supportingcharacteristicsofsoldierpilesforfoundation...

Research ArticleSupporting Characteristics of Soldier Piles for FoundationPits under Rainfall Infiltration

Changxi Huang 1 Xinghua Wang 1 Min Zhang2 Hao Zhou 1 and Yan Liang1

1School of Civil Engineering Central South University Changsha 410075 China2School of Architecture and Civil Engineering Taiyuan University of Technology Taiyuan 030024 China

Correspondence should be addressed to Xinghua Wang xhwangmailcsueducn

Received 14 November 2018 Accepted 12 June 2019 Published 26 June 2019

Academic Editor Khalid Abdel-Rahman

Copyright copy 2019 Changxi Huang et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

e evolution of supporting characteristics of soldier piles for foundation pits under rainfall infiltration is described in this paperBased on the Richards seepage equation and earth pressure theory considering the effect of intermediate principal stress ofunsaturated soil the approximate solution of permeability coefficient in unsaturated soil is developed using the Laplace integraltransform and residue theorem e transition of the boundary condition from flux boundary to head boundary is taken intoaccount and the rationality of the solution is verifiede safety factor of soldier pilesrsquo stability against overturning is derived andits displacements are obtained by the finite-difference method e results show that the stability of soldier piles graduallydecreases with rainfall duration and the actual embedded depth of the pile should be greater than the minimum embedded depthspecified by the design codee displacement response of pile top lags behind the rainfall and increases later in the rainfall periode hysteresis time is shortened with the increase of soil permeability Considering moisture absorption the effect of rainfall onthe properties of retaining structure is limited for the homogeneous soils with low suction or permeability

1 Introduction

For foundation pit excavation in rainy areas the mostprominent problem is foundation pit instability caused byrainwater infiltration In rainy season rainfall increases thewater content of shallow soil of foundation pits which formsa transient saturation zone and expands downward thechange of pore water pressure in the soil will cause seepageand additional permeable pressure which leads to the in-crease of soil sliding force or soil loss between piles Inaddition with the infiltration of rainwater the contributionof the matric suction to the shear strength of soil decreases oreven completely loses which results in the decrease of re-sistance in the potential slip surface As a result the safety ofthe foundation pit declines

Natural foundations are mostly unsaturated ere arecapillary pressure phenomena and coupling effects betweenthree phases which are soil particles water and gases in theunsaturated soil Wu et al [1] pointed out that the internalforce of the retaining structure of the foundation pit

significantly reduced considering the suction of unsaturatedsoil which should be taken advantage of in design Aqtashand Bandini [2] predicted the unsaturated shear strength ofthe adobe soil which increases obviously with soil suctionKhaboushan et al [3] studied the relation between soilcharacteristics and unsaturated shear strength parameterse shear strength parameters can be predicted based on thesoil properties Moreover Leong and Abuel-Naga [4] foundthat there is little effect of osmotic suction on the shearstrength of unsaturated silt However matric suction affectsthe shear strength of unsaturated soil Kim et al [5] analyzedhysteresis effects on the mechanical property of unsaturatedgranite soil by the finite element method It shows that thematric suction is small with the main wetting curve

e establishment of the rainfall infiltration model ofunsaturated soil and the determination of matric suctiondistribution at different times are the prerequisites foranalysis of foundation pits or slopes under rainfall condi-tions Wu and Huang [6] established one-dimensionaltransient and steady state models under rainfall infiltration

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 1053576 10 pageshttpsdoiorg10115520191053576

with the ux and pore pressure boundaries in unsaturatedsoils taking into account the variation of rainfall intensityWu et al [7] studied the seepage problem under the couplingeect of deformation and seepage Oh and Lu [8] analyzedthe transient slope stability under rainfall in a generalizedeective stress framework which increases three un-saturated parameters to the saturated conditions e theoryframework was used to reproduce the slope failure inpractice Dou et al [9] obtained the failure probability ofunsaturated slope over time with rainfall by the Monte Carlomethod Furthermore Song et al [10] investigated theunsaturated slope stability in real time according to thesuction stress attained from monitoring data Sun et al [11]provided a three-phase model to simulate the dynamicchange of safety factor with a given slip surface of un-saturated slope under rainfall inltration Cho [12] studiedthe inuence of pore-air pressure variation on the slopestability due to rainfall A numerical study shows that airow contributes to the silt slope stability but air ow isadverse to the stability of sandy slope In addition otherscholars carried out relevant studies on two-dimensionalrainfall inltration [13] and double-layer foundation [14]

