + 𝛼2

(𝜅 + 1) 𝑎0.

(17)

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

0, since 𝑎

0represents

rigid body displacement; thus, no stress will be generatedand it can be deemed as 0. By 𝑘 times of iteration, one canobtain 𝑎

𝑘; thus, to get 𝜑

0(𝜉) and 𝜓

0(𝜉), one must substitute

𝜑0(𝜉) and 𝜓

0(𝜉) in the complex function of the fundamental

equation, to ascertain the stress and displacement field ofthe soil around a pile of micropiles under horizontal load.According to the assumed conditions specified in Sections2.1 and 2.2, the relationship between horizontal load and thehorizontal displacement can be obtained from 𝐾 = 𝐹/𝑢

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,


Micropile

Sloping surface

Roof beam

Land

slide s

urface

Figure 3: Micropile composite structure set on step of slope.

a pile groups’ layout will be adopted. Especially for multistageslopes, platforms are available for each grade. Miniature pilescan be set at the platform for reinforcement. The multiple,miniature piles that are exposed on the platform can be fixedwith a top beam to enhance their sliding resistance force.Themicropile layout is shown in Figure 3.

3.1. 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 calculation,theWinkler low econometricmodel is adopted.This researchassumes that the soil around the micropile has a series ofdiscrete springs and a rigidity coefficient of 𝐾 = 𝐹/𝑢

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.

3.2. 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, namely,with 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 structure,the 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).

3.2.1. 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 𝑧 = 0; the bendingmoment is 𝑀 = 0; and 𝐹 is horizontal shear. The boundarycondition on the bottom of the micropile is 𝑧 = 𝐻, the angleis 𝜑 = 0, and the horizontal displacement is 𝑢 = 0. Theflexural differential equation of an Euler-Bernoulli beamwithone fixed and one sliding end is [19]

𝐸𝐼𝑑4

𝑢

𝑑𝑧4+ 𝐾𝑢 (𝑧) = 0. (18)

In the formula above, 𝐸 is the elasticity modulus of theminiature pile, 𝐼 is the inertia moment of a micropile crosssection,𝐾 is the rigidity coefficient of an assumed spring, and𝑢(𝑧) is the horizontal displacement of the soil around the pilealong the pile body.


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# 3#

4# 5# 6#meter

0.5m

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:

𝑢 = 𝑒𝜆𝑧

[𝑐1cos (𝜆𝑧) + 𝑐

2sin (𝜆𝑧)]

+ 𝑒−𝜆𝑧


4sin (𝜆𝑧)] ,

𝑀 = 2𝜆2

𝐸𝐼𝑒𝜆𝑧

[𝑐2cos (𝜆𝑧) − 𝑐

1sin (𝜆𝑧)]

+ 2𝜆2

𝐸𝐼𝑒−𝜆𝑧

[𝑐4cos (𝜆𝑧) − 𝑐

3sin (𝜆𝑧)] ,

(19)

where 𝜆 = (𝐾/4𝐸𝐼)1/4 and 𝑐1, 𝑐2, 𝑐3, and 𝑐

4are the integral

constant.

3.2.2. 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]

𝐸𝐼𝑑4

𝑢

𝑑𝑧4+ 𝐾𝑢 (𝑧) = 𝑞 (𝑧) = 0, (20)

where 𝑞(𝑧) is the uniform load of the soil around the pile. Tosolve this equation, 𝜆 = (𝐾/4𝐸𝐼)1/4 can be substituted intoformula (19) to get the following:

𝑑4

𝑢

𝑑𝑧4+ 4𝜆𝑢 (𝑧) =

𝑞

𝐸𝐼. (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

∅150

steel pipe

Diameter 50mm-pipe

B4000

Figure 7: Micropiles sectional drawing.

Figure 8: The field tests of micropile.

be obtained. The general solution 𝑞(𝑧) = 0 and specialconnection is 𝑞(𝑧); that is,

𝑢 = 𝑒𝜆𝑧


2sin (𝜆𝑧)]

+ 𝑒−𝜆𝑧


4sin (𝜆𝑧)] +

𝑞

𝐾.

(22)


Figure 9: The horizontal static load test of micropiles.

3.2.3. The Determination of the Elastic Modulus (𝐸). In theprevious calculation, 𝐸 is a crucial parameter, usually deter-mined by laboratory experiment and experimental relation-ship of previous practice. However, as laboratory experiment,it is very difficult to reproduce the soil pile in live loadeffect on the stress path, and experiential relationship cannotdetermine the value of 𝐸 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 applied.The 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

4.1. Engineering Examples. In this example, the landslide ofa highway is medium type, about 80–100m wild and 160mlong. The front is about 5m thick, whereas the central partis 10–15m thick and the back part is 3.5–8.0m thick. Theaverage thickness of the landslide is about 10m and thetotal volume of landslide is about 160,000m3. 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 cracks.Specific to the characteristics of this landslide, it is proposedto adopt a light, retaining structure with a ministeel tub pilescomposite structure for reinforcement.

4.2. 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 sides.Figure 6 shows the model demonstration diagram.

