research article three-dimensional stability...

Research ArticleThree-Dimensional Stability Analysis ofa Homogeneous Slope Reinforced with Micropiles

Shu-Wei Sun Wei Wang and Fu Zhao

Faculty of Resources and Safety Engineering China University of Mining and Technology Beijing 100083 China

Correspondence should be addressed to Shu-Wei Sun ssw1216163com

Received 12 August 2014 Accepted 30 September 2014 Published 21 October 2014

Academic Editor Evangelos J Sapountzakis

Copyright copy 2014 Shu-Wei Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Micropiles are widely used to reinforce slopes due to their successful performance and fast construction In this study a simplenonlinear method is proposed to analyze the stability of a homogeneous slope reinforced with micropiles This method is based onshear strength reduction technique in which the soil behavior is described using the nonassociated Mohr-Coulomb criterion andmicropiles are modeled as 3D pile elements A series of slope stability analyses is performed to investigate the coupled mechanismof micropile system and the optimum of pile position depth of embedment and length of truncation are analyzed Results showthat the position of micropile system plays an important role not only in the calculation of the safety factor but also in locating thefailure surface which demonstrates the dominating coupled effect exists between micropiles and slope The critical embedmentdepth of the micropile is about 2 times the length of micropile above the critical slip surface and the micropiles flexure rather thanrotation becomes increasingly prevalent as the depth of micropiles embedment increases Truncation of micropiles may improvethe capacity of the micropile system and the largest truncation length of micropile is about 14 depth of critical slip surface in thisstudy

1 Introduction

The use of micropiles to stabilize an unstable slope has beenwidely adopted in recent years (eg [1ndash6]) A micropile isa small-diameter (typically less than 300mm) drilled andgrouted nondisplacement pile that is typically reinforcedMicropiles can withstand relatively significant axial loads andmoderate lateral loads and can be constructed in almost anytype of soilrock conditions Compared to conventional anti-sliding piles micropiles construction is relatively simple fastenvironmental-friendly and economic Besides micropilescan be easily installed in areas with limited equipment accesssuch as for landslides located in steep hilly or mountainousareas (see Figure 1) Drilled micropiles can also be combinedwith other slope-stabilization techniques such as retainingwall and use of ground anchors

Different methods have been used to evaluate the per-formance and design of the micropiles which are used asreinforcement in slopes Lizzi [1] suggests that micropilescan be used as the reticulated network system which creates

a stable reinforced soil as ldquogravity-retaining wallrdquo and thereinforced soil gravity mass supplies the essential lateralloads due to the movement of the unstable slope Reeseel al [7] described a procedure for the calculating of theresistance provided by micropiles assuming that the limitstate is the failure of the micropile in bending Loehr et al[4] also proposed a simplifiedmethod for predicting the limitresistance of recycled plastic reinforcement for applicationto stabilization of minor slopes In their method two failuremechanisms are considered to determine the distribution oflimit lateral resistance along the reinforcing members failureof soil around or between reinforcingmembers and structuralfailure of the reinforcing member due to mobilized forcesfrom the surrounding soil The comparative tests betweenmicropiles and conventional antisliding piles conducted bySun et al [8] showed that micropiles were totally differentfrom antisliding pile in loading mechanism Therefore thesolution for conventional piles cannot be easily adaptedfor the situation of micropiles Sun et al [9] developed alimit equilibrium approach for the design of micropiles and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 864017 11 pageshttpdxdoiorg1011552014864017

2 Mathematical Problems in Engineering

(a) (b)

Figure 1 Micropiles for slope stabilization

the design procedure includes (1) choosing a location for themicropiles within the existing slope (2) selecting micropilecross section (3) estimating the length of the micropile (4)evaluating the shear capacity of the micropiles group (5)calculating the spacing required to provide force to stabilizethe slope and (6) the design of the concrete cap beam

Esmaeili et al [6] presented three experimental modelsof embankments of 10m in height on a scale of 120 to set upa number of loading tests For comparison three numericalmodels were developed by using the PLAXIS-3D code basedon the FEM Sensitivity analysis on the geometric parametersof micropiles was conducted to investigate the numberspacing diameter and length with the embankment safetyfactor It was shown that with the optimum arrangement ofthemicropiles the safety factor of the slope stability and staticload-bearing capacity of the embankments will increase bymore than 30 and 65 respectively and also the settlementof the embankment crest will decrease by approximately 35Isam et al [10] presented a numerical method for the analysisof inclined micropiles In this method the micropiles aremodeled as 3D elastic beam elements rigidly connected toa cap which is free of contact with the soil Then it is showedthat the inclination ofmicropiles leads to a goodmobilizationof their axial component and consequently to an importantdecrease in the shearing and bending loading In general thesoil-pile interaction for such micropiles is very complex andthere are many factors whose influence on the final behaviorof the structure cannot be conveniently assessed Thus afurther study of the behavior of micropiled-reinforced slopesis necessary

The shear strength reduction technique has been used inthe stability analysis of slopes without piles by Zienkiewiczet al [11] Matsui and San [12] Ugai and Leshchinsky [13]Dawson et al [14] Griffiths and lane [15] Won et al [16] Linand Cao [17] and others For piled reinforced slopesWei andCheng [18] have considered the effects of piles on the stabilityof a slope by a three-dimensional finite element analysis usingthe shear strength reduction technique Won et al [16] andCai and Ugai [19] have analyzed the same slope by Wei andCheng [18] In the piled slope analysis by Won et al [16] andWei and Cheng [18] the location of the critical slip surfacewas determined by the maximum shear force in the pile so