From the literature review the research on the un-saturated soil stability under rainfall inltration mainly fo-cused on slopes or landslides [14ndash16] and less on foundationpits Although they had similarities in stability checking thefoundation pits were dierent from slopes in support forminstability mechanism and analysis methods At presentthere is lack of comprehensive quantitative study on theevolution law of stability and deformation characteristics ofsupporting structures of foundation pits with rainfall which isa worthy theoretical and engineering subject

In this paper based on the Laplace integral trans-formation and nite-dierence method approximate solu-tion of permeability coecient in unsaturated soil duringrainfall inltration is obtained considering the boundarytransition from ux boundary to head boundary A theo-retical model for calculating the stability and deformationcharacteristics of soldier piles of foundation pit is establishede rainfall inltration eect on the supporting structure offoundation pit is discussed through parameter analysis

2 Calculation Model and UnsaturatedSoil Pressure

Figure 1 is the calculation model of soldier piles of foun-dation pit under rainfall e coordinate origin is located onthe soil surface z axis positive direction points to the soilvertically e excavation depth of foundation pit is Hembedded depth is D and pile length is L H +D egroundwater level in active and passive earth pressure areasare La and Lp respectively qA is the initial steady ow intothe soil surface qB is the rainfall intensity

In order to simplify the calculation the assumptions areintroduced as follows (1) the soil is a homogeneous isotropicunsaturated medium and soil surface is a permeableboundary (2) supporting piles are equal in diameter re-gardless of their friction and spatial eects (3) rainfall

intensity is constant neglecting the seepage eect caused bythe groundwater level dierence between the active andpassive zones (4) the bottom of the foundation pit and thesoil behind the pile are at the Rankine limit stress state Inaddition the purpose of this paper is to study the inuenceof matric suction on shear strength and earth pressure ofsoil without considering the soil stiness weakening causedby long-term immersion or periodic soaking-air dryingunder natural environment

According to [17] the shear strength of unsaturated soilscan be obtained on the basis of unied strength theory andBishop eective stress principle

τfprime ctprime + σ minus ua( ) + χ ua minus uw( )[ ]tanφtprime (1)

where σ is the total normal stress τfprime is the eective shearstrength and χ is the eective stress coecient of un-saturated soil ua and uw are the air pressure and waterpressure respectively ua minus uw is the matric suction ctprime andφtprime are the unied eective cohesion and unied eectiveinternal friction angle respectively which can be calculatedby ctprime (2(1 + b)cprime cosφprime)2 + b(1 + sinφprime)(1cos φtprime) andsin φtprime (2(1 + b)sin φprime2 + b(1 + sinφprime)) cprime and φprime are theeective cohesion and eective internal friction of soilrespectively b is the unied strength theory parameterwhich reects the inuence of intermediate principal stresson the yield or failure of materials e range of value is0le ble 1

ere is a highly nonlinear relationship between χ andmatric suction or saturation e range of χ is from 0 to 1which reects the contribution of suction stress to soilshear strength It also shows that there is a nonlinearfunctional relationship between unsaturated soil shearstrength and matric suction Based on equation (1) un-saturated Rankine active and passive earth pressure can beobtained which considers intermediate principal stress asfollows [18]

σh minus ua σv minus ua( )ka minus 2ctprimeka

radicminus χ ua minus uw( ) 1minus ka( ) (2a)

σh minus ua σv minus ua( )kp + 2ctprimekpradic

+ χ ua minus uw( ) kp minus 1( ) (2b)

qB

L p

(σh ndash ua)p

(σh ndash ua)a

Water table

Zot

0

qB

L a

HD

Z

Figure 1 Computation model for deep excavation under rainfall

2 Advances in Civil Engineering

where σh and σv are the horizontal and vertical stressesrespectively ka tan2((45minusφtprime)2) is the active earthpressure coefficient and kp tan2((45 + φtprime)2) is the pas-sive earth pressure coefficient

Equations (1) and (2) indicate that soil shear strengthreduces with the decrease ofmatric suction Rainfall infiltrationincreases the degree of saturation which leads to a disad-vantageous situation e earth pressure can be calculatedaccording to limit stress state In addition the matric suction isnonlinear along the depth direction in practice especiallyunder the effect of water infiltration and moisture absorptionIt means that the distribution of lateral earth pressure hascomplex space-time characteristics under rainfall conditions