4.2.1. 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 50mm.Themainreinforcement pile contains 3 roots, made up of 28 reinforcedsteel pipes.Themicrocap sets the C30 concrete capping beamwith a beam that is 0.5m high and 1.5m 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 damaged,indicating 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 load.Themicropile section and testprocess are shown in Figures 7, 8, and 9.

4.2.2. 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 surface,the horizontal displacement is basically 0, which indicatesthat the anchorage effects at the anchorage section are com-paratively better.When consistently increasing the horizontalload, the variable quantities of the displacement of pile top ofeach row of piles are the same.This 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 load.The bending momentand the displacement distribution laws of the three-row pilesare similar. A bendingmoment above 0.25m is 0. No bendingdeflection of the micropile is generated, due to the constraint


146.7KPa133KPa119.7KPa106.4KPa93.1KPa

79.8KPa66.5KPa53.2KPa39.9KPa26.6KPa13.3KPa

−8

−6

−4

−2

0H

igh

(m)

2 4 6 80Displacement (mm)

(a) The first row of piles



0Displacement (mm)

2 4 6 8

−8

−6

−4

−2

0

Hig

h (m

)

(b) The second row of piles



2 4 6 80Displacement (mm)

−8

−6

−4

−2

0

Hig

h (m

)

(c) The third row of piles

Figure 10: Comparative curves of distribution of pile deflection.


Landslide surface



−8

−6

−4

−2

0Pi

le le

ngth

(mm

)

0 5 10−5−10Moment (KN, m)


Landslide surface



−8

−6

−4

−2

0

Pile

leng

th (m

)

0 5 10−5−10Moment (KN, m)


Landslide surface

−8

−6

−4

−2

0

Pile

leng

th (m

)

5−5 0 10−15 −10Moment (KN, m)




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

The first row pile −The second row pile −The third row pile −

−10

−5

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.


−12

−10

−8

−6

−4

−2

0H

igh

(m)

1 2 3 4 50Displacement (mm)

119.7KPa (calculation)119.7KPa (test)79.8KPa (calculation)

79.8KPa (test)26.6KPa (calculation)26.6KPa (test)



−12

−10

−8

−6

−4

−2

0

Hig

h (m

)




−12

−10

−8

−6

−4

−2

0

Hig

h (m

)





Figure 12: Comparative curves of the distribution of pile deflection.

of the cap lid. The point of contraflexure occurred at 0.9mabove the sliding surface and 2.5m 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 0.5m below the sliding surface. The value ofbending moment increases along with the horizontal load.As the horizontal load increased to grades 9–120KPa (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.

4.3. 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 26.6 KPa,79.8 KPa, and 119.7 KPa, as well as the bendingmoment under26.6 KPa and 119.7 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


119.7KPa (test) 119.7KPa (calculate)26.6KPa (test) 26.6KPa (calculate)

−8

−6

−4

−2

0

Hig

h (m

)

20 4 6−4−6 −2−8Moment (KN, m)

0.4

0.2

0.0

0.6

−0.

2

−0.

8−

0.6

−0.

4

Moment (KN, m)

0.4

0.2

0.0

0.6

−0.

2

−0.

8−

0.6

−0.

4

Moment (KN, m)

−8

−6

−4

−2

0

Hig

h (m

)

−8

−6

−4

−2

0

Hig

h (m

)

−8

−6

−4

−2

0

Hig

h (m

)

20 4 6−4−6 −2−8Moment (KN, m)



−8

−6

−4

−2

0H

igh

(m)

−8

−6

−4

−2

0

Hig

h (m

)−8

−6

−4

−2

0

Hig

h (m

)

−8

−6

−4

−2

0

Hig

h (m

)

−6 −4 −2 0 2 4 6−8Moment (KN, m)

−6 −4 −2−8 2 4 60Moment (KN, m)0

.0 0.2

0.4

−0.

4−

0.6

−0.

2

−0.

8

Moment (KN, m)

−0.

8−

0.6

Moment (KN, m)

−0.

4−

0.2

0.0

0.2

0.4



Moment (KN, m)

−10

−5

0

Hig

h (m

)

−8

−6

−4

−2

0

Hig

h (m

)

−8

−6

−4

−2

0

Hig

h (m

)

−8

−6

−4

−2

0

Hig

h (m

)

20 4 6−4−6 −2−8Moment (KN, m)

20 4 6−4−6 −2−8Moment (KN, m)−

0.8

−0.

6

Moment (KN, m)

−0.

4−

0.2

0.0

0.2

0.4

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4


Figure 13: Comparative curves of the distribution of the pile model.


in simulated field experiments under each grade of load,which 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 beam,the pile deformation and internal force calculationmethods of micropile composite structures underhorizontal loads can be derived using two modes,which 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

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[2] D. A. Bruce, A. F. Dimillio, and I. Juran, “Introduction tomicropiles: an international perspective,” in Proceedings of theConference on Geotechnical Engineering Division of the ASCEin Conjunction with the ASCE Convention, pp. 1–26, San Diego,Calif, USA, 1995.

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