A

B C

1205903

120590 3minus 120590

1= 0

120590tfs = 0 ft = 0

2c

radicN120593

c

tan120593

1205901

Figure 2 Composite Mohr-Coulomb failure criterion

that a very deep critical slip surface was determined whilethe maximum shear strain in the soil was not considered CaiandUgai [19] have discussed the use ofmaximum shear strainand maximum shear strain rate to determine a more realisticcritical slip surface

In this paper an analyticalmodel is developed to calculatethe safety factor of a slope reinforced with micropiles basedon the shear strength reduction method The homogeneousslope considered by Won et al [16] and Wei and Cheng[18] and Cai and Ugai [19] is employed for the analysis Thesoil behavior is described using the nonassociated Mohr-Coulomb criterion and micropiles are modeled as 3D pileelements As such a series of slope stability analyses isperformed and the optimization of pile position depth ofembedment and length of truncation are discussed

2 Analysis Method

21 Model of Materials In the present study the soil materialof the slope is simulated with an elastic perfectly plasticmodel with Youngrsquos modulus being 119864

119904before yielding

The yielding is described by a composite Mohr-Coulombcriterion with a tension cutoff as shown in Figure 2

The failure envelope from point 119860 to 119861 is defined by theMohr-Coulomb criterion 119891119904 = 0 with

119891119904

= 1205901minus 1205903119873120593+ 2119888radic119873

120593 (1)

Mathematical Problems in Engineering 3

1

|Fs|

L

|Fmaxs |

L

ks

|us|

(a) Shear forcelength versus relative shear displace-ment 119906

119904

(tension) (compression)

|Fmaxs |

L

120593s

120590mp

Cs

(b) Shear strength criterion

Figure 3 Shear-directional material behavior for pile elements

1

|Fn|

L

|Fmaxn |

L

kn

|un|

g = on g = off

(a) Normal forcelength versus relative nor-mal displacement 119906

119899


|Fmaxn |

L

120593n

120590mp

Cn

(b) Normal-strength criterion

Figure 4 Normal-directional material behavior for pile elements

where 1205901and 120590

3= major and minor principal effective

stresses respectively 119888 = effective cohesion of soil 120593 =internal friction angle of soil and119873

120593= function of 120593

119873120593=1 + sin1205931 minus sin120593

(2)

The failure envelope from point 119861 to 119862 is represented bythe tension failure criterion of the form 119891119905 = 0 with

119891119905

= 1205903minus 120590119905

(3)

where 120590119905 = effective tensile strength whose maximum value120590119905

max is given by

120590119905

max =119888

tan120593 (4)

A ldquopilerdquo element is used to simulate a micropile behaviorThe ldquopilerdquo element offers the combination features of beamand cable In this sense the ldquopilerdquo element can simulate thecombination of tension shearing and bending behavior ofmicropiles Piles interact with the grid through shear andnormal coupling springs which are cohesive and frictional innature as well as nonlinear

The mechanical characteristics of shear and normalcoupling springs are shown in Figures 3 and 4 respectivelywhich are located at the nodal points along the pile axisto describe relative shear and normal displacement betweenthe pilegrout interface and the groutrock interface Thesesprings present numerically as spring-slider connecterstransferring forces and motion between the pile and thegrid at the pile nodes The shear behavior of the micropile-grid interface during relative shear displacement 119906

119904between


the micropile and the soil is described numerically by theshear coupling spring properties of stiffness 119896

119904 cohesive

strength 119862119904 friction angle 120593

119904 and exposed perimeter 119901

The normal behavior of the micropile-grid interface duringrelative normal displacement 119906

119899between the micropile and

the soil is described by the normal coupling spring propertiesof stiffness 119896

119899 cohesive strength119862

119899 friction angle120593

119904 exposed

perimeter 119901 and gap-use flag 119892

22 Determination of the Factor of Safety For slopes thefactor of safety (FS) often is defined as the ratio of the actualshear strength to the minimum shear strength required toprevent failure A logical way to compute the factor of safetywith a finite element or finite difference program is to reducethe shear strength until collapse occurs The factor of safetyis the ratio of the rockrsquos actual strength to the reduced shearstrength at failure This shear strength reduction techniquewas used first with finite elements by Zienkiewicz et al [11]to compute the safety factor of a slope composed of multiplematerials

The shear strength reduction technique has two mainadvantages over limit equilibrium slope stability analysesFirst the critical slide surface is found automatically andit is not necessary to specify the shape of the slide surface(eg circular log spiral and piecewise linear) in advanceIn general the failure surface geometry for slopes is morecomplex than simple circles or segmented surfaces Secondnumerical methods automatically satisfy translational androtational equilibrium whereas not all limit equilibriummethods do satisfy equilibrium Consequently the shearstrength reduction technique usually will determine a safetyfactor equal to or slightly less than limit equilibriummethods

To perform slope stability analysis with the shear strengthreduction technique a series of stability analyses are per-formed with the reduced shear strength parameters 119888trial and120593trial defined as follows

119888trial=1

119865trial119888 120593

trial=1

119865trial120593 (5)

where 119888 and 120593 are real shear strength parameters and 119865trial isa trial factor of safety

Figure 5 shows the proposed analytical model to calculatesafety factor of slopes reinforced with micropiles In Figure 5the initial 119865trial is set to be 10 so as to judge the stability of thesystemThen the value of 119865trial is increased or decreased untilthe slope fails After the slope fails the 119865up or 119865low is replacedby the previous119865trial Next a pointmidway between the upperand lower brackets is tests If the simulation converges theupper bracket is replaced The process is repeated until thedifference between upper and lower brackets is less than aspecified tolerance 120576

There are several possible definitions of failure for exam-ple some test of bulging of the slope profile [20] limitingof the shear stresses on the potential failure surface [21] ornonconvergence of the solution [22] These are discussed inAbramson et al [23] but without resolution In this study thenonconvergence option is taken as being a suitable indicatorof failure