3 Model Solution

31 Rainfall Intensity Is Less than Saturated PermeabilityCoefficient For homogeneous isotropic soils the perme-ability coefficient and soil-water characteristic curve ofunsaturated soils can be described by the exponentialfunction proposed by Gardner [19]

k hm( 1113857 ks exp βhm( 1113857 (3a)

θminus θrθs minus θr

exp βhm( 1113857 (3b)

where hm is the suction head which is calculated byhm (minus(ua minus uw)cw) cw is the unit weight of water k isthe unsaturated permeability coefficient related to matricsuction ks is the saturated permeability coefficient θ is thevolume water content of soil θs and θr are the saturatedand residual water content respectively and β is the de-creasing rate of permeability coefficient with the increaseof suction It ranges from 0 to 5 per meter For coarsegrained soil β can have large values while for clay and siltit would be small

Combined with boundary and initial conditions Sri-vastava and Yeh [20] derived the permeability coefficientbetween the ground surface and groundwater level by theLaplace transform on the one-dimensional transient un-saturated Richards seepage equation and inverse trans-formation based on residue theorem

K1(Z T) QB minus QB minus eminusβh01113872 1113873e

minus LaminusZ( ) minus 4 QB minusQA( 1113857eZ2

eminusT4

middot 1113944+infin

n1

sin λn La minusZ( 11138571113858 1113859sin λnLa( 1113857eminusλ2nT

1 + La2 + 2λ2nLa1113872 1113873

(4)

where λn is the n th root of the following equation

tan λLa( 1113857 + 2λ 0 (5)

where Z β middot z K kks QA qAks QB qBksT (β middot ks middot t)(θs minus θr) La βLa z is the soil depth and t

the is rainfall duration h0 is the suction head at groundwaterlevel and it is usually assumed to be zero

32 Rainfall Intensity Is Greater than Saturated PermeabilityCoefficient When rainfall intensity is greater than the satu-rated permeability coefficient ponding or runoff occurs on theground surface and pit bottom after a period of rainfall Itmeans that there is a totally saturated stateWhen suction headdecreases to zero the time (T0 or t0) when it begins to saturateat ground surface can be calculated from equation (4) Fourierintegral transformation was used to solve the problem in [21]However the initial condition in this paper is different fromequation (4) erefore Laplace integral transformation isused to deduce the approximate solutions of the permeabilitycoefficient

When rainfall duration (t) is less than t0 the rainwaterinfiltrates completely ere is no rainwater accumulationon the ground surface and foundation pit bottome upperboundary is still the flux boundary condition

minuszK

zZ+ K1113890 1113891

Z0 QB (6)

At this stage the permeability coefficient can be obtainedby equation (4) When t is greater than t0 rainwater ac-cumulates on the ground surface and at the bottom of thepit erefore soil suction disappears e upper boundarychanges into head boundary as follows

K21113868111386811138681113868Z0 1 (7)

At this time the law of rainfall infiltration is not affectedby rainfall intensity any more It only plays the role of headboundary Combining with boundary condition equation (6)permeability coefficient in unsaturated region is obtained byLaplace integral transformation of time T in Richardsequation in [18]

K2(Z s) QA minus QA minus eminusβh0( 1113857eminus LaminusZ( )

s

+ 1minusQA + QA minus eminusβh01113872 1113873e

minusLa1113876 1113877eZ2

middot F(s)

(8)

where F(s) is calculated by F(s) (1s)((sinh(

s + (14)

1113968(La minusZ)))(sinh(

s + (14)

1113968La))) p(s)q(s) and

s is the Laplace transform coefficient It should be noted thatthe second term on the left side of the original formulashould be changed to minus in calculation since the z axispositive direction (downward) of the model in this paper isexactly opposite to that of Richards governing equation (5)in reference [20]

Using the residue theorem the result of Laplace inversetransformation of F(s) is as follows

F(T) 1113944n

k1Res F(s)e

sT sk1113960 1113961 1113944

n

k1

p sk( 1113857

qprime sk( 1113857 (9)

where qprime(s) represents the first derivative of q(s) and sk isthe k th isolated singularity

According to the above theorem the inverse trans-formation expression of F(s) can be obtained