23 Method Execution The proposed method was executedby using a three-dimensional explicit-finite difference pro-gram FLAC3D [24] which is a three-dimensional explicitfinite-difference program that has the ability to simulatethe behavior of soil rock or other materials that mayundergo plastic flow when their yield limits are reachedThe explicit Lagrangian calculation scheme and the mixed-discretization zoning technique used in FLAC3D facilitateaccurate modeling of plastic collapse and flow The explicitLagrangian solution scheme involves a large number ofcalculation steps each progressively redistributing throughthe mesh the unbalanced force resulting from changes to thestresses or boundary displacements

The convergence criterion for FLAC3D is the ratiodefined to be the maximum unbalanced force magnitudefor all the gridpoints in the model divided by the averageapplied force magnitude for all the gridpointsThemaximumunbalanced force is the magnitude of the vector sum ofnodal forces for all of the nodes within the mesh Themodel is considered to be in equilibriumwhen themaximumunbalanced force is small in comparison with the totalof the applied forces associated with stress or boundarydisplacement changes If the unbalanced force approaches aconstant nonzero value this usually indicates that failure andplastic flow are occurring within the model The model isnormally assumed to be in equilibrium when the maximumunbalanced force ratio is below 1 times 10minus5

3 Validation and Application of the Method

Wei and Cheng [18] performed a numerical analysis toinvestigate the effect of stabilizing piles on the stability ofa slope For the slope as shown in Figure 6 it is 10m inheight with a gradient of 1 V 15H and a ground thicknessof 10m The material properties for prediction purposeswere selected based on Cai and Ugairsquos assumptions and thecohesive strength friction angle elastic modulus Poissonratio and unit weight of the soil are 10 kPa 20∘ 200MPa025 and 20 kNm3 respectively Two symmetric boundariesare used and nodes on the symmetric faces were preventedfrom moving in the y-direction but free to move in thex- and z-directions The nodes on the right- and left-handfaces were restrained from moving in the x-direction whileboth horizontal and vertical displacements are fixed along thebottom boundary

The factor of safety of the slope is 116 using the proposedmethod with program FLAC3D and the critical slip surfaceis shown in Figure 7 Cai and Ugai [19] have discussed thatthe use of maximum shear strain and maximum shear strainrate in the nonlinear iteration will give virtually the sameresult in most cases and the adoption of the maximum shearstrain in the present study is in line with the flow rule relationThe Cai and Ugairsquos shear strength finite element method andthe Bishoprsquos simplified method gave safety factors of 114 and113 respectively It should be noted that the final calculatedfactor of safety by finite difference methods depends highlyon the size of element unlike finite elementmethods in whicha shape function can be used within the elements In general


No

Yes

Yes

No

No

No

Yes

Yes

No

Yes

Input c 120593 T 120576

Ftrial = 1

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

Flow = Ftrial

Flow = Ftrial

Flow = Ftrial

Ftrial = 2Ftrial

Fup = Ftrial

Fup = Ftrial

Fup = Ftrial

Ftrial =Ftrial

2

Fup = Fup

Fup = Fup

Flow = Flow

Flow = Flow

Ftrial =Flow + Fup

2

Fup minus Flow lt 120576

F = Ftrial

Figure 5 Analytical model for obtaining the 119865119904of the reinforced slope

L = 15m

10m

10m

20m

10 m

x

y

z

Lx

Figure 6 Slope model and finite difference mesh

the finer the size of element is the more precise the resultis So the authors view that the factor of 116 is a slightlybetter result for the present problemThus the overall modelperformance in predicting the factor of safety of the slope isquite satisfactory adding to confidence in its validity

Figure 7 Slip surface of the slope (119865119904= 116)

Four steel tube micropiles with an outer diameter of119863 = 015m cover a group in slope stabilization as shownin Figure 8 The micropiles are installed in the middle of theslope that is the horizontal distance between the slope toe


Figure 8 Slip surface of the reinforced slope (119865119904= 145)

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

09 10 11 12

120575

Dep

th (m

)

Figure 9 Values of 120575 for different depth

and the pile position 119871119909= 75 and the center-to-center

spacing among micropiles 119878 = 1198781015840 = 3D Micropiles aremodeled as 3D elastic pile elements as above and a roughmicropile surface is assumed of stiffness 119896

119904= 119896119899= 13 times

1011 Pa cohesive strength 119862

119904= 15 times 10

10 Pa and 119862119899=

10 times 104 Pa and friction angle 120593

119904= 120593119899= 20∘

When the slope is reinforcedwithmicropiles the factor ofsafety is 145 using an associated flow rule with the proposedmethod The approximate critical slip surface can be visuallyobserved from the shear strain contour as shown in Figure 8

The design of micropiles particular for slope stabilizationusually dictates the need for groups of closely spaced pilesWith conventional micropiles there is a compromise to beresolved between the desire to select a closemicropile spacingand on the other hand the need to maintain a certainminimum interpile spacing so as to avoid the group effectnecessitating a reduction in the nominal capacity of eachpile On the basis of the experimental studies conducted by

Lieng [25] for 119878119863 values greater than 3 which are generallyused in micropile design practice the reduction accountfor the group effect is negligible However if the directionof loading is in line with the interaction of micropiles theinteraction is not a simple function but depends greatly onthe relative positions of the micropiles Such effects have notbeen examined in this paper

Special analysis was performed to verify the micropilespacing that can generate sufficient soil arching and achievebetter deformation compatibility between the micropiles andthe enclosed soil mass The index 120575 was defined with adimensionless ration of interpile ground displacement 119906