Advances in Civil Engineering 3

F(T) sinh La minusZ( 111385721113858 1113859

sinh La2( 1113857

+ 1113944+infin

n0

(minus1)n8πn middot sin πn La minusZ( 1113857La( 1113857

4π2n2 + L2a1113872 1113873

eminus π2n2L

2a( 1113857+(14)( 1113857T

(10)

e first term on the right-hand side of the equation isthe residue as s equals zero e summation terms corre-spond to residues at the isolated singular point

sn + (14)1113968

La inπ in the complex fieldPutting equation (10) into equation (8) the permeability

coefficient at any time and depth can be obtained for thesecond stage (tge t0)

K2(Z T) QA minus QA minus eminusβh01113872 1113873e

minus LaminusZ( )


minusLa1113876 1113877eZ2

middot F(T)

(11)

It is worth noting that the initial condition is tlt t0during the above derivation

K(Z 0) QA minus QA minus eminusβh01113872 1113873e

minus LaminusZ( ) K0(Z) (12)

e initial condition of tgt t0 can be attained by equation (4)with T T0 However it is difficult to get the exact expressionof permeability coefficient because of the complexity ofequation (4) Equation (11) is only an approximate solutionexpression e result of trial calculation shows that thedistribution of K1 and K2 are approximately same at T0 K2 isgenerally larger than K1 except for ground surface andgroundwater level where both K2 and K1 equal one K2 tendsto be 10 while T is close to infinity erefore K2 is modifiedon the basis of the linear proportional distribution of thedifference between K2 and K1 at T0 e result is as follows

K2(Z T) K2(Z T)minusΔ0(Z)1 minusK2(Z T)

1 minusK2 Z T0( 1113857 (13)

e above deductions are all aimed at the solution of thepermeability coefficient in the active zone while the solutionin the passive zone can be attained by replacing La with Lp

33 Finite-Difference Solution of Permeability CoefficientIn order to verify the rationality and accuracy of the ap-proximate solution in Section 32 permeability coefficient inthe second stage of rainfall is calculated by finite-differencemethod under the actual initial conditions (permeabilitycoefficient is K1 at the time of T0 ) e unsaturated soilregion is discretized by space and time e depth of the soilis divided into n equal segments e time step and spacestep are recorded as ΔT and ΔZ which are dimensionlessWhen the time step is very small the Richardsrsquo percolationequation in [18] can be discretized into the following dif-ferential form

aKijminus1 + bKij + cKij+1 dKiminus1j (14)

where a ΔT times (1 + ΔZ) b minusΔZ times ΔTminus 2 times ΔTminusΔZtimes

ΔZ and c ΔT d minusΔZ times ΔZ i and j represent time andspace nodes respectively i 1 2 j 1 2 n+ 1Ti1 0 represents the time T0 (or t0) that rainwater beginsto accumulate on the ground surface Zj1 0 Zjn+1 LaKij is the dimensionless permeability coefficient with Ti

i times ΔT and Zj j times ΔZ e corresponding boundaryconditions can be discretized into the following equation

Ki1 1

Kin+1 1(15a)

After putting T T0 into equation (4) the permeabilitycoefficient is taken as the initial condition of the secondstage e discrete form is as follows

K1j K1 Zj T01113872 1113873 (15b)

It is clear that the difference form for the flux boundarycondition in the first stage of rainfall is as follows

Ki1 Ki2 + ΔZ times QB

1 + ΔZ (16)

Different differential equations are set up for each nodein the differential grid respectively

b c 0 0 middot middot middot 0

a b c 0 middot middot middot 0

⋮ ⋮ ⋮ ⋮ ⋮ ⋮

0 middot middot middot a b c 0

0 middot middot middot 0 a b c

0 0 middot middot middot 0 a b

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(nminus1)times(nminus1)

Ki2

Ki3

⋮

Kinminus1

Kin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

dKiminus12 minus aKi1

dKiminus13

⋮

dKiminus1nminus1

dKiminus1n minus cKin+1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)

Accordingly the Richards seepage equation is trans-formed into algebraic equations

It can be seen that equation (17) is a set of linearequations about the unknown numbers of Kij e co-efficient matrix is a strictly diagonally dominant triangularmatrix which can be solved recursively by the pursuitmethod

34 Stability and Deformation of Embedded Soldier PilesOn the basis of the permeability coefficients of active andpassive zones suction head and effective stress coefficientscan be obtained by equations (3a) and (3b) By bringingthem into equations (2a) and (2b) the distribution of activeand passive earth pressures of unsaturated soils can beobtained finally at any time and depth When effective stresscoefficient χ is expressed by a general van Genuchten model[22] equations (2a) and (2b) can be written as follows