119904

to the maximum displacement of the micropile heads 119906pi atcollapse If this ratio is maintained between 1 and 2 (at most)micropiles and the inner-micropile soil are displaced bynearly the same amount and micropiles could be consideredto be effective in terms of arching

Figure 9 shows the relationship of the index 120575 of thesystem for different depth of the slope in Figure 8 The 120575values range between 10 and 11 along the depth whichdemonstrated that a highly composite system could beformed with 119878 = 1198781015840 = 3D in this study Micropiles andthe micropile enclosed soil mass behave as a coherent bodyduring the reinforced slope failure

4 Results and Discussions

To simplify the representations of results the factor of safetyof micropile reinforced slopes is defined with a nondimen-sional parameter called improvement factor (If) given by

119868119891=

FSreinforcedFSunreinforced

(6)

where FSreinforced and FSunreinforced are factors of safety forreinforced and unreinforced slopes respectively

41 Effect of Micropile Position The position of the pile isvery important and its effect has been discussed by severalauthors [18 19] who came to the similar conclusion thatthe improvement of the safety factor will be largest whenthe piles are installed in the middle of the slopes Howeverthe comparative tests between micropiles and conventionalantisliding piles conducted by Sun et al [8] showed thatmicropiles were totally different from antisliding pile withregard to the loading mechanism With larger flexural rigid-ity conventional piles suffered inclination deformation resultfrom compression fracture of soil behind pile With smallflexural rigidity micropiles suffered flexible deformationwhich also made the plastic zone of soil among micropilescross and overlap so the larger lateral displacement occurredat the sliding surface and on the top of the micropileTherefore the conclusion for conventional piles cannot beeasily adapted to micropiles

The micropile system positions in the slope are indicatedwith a dimensionless ratio of the horizontal distance betweenthe slope toe and themicropile positions119871

119909 to the horizontal

distance between the slope toe and slope shoulder 119871 asshown in Figure 6 The influence of the system positions


00 02 04 06 08 10

110

115

120

125

130

I f

LxL

Figure 10 Effects of system positions on slope stability

on the safety factor of the homogeneous slope is shown inFigure 10 The numerical results obtained with the proposedmethod show that the improvement of the safety factor ofslopes reinforcedwithmicropiles is largest when the system isinstalled in the middle-upper part of the slope In the presentstudy the factor of safety will be the optimal solution whenthe system position is 119871

119909119871 = 07 Therefore the system in

slope stabilization should be placed slightly closer to the topof the slope for the largest safety factor The reason for this isthat when the system is placed in the middle-upper part ofthe slopes the shear strength of the soil-micropile interface issufficiently mobilized

The critical slip surfaces and the factors of safety for differ-ent system positions are shown in Figure 11The approximatecritical slip surface can be visually observed from the shearstrain contour When 119871

119909119871 = 01 a clear single critical slip

surface is shown in Figure 11(a) which is nearly the same asthe critical slip surface for unreinforced slope This meansthat micropiles placed in the toe of slope has little effect onthe overall stability and the slope occurs sliding over the pileheadWhen 119871

119909119871 = 03 (Figure 11(b)) the critical slip surface

is divided into two parts due to the presence of the micropilebut the shear strain in the upper part is mobilized at the limitstateThismeans that the overall safety factor of the slope willbe controlled by the upper part of the critical slip surfaceWhen 119871

119909119871 increases from 05 to 07 (Figures 8 and 11(c))

the critical slip surface is also divided into two parts but thetwo parts of the critical slip surfaces get deeper due to theshear strain mobilization along the vertical direction at theinterface between the micropile and soil A clear shear strainmobilization in the vertical direction is found at themicropilelocation When 119871

119909119871 = 09 (Figure 11(d)) with the micropile

located in the vicinity of the crest of the slope the lower partof slope away from themicropiles occurs overall failure Andit can be seen that the critical slip surface is clearly one singlecritical slip surface but the critical slip surface is shallowerthan that with no micropile

It is obvious that the critical slip surface can change dueto the addition of the micropile which demonstrates the

coupled effect exists between micropiles and slope systemTherefore the uncoupled analysis which can only considera fixed failure surface should be limited in its application inhomogeneous slope stabilization with micropiles

42 Effect of Micropile Embedment It should be noted thatmicropile embedment influences the ultimate resistance forceoffered by the system thus the depth ofmicropile embedmentwill have a great influence on the stability of the micropiled-slope The critical embedment depth of micropile will beinvestigated in the present study

The length of micropile above the critical slip surface119867119886and below the critical slip surface 119867

119887 can be obtained

approximately from Figure 7 and based on the micropilelocation as shown in Figure 8 respectivelyThe relative depthof micropile embedment is indicated with a dimensionlessratio of the119867

119887and119867

119886

When micropile system is installed with the horizontaldistance between the slope toe and the micropile position119871119909of 75m the effect of micropile embedment on safety

factor is shown in Figure 12 As expected the safety factorincreases significantly as the depth of micropile embedmentincreases It is noticed that the safety factor of the reinforcedslope increases rapidly when the relative depth of micropileembedment 119867

119887119867119886increases from 10 to 20 But the safety

factor increases slightly when the relative depth of micropileembedment 119867

119887119867119886increases beyond 20 This implies that

there exists a critical embedment depth 119867119887 which is of the

order of 20 119867119886in this case This is explained by the fact

that the embedment is enough to provide adequate fixityconditions Therefore for homogeneous slope stabilizingdesign the micropile length in the stable layer should notexceed the critical embedment length of 2 times the lengthof micropile above the critical slip surface But it is notedthat the strength of the stable layer influences the criticalembedment depth and the increase of strength of the stablelayer will unavoidably be associated with a decreased criticalembedment depth

It is clear from Figure 13 that micropile system behavesdifferent from the embedment length Snapshots of deformedmesh are compared in Figure 13 for the four extremes119867119887119867119886= 10 15 20 and 25 The deformed mesh gives a

rather diffuse indication of the failure mechanism For thesmall embedment length (Figure 13(d)) the system behavesrigid and its response resembles that of a caisson The defor-mation of the system is dominated by rigid-body rotationwithout substantial flexural distortionThis finding is consis-tent with Poulosrsquos [26] description of the short conventionalpile mode of failure which involves mobilization of the stablesoil strength and failure of the soil underneath the pileThere-fore the micropile capacity is not adequately exploited in thiscase and such a design would not be economical To utilizethe full micropile system capacity a larger embedment depthis required so does the ability of the stable layer to providefixity conditions As the length of micropiles embedmentincreases micropiles flexure rather than rotation becomesincreasingly prevalent as vividly portrayed in Figures 13(a)and 13(b)


Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09

Figure 11 Effects of system positions on failure mode of slope

05 10 15 20 25 3010

11

12

13

HbHa

I f

Figure 12 Effects ofmicropiles embedment length on slope stability

43 Effect of Truncation of Micropiles The truncation ofmicropiles likely increases the capacity of the micropilesystem (Figure 14) because the moment development in atruncated micropile is reduced due to the shortened momentarm on which the load acts But truncation of micropiles willdecrease the safety factor of reinforced slope Therefore thelargest truncation length of a micropile beyond which theslopemay occur sliding over the pile head is verymeaningful

and important for the detailed design of a micropile systemsupported slope In this paper the relative length ofmicropiletruncation is investigated with a dimensionless ratio of thelength of micropile truncation 119862

119909 to the length of micropile

above the potential slip surface119867119886as shown in Figure 14

The effect of the truncation of micropiles on the safetyfactor of the slope stabilized with micropiles is shown inFigure 15 As expected the rate of decrease in the safety factorincreases with increasing the truncation length of micropilesIt can be seen that the safety factor of the reinforced slopedecreases slightly when the relative length of micropilestruncation 119862

119909119867119886increases from 0 to 14 However the

safety factor of the reinforced slope decreases rapidly whenthe relative length of micropiles truncation 119862

119909119867119886increases

above 14This implies that there exists the largest truncationlength of micropiles which is of the order of 14 119867

119886in this

caseThedifferences in the safety factor between different trun-

cation lengths conditions can be explained by the shearingforce 119876 in the micropiles at collapse as shown in Figure 16The shearing force is positive when its direction is identicalto that of the sliding of the slope that is opposite to the x-direction in Figure 6

The lateral shearing force along the x-direction in themicropile reaches the first extreme point at a critical depthwhich can be regarded as the level of the slip surfaceaccording to results of Cai and Ugai [19] and Won et al [16]It can be seen that the shearing force in the micropiles at


(a) (b)

(c) (d)

Figure 13 Deformation of reinforced slopes with micropiles

Ha

Cx

Figure 14 Truncation of piles embedded within a slope

the potential slip surface decreases slightly when the relativelength of micropiles truncation 119862

119909119867119886increases from 18 to

14 but it decreases rapidly when 119862119909119867119886increases beyond

14 It is noticed that the shearing force in themicropiles at thepotential slip surface is identical to the resisting reaction forceto the sliding body Therefore the larger the shearing forcein the micropile at the potential slip surface the larger thereaction force to the sliding body supplied by the micropileand the higher the safety factor of the slope reinforced withmicropilesThis conclusion agrees well with the above resultsof safety factor

Figure 16 also shows that themaximum bendingmomentoccurs below the potential slip surface for the micropile The

00 02 04 06 08 1010

11

12

13

I f

CxHa

Figure 15 Effect of truncation of micropiles on slope stability

value and depth of the maximum bending moment decreasewith increasing the truncation length of micropile Becauseit is more possible for the micropile to be yielded by thebending moment than by the shearing force this benefitwill be particularly important for the slope reinforced withmicropiles For truncated micropiles the portion of boreholecould be filled with surrounding soil and compacted inthe standard procedure for lesser consumption of reinforcingsteel bar


16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)

minus250 minus200 minus150 minus100 minus50

Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)

minus60 minus40 minus20

Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)

Figure 16 Micropile behavior characteristics

5 Conclusions

A simple nonlinear method has been proposed to analyze thestability of a homogeneous slope reinforced with micropilesThis method is based on shear strength reduction techniquein which the soil behavior is described using the nonassoci-atedMohr-Coulomb criterion andmicropiles are modeled as3D pile elements

The reliability of the proposed three-dimensional modelhas been tested using an example fromWon et al [16] andCaiand Ugai [19] andWei and Cheng [18]The value of the factorof safety of the nonreinforced slope is equal to 116 which isclose to the Cai and Ugairsquos results with safety factors of 114and 113 for finite element and the Bishoprsquos simplifiedmethodrespectively It was found that micropiles could improve sig-nificantly the stability of the slope and the micropile spacing119878 = 3D can generate sufficient soil arching and achieve betterdeformation compatibility between the micropiles and theenclosed soil mass Micropiles and the micropiles enclosedsoil mass behave as a coherent body during the reinforcedslope collapse The authors have proposed an index 120575 toinvestigate the arching and deformation compatibility of thesystem

Our analysis demonstrates that the optimal position ofmicropile system is located in the middle-upper part of theslopes Besides it was found that the position of micropilesystem plays an important role not only in the calculation ofthe safety factor but also in locating the failure surface whichdemonstrates the dominating coupled effect exists between

micropiles and slope Therefore the uncoupled analysiswhich can only consider a fixed failure surface should belimited in its application in homogeneous slope stabilizationwith micropiles