σh minus ua( 1113857a czka minus 2ctprime

ka

1113969

minusua minus uw( 1113857 1minus ka( 1113857

1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18a)

σh minus ua( 1113857p czkp + 2ctprimekp

1113969+

ua minus uw( 1113857 kp minus 11113872 1113873

1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18b)

where a0 and d0 are the fitting parameters of V-Gmodel and(ua minus uw) minuscw In[K(Z T)]β e subscripts a and prepresent the active and passive limit state of soil re-spectively c is unit weight of soil above groundwater

By substituting equations (4) or (13) into equation (18a)the critical depth zot can be obtained as active earth pressure(σh minus ua)a equals zero Because the solution of transcen-dental equation is involved numerical calculation methodssuch as Newton iteration can be used

e safety factor of supporting piles stability againstoverturning can be expressed by the ratio of the resistancemoment Mp formed by passive earth pressure around thepile bottom to the overturning moment Ma formed by activeearth pressure Both include the moments generated byunsaturated soils and effective unit weight and hydrostaticpressure of saturated soils erefore the safety factor ofpiles stability against overturning can be expressed as

Kq Mp

Ma

Mpun + Mpsa

Maun + Masa (19)

where

Maun 1113946La

zot

σh minus ua( 1113857a(Lminus z)dz

Masa cw Lminus La( 1113857

3

6+ Lminus La( 1113857

2 ka 3cLa + cprime Lminus La( 11138571113858 1113859

6minus ctprimeka

11139681113896 1113897

Mpun 1113946Lp

0σh minus ua( 1113857p(Dminus z)dz

Mpsa cw Dminus Lp1113872 1113873

3

6+ Dminus Lp1113872 1113873

2 kp 3cLp + cprime Dminus Lp1113872 11138731113960 1113961

6 + ctprimekp

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

cprime csat minus cw

(20)

where cprime is the effective unit weight of the soil and csat is thesaturated unit weight of the soil

For calculation of horizontal displacement of supportingpiles pile section above the excavation surface can beregarded as the cantilever section with embedded bottomPile section below the excavation surface can be taken aselastic foundation beam So the deflection of each sectioncan be calculated separately e actual active earth pressurecalculated during the rainfall process is considered as lateral

load of the cantilever section while the load of anchoragesection is rectangular load e elastic deflection differentialequation of soldier piles is as follows

EIdy4

d4z+ b0K(z)yminus b0pa(z) 0 (21)

where y is the horizontal displacement pa(z) is the dis-tributed load acting on the supporting pile EI is the flexuralrigidity of the pile and b0 is the calculation spacing of thepiles K(z) is the horizontal resistance coefficient of thefoundation Above the excavation surface K(z) equals zerowhile K(z) equals mz below the excavation surface m is theproportion coefficient of the foundation which can beachieved by the relevant tables

For elastic foundation beam the bending moment andshear force at pile end are 0 (EI middot yPrime 0 EI middot yprimePrime 0)Equation (21) can be solved directly by the bvp5c functionembedded in MATLAB software For the cantilever sectionabove the excavation face it is difficult to solve equation (21)directly considering the nonlinear distribution of earthpressure e cantilever section is discretized into N seg-ments with a same length of Δh Δh HN e number ofthe nodes are 0 1 2 N e displacement and rotationangle of the fixed end are zero (yN 0 yN

prime 0) ConsideringyPrime M(z)EI the displacement of the bottom node satisfiesthe following relationship

yN+1 minusyNminus12Δh

0 (22a)

yN+1 minus 2yN + yNminus1 (Δh)2MN

EI (22b)

where yN+1 is the horizontal displacement of the virtualnode N + 1 and MN is the bending moment at the fixed endwhich can be attained from the earth pressure above theexcavation face

According to equations (22a) and (22b) yNminus1 (Δh)2 middot

MN(2EI) can be obtained e displacements of othernodes can be calculated by the deduced difference form in[23] e final displacement distribution of the pile body canbe obtained by accumulating the translation and rotationdisplacement of the vertex of the embedded segmentFurthermore internal force distribution of the pile body canbe obtained

4 Results and Discussion

A foundation pit is supported by bored piles with a diameterof 10m length of 10m and spacing of 14m e elasticmodulus of concrete is 30GPa e depth of excavation is6m (H 6m ) and embedded depth of pile is 4m(D 4m ) e depth of water table in active and passiveearth pressure areas is 8m and 2m respectively (La 8mLp 2m ) e stratum is mainly silt e unit weight c ofsoil above the groundwater level is 18 kNmiddotmminus3 the saturatedunit weight csat is 20 kNmiddotmminus3 e saturated permeabilitycoefficient ks is 10times10minus6ms θs 04 θr 015 andβ 05 mminus1 e proportion coefficient of foundation m is10MN middotmminus4 the unified strength theoretical parameter b is