The results of this study demonstrate that to obtain thefull micropile system capacity a larger embedment depthof the micropile is required so does the ability of thestable layer to provide fixity conditions The value of safetyfactor will increase with the increase of embedment depthwhen the relative depth of micropile embedment 119867

119887119867119886le

20 However the safety factor of reinforced slope reachesalmost stable values when 119867

119887119867119886gt 20 Therefore for the

homogeneous slope in this study the micropile length inthe stable layer should not exceed the critical embedmentlength of 2 times the length ofmicropile above the critical slipsurface Moreover it was found that the micropiles flexurerather than rotation becomes increasingly prevalent as thedepth of micropiles embedment increases

It should be note that truncation of the micropile mayimprove the capacity of the micropiles system and thelargest truncation length of micropiles is about 14 depth ofcritical slip surface in this study In practice engineering theupper boreholes for truncated micropiles could be filled withsurrounding soil and compacted in the standard procedurefor lesser consumption of reinforcing steel bar

The current three-dimensional slope stability study isbased on a simple homogeneous slope However the geom-etry of real slopes is more complex For example a naturalslope often has curvature and irregular surfaces appear in


open-pit and roadside design Therefore further research isrequired to consider the effect of complex geometries onthree-dimensional numerical analysis

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 Chinese National ScienceFund (no 41002090 and no 51034005) and the FundamentalResearch Funds for the Central Universities (no 800015T5)They would like to express their gratitude to the editorsand reviewers for their constructive and helpful reviewcomments

References

[1] F Lizzi ldquoReticulated Root Piles to correct landslidesrdquo inProceedings of the ASCE Convention Preprint 3370 Chicago IllUSA October 1978

[2] R Cantoni T Collotta and V Ghionna ldquoA design methodfor reticulated micropiles structure in sliding slopesrdquo GroundEngineering vol 22 no 4 pp 41ndash47 1989

[3] S L Pearlman B D Campbell and J L Withiam ldquoSlopestabilization using in-situ earth reinforcementrdquo in Proceedingsof the ASCE Conference on Stability and Performance of Slopesand Embankments vol 2 Berkeley Calif USA 1992

[4] J E Loehr J J Bowders J W Owen L Sommers andW LiewldquoSlope stabilization with recycled plastic pinsrdquo TransportationResearch Record no 1714 pp 1ndash8 2000

[5] W K Howe ldquoMicropiles for slope stabilizationrdquo in Proceedingsof the Biennial Geotechnical Seminar ASCE New York NYUSA 2000

[6] M Esmaeili M G Nik and F Khayyer ldquoExperimentaland numerical study of micropiles to reinforce high railwayembankmentsrdquo International Journal of Geomechanics vol 13no 6 pp 729ndash744 2013

[7] L C Reese S T Wang and J L Fouse ldquoUse of drilled shaftsin stabilizing a sloperdquo in Stability and Performance of Slopes andEmbankments II Geotechnical Special Publication (GSP) no 31pp 1318ndash1332 1992

[8] S-W Sun B-Z Zhu H-M Ma and R-H Yang ldquoModel testson anti-slidingmechanism of micropile groups and anti-slidingpilesrdquoChinese Journal of Geotechnical Engineering vol 31 no 10pp 1564ndash1570 2009 (Chinese)

[9] S-W Sun B-Z Zhu and J-CWang ldquoDesignmethod for stabi-lization of earth slopes with micropilesrdquo Soils and Foundationsvol 53 no 4 pp 487ndash497 2013

[10] S Isam A Hassan and S Mhamed ldquo3D elastoplastic analysisof the seismic performance of inclined micropilesrdquo Computersand Geotechnics vol 39 pp 1ndash7 2012

[11] O C Zienkiewicz C Humpheson and R W Lewis ldquoAssoci-ated and non-associated visco-plasticity and plasticity in soilmechanicsrdquo Geotechnique vol 25 no 4 pp 671ndash689 1975

[12] T Matsui and K C San ldquoFinite element slope stability analysisby shear strength reduction techniquerdquo Soils and Foundationsvol 32 no 1 pp 59ndash70 1992

[13] K Ugai and D Leshchinsky ldquoThree-dimensional limit equilib-rium and finite element analyses a comparison of resultsrdquo Soilsand Foundations vol 35 no 4 pp 1ndash7 1995

[14] E M Dawson W H Roth and A Drescher ldquoSlope stabilityanalysis by strength reductionrdquo Geotechnique vol 49 no 6 pp835ndash840 1999

[15] D V Griffiths and P A Lane ldquoSlope stability analysis by finiteelementsrdquo Geotechnique vol 49 no 3 pp 387ndash403 1999

[16] JWon K You S Jeong and S Kim ldquoCoupled effects in stabilityanalysis of pile-slope systemsrdquo Computers and Geotechnics vol32 no 4 pp 304ndash315 2005

[17] H Lin and P Cao ldquoA dimensionless parameter determiningslip surfaces in homogeneous slopesrdquo KSCE Journal of CivilEngineering vol 18 no 2 pp 470ndash474 2014

[18] W B Wei and Y M Cheng ldquoStrength reduction analysisfor slope reinforced with one row of pilesrdquo Computers andGeotechnics vol 36 no 7 pp 1176ndash1185 2009

[19] F Cai and K Ugai ldquoNumerical analysis of the stability of a slopereinforced with pilesrdquo Soils and Foundations vol 40 no 1 pp73ndash84 2000

[20] N Snitbhan and W-F Chen ldquoElastic-plastic large deformationanalysis of soil slopesrdquo Computers and Structures vol 9 no 6pp 567ndash577 1978

[21] J M Duncan and P Dunlop ldquoSlopes in stiff fissured clays andsoilsrdquo Journal of SoilMechanics and Foundation Engineering volSM5 pp 467ndash492 1969

[22] O C Zienkiewicz and R L TaylorThe Finite Element Methodvol 1 McGraw-Hill London UK 4th edition 1989

[23] L W Abramson T S Lee S Sharma and G M BoyceSlope Stability and Stabilization Methods John Wiley amp SonsChichester UK 1995

[24] FLAC3D Fast LagrangianAnalysis of Continua in 3DimensionsVersion 30 Itasca Consulting GroupMinneapolis Minn USA2005

[25] J T Lieng Behavior of laterally loaded piles in sand large scalemodel tests [PhD thesis] Department of Civil EngineeringNowegian Institute of Technology 1988

[26] H G Poulos ldquoDesign of reinforcing piles to increase slopestabilityrdquo Canadian Geotechnical Journal vol 32 no 5 pp 808ndash818 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(a) (b)

Figure 1 Micropiles for slope stabilization

the design procedure includes (1) choosing a location for themicropiles within the existing slope (2) selecting micropilecross section (3) estimating the length of the micropile (4)evaluating the shear capacity of the micropiles group (5)calculating the spacing required to provide force to stabilizethe slope and (6) the design of the concrete cap beam

Esmaeili et al [6] presented three experimental modelsof embankments of 10m in height on a scale of 120 to set upa number of loading tests For comparison three numericalmodels were developed by using the PLAXIS-3D code basedon the FEM Sensitivity analysis on the geometric parametersof micropiles was conducted to investigate the numberspacing diameter and length with the embankment safetyfactor It was shown that with the optimum arrangement ofthemicropiles the safety factor of the slope stability and staticload-bearing capacity of the embankments will increase bymore than 30 and 65 respectively and also the settlementof the embankment crest will decrease by approximately 35Isam et al [10] presented a numerical method for the analysisof inclined micropiles In this method the micropiles aremodeled as 3D elastic beam elements rigidly connected toa cap which is free of contact with the soil Then it is showedthat the inclination ofmicropiles leads to a goodmobilizationof their axial component and consequently to an importantdecrease in the shearing and bending loading In general thesoil-pile interaction for such micropiles is very complex andthere are many factors whose influence on the final behaviorof the structure cannot be conveniently assessed Thus afurther study of the behavior of micropiled-reinforced slopesis necessary

The shear strength reduction technique has been used inthe stability analysis of slopes without piles by Zienkiewiczet al [11] Matsui and San [12] Ugai and Leshchinsky [13]Dawson et al [14] Griffiths and lane [15] Won et al [16] Linand Cao [17] and others For piled reinforced slopesWei andCheng [18] have considered the effects of piles on the stabilityof a slope by a three-dimensional finite element analysis usingthe shear strength reduction technique Won et al [16] andCai and Ugai [19] have analyzed the same slope by Wei andCheng [18] In the piled slope analysis by Won et al [16] andWei and Cheng [18] the location of the critical slip surfacewas determined by the maximum shear force in the pile so

A

B C

1205903

120590 3minus 120590

1= 0

120590tfs = 0 ft = 0

2c

radicN120593

c

tan120593

1205901

Figure 2 Composite Mohr-Coulomb failure criterion

that a very deep critical slip surface was determined whilethe maximum shear strain in the soil was not considered CaiandUgai [19] have discussed the use ofmaximum shear strainand maximum shear strain rate to determine a more realisticcritical slip surface

In this paper an analyticalmodel is developed to calculatethe safety factor of a slope reinforced with micropiles basedon the shear strength reduction method The homogeneousslope considered by Won et al [16] and Wei and Cheng[18] and Cai and Ugai [19] is employed for the analysis Thesoil behavior is described using the nonassociated Mohr-Coulomb criterion and micropiles are modeled as 3D pileelements As such a series of slope stability analyses isperformed and the optimization of pile position depth ofembedment and length of truncation are discussed

2 Analysis Method

21 Model of Materials In the present study the soil materialof the slope is simulated with an elastic perfectly plasticmodel with Youngrsquos modulus being 119864

119904before yielding

The yielding is described by a composite Mohr-Coulombcriterion with a tension cutoff as shown in Figure 2

The failure envelope from point 119860 to 119861 is defined by theMohr-Coulomb criterion 119891119904 = 0 with

119891119904

= 1205901minus 1205903119873120593+ 2119888radic119873

120593 (1)


1

|Fs|

L

|Fmaxs |

L

ks

|us|


119904


|Fmaxs |

L

120593s

120590mp

Cs



1

|Fn|

L

|Fmaxn |

L

kn

|un|

g = on g = off


119899


|Fmaxn |

L

120593n

120590mp

Cn



where 1205901and 120590



120593= function of 120593

119873120593=1 + sin1205931 minus sin120593

(2)


119891119905

= 1205903minus 120590119905

(3)


max is given by

120590119905

max =119888

tan120593 (4)



119904between



119904 cohesive








119904 exposed





119888trial=1

119865trial119888 120593

trial=1

119865trial120593 (5)










No

Yes

Yes

No

No

No

Yes

Yes

No

Yes

Input c 120593 T 120576

Ftrial = 1

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

Flow = Ftrial

Flow = Ftrial

Flow = Ftrial

Ftrial = 2Ftrial

Fup = Ftrial

Fup = Ftrial

Fup = Ftrial

Ftrial =Ftrial

2

Fup = Fup

Fup = Fup

Flow = Flow

Flow = Flow

Ftrial =Flow + Fup

2


F = Ftrial


L = 15m

10m

10m

20m

10 m

x

y

z

Lx







minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

09 10 11 12

120575

Dep

th (m

)




119904= 119896119899= 13 times


119904= 15 times 10

10 Pa and 119862119899=


119904= 120593119899= 20∘





119904





119868119891=


(6)







00 02 04 06 08 10

110

115

120

125

130

I f

LxL




















119887and119867

119886












Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09


05 10 15 20 25 3010

11

12

13

HbHa

I f









119909119867119886increases


119886in this





(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

|Fs|

L

|Fmaxs |

L

ks

|us|


119904


|Fmaxs |

L

120593s

120590mp

Cs



1

|Fn|

L

|Fmaxn |

L

kn

|un|

g = on g = off


119899


|Fmaxn |

L

120593n

120590mp

Cn



where 1205901and 120590



120593= function of 120593

119873120593=1 + sin1205931 minus sin120593

(2)


119891119905

= 1205903minus 120590119905

(3)


max is given by

120590119905

max =119888

tan120593 (4)



119904between



119904 cohesive








119904 exposed





119888trial=1

119865trial119888 120593

trial=1

119865trial120593 (5)










No

Yes

Yes

No

No

No

Yes

Yes

No

Yes

Input c 120593 T 120576

Ftrial = 1

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

Flow = Ftrial

Flow = Ftrial

Flow = Ftrial

Ftrial = 2Ftrial

Fup = Ftrial

Fup = Ftrial

Fup = Ftrial

Ftrial =Ftrial

2

Fup = Fup

Fup = Fup

Flow = Flow

Flow = Flow

Ftrial =Flow + Fup

2


F = Ftrial


L = 15m

10m

10m

20m

10 m

x

y

z

Lx







minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

09 10 11 12

120575

Dep

th (m

)




119904= 119896119899= 13 times


119904= 15 times 10

10 Pa and 119862119899=


119904= 120593119899= 20∘





119904





119868119891=


(6)







00 02 04 06 08 10

110

115

120

125

130

I f

LxL




















119887and119867

119886












Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09


05 10 15 20 25 3010

11

12

13

HbHa

I f









119909119867119886increases


119886in this





(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904 cohesive








119904 exposed





119888trial=1

119865trial119888 120593

trial=1

119865trial120593 (5)










No

Yes

Yes

No

No

No

Yes

Yes

No

Yes

Input c 120593 T 120576

Ftrial = 1

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

Flow = Ftrial

Flow = Ftrial

Flow = Ftrial

Ftrial = 2Ftrial

Fup = Ftrial

Fup = Ftrial

Fup = Ftrial

Ftrial =Ftrial

2

Fup = Fup

Fup = Fup

Flow = Flow

Flow = Flow

Ftrial =Flow + Fup

2


F = Ftrial


L = 15m

10m

10m

20m

10 m

x

y

z

Lx







minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

09 10 11 12

120575

Dep

th (m

)




119904= 119896119899= 13 times


119904= 15 times 10

10 Pa and 119862119899=


119904= 120593119899= 20∘





119904





119868119891=


(6)







00 02 04 06 08 10

110

115

120

125

130

I f

LxL




















119887and119867

119886












Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09


05 10 15 20 25 3010

11

12

13

HbHa

I f









119909119867119886increases


119886in this





(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










No

Yes

Yes

No

No

No

Yes

Yes

No

Yes

Input c 120593 T 120576

Ftrial = 1

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

R lt 10e minus 5

Flow = Ftrial

Flow = Ftrial

Flow = Ftrial

Ftrial = 2Ftrial

Fup = Ftrial

Fup = Ftrial

Fup = Ftrial

Ftrial =Ftrial

2

Fup = Fup

Fup = Fup

Flow = Flow

Flow = Flow

Ftrial =Flow + Fup

2


F = Ftrial


L = 15m

10m

10m

20m

10 m

x

y

z

Lx







minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

09 10 11 12

120575

Dep

th (m

)




119904= 119896119899= 13 times


119904= 15 times 10

10 Pa and 119862119899=


119904= 120593119899= 20∘





119904





119868119891=


(6)







00 02 04 06 08 10

110

115

120

125

130

I f

LxL




















119887and119867

119886












Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09


05 10 15 20 25 3010

11

12

13

HbHa

I f









119909119867119886increases


119886in this





(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

09 10 11 12

120575

Dep

th (m

)




119904= 119896119899= 13 times


119904= 15 times 10

10 Pa and 119862119899=


119904= 120593119899= 20∘





119904





119868119891=


(6)







00 02 04 06 08 10

110

115

120

125

130

I f

LxL




















119887and119867

119886












Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09


05 10 15 20 25 3010

11

12

13

HbHa

I f









119909119867119886increases


119886in this





(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 10

110

115

120

125

130

I f

LxL




















119887and119867

119886












Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09


05 10 15 20 25 3010

11

12

13

HbHa

I f









119909119867119886increases


119886in this





(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Fs = 127

(a) 119871119909119871 = 01

Fs = 142

(b) 119871119909119871 = 03

Fs = 148

(c) 119871119909119871 = 07

Fs = 132

(d) 119871119909119871 = 09


05 10 15 20 25 3010

11

12

13

HbHa

I f









119909119867119886increases


119886in this





(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)


Ha

Cx







00 02 04 06 08 1010

11

12

13

I f

CxHa




16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










16

14

12

10

8

6

4

2

0

0M (kN-m)

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(a)

0 20 40 60 80 100Q (kN)

16

14

12

10

8

6

4

2

0

H(m

)


Cx = 18Ha

Cx = 14Ha

Cx = 38Ha

Cx = 12Ha

Cx = 58Ha

Cx = 34Ha

(b)


5 Conclusions






119887119867119886le










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article three-dimensional stability...

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