05 In order to simplify the calculation the eective co-hesion cprime and eective internal friction angle φprime of the soilsare 10 kPa and 20deg respectively e parameters of vanGenuchten model a0 and d0 are 005 and 2 respectivelyInitial steady ow qA is 0m middot sminus1 rainfall intensity qB is2315times10minus6mmiddotsminus1 (equivalent to 200mm middotdminus1)

Figures 2(a) and 2(b) are the distributions of suction headalong depth when rainfall intensity is less and greater thansaturated permeability coecient respectively It can be seenthat suction head has a linear distribution along the depthdirection at the initial time e suction at the groundwaterlevel is 0 When rainfall begins suction head near the groundsurface decreases rapidly (absolute value) e suction re-duction area gradually expands to the groundwater level withrainfall Meanwhile the wetting front advances downward Inthe early stage of inltration rainfall mainly aects the suctionin the upper part of the wetting front From Figure 2(a)rainwater inltrates into the soil completely So the suctionhead at the ground surface cannot be 0 In Figure 2(b) thepermeability of the soil is too small and thus rainwater cannotinltrate totally Hence the ground surface is saturated withthe water accumulation or runo on the ground surface As aresult suction head at the ground surface can be 0 as shown inFigure 2(b) In addition the suction head calculated by theanalytical method and the nite-dierence method are ba-sically in agreement which veries the rationality of theapproximate solution It provides a guarantee for the sub-sequent calculation of earth pressure

Figure 3 shows that the relationship between safetyfactor against overturning and time under dierent rainfallintensities e safety factor gradually decreases with thecontinuous rainfall e safety factor declines obviously withgreater rainfall intensity With rainwater inltration degreeof saturation gradually increases and matric suction of thesoil decreases erefore active earth pressure increaseswhile passive earth pressure decreases which leads to theincrease of overturning moment acting on the supportingpile and decrease of resistance moment Consequently thesafety factor decreases

Figure 4 displays that the relation between safety factoragainst overturning and time under dierent saturatedseepage coecients e selected permeability coecient isless than rainfall intensity e safety factor has a signicantdecline with the increase of permeability coecient andrainfall duration It is pointed out that supporting offoundation pit in a highly permeable foundation shouldcause attention e increase of rainfall inltration ratequickens up the downward expansion rate of the saturatedregion which accelerates the reduction of thematric suctionIt should be noted that the inuence of rainfall inltration onthe foundation pit involves many factors including soilmoisture absorption groundwater seepage and waterweakening Reduction of suction additional seepage forceand stiness degradation of soil will reduce the safety of thefoundation pit In this paper only the soil moisture ab-sorption is analyzed considering the complexity of theproblem Hence the actual safety factor of piles stabilityagainst overturning should be lower than the result inFigure (4)

e inuence of groundwater level depth on the safetyfactor against overturning is shown in Figures 5(a) and 5(b)Figure 5(a) shows that the groundwater level is the same onboth sides of active and passive earth pressure zones whileFigure 5(b) illustrates that the groundwater level is dierenton both sides of active and passive earth pressure areas Itcan be seen that the safety of foundation pit is graduallyimproved with the decrease of the groundwater level Itindicates that the less the groundwater level is the better thestability of soldier piles will be e reason is that the de-crease of the water level increases the initial distributionrange and amplitude of the suction e suction contribu-tion to soil shear strength is also fortied Meanwhile theactive earth pressure decreases and the passive earth pres-sure increases which improve the properties of retaining

0

1

2

3

4

5

6

7

8

z (m

)

ndash8 ndash7 ndash6 ndash5 ndash4 ndash3 ndash2 ndash1 0

ks = 10 times 10ndash6msqB = 05 times 10ndash6ms

hm (m)

1h

10h

20h

80h

Finite-difference solution+Analytical solution

(a)

z (m

)0

1

2

3

4

5

6

7

8ndash5 ndash45 ndash4 ndash35 ndash3 ndash25 ndash2 ndash15 ndash1 ndash05 0


hm (m)

0h20h

40h70h


(b)

Figure 2 Distribution of suction head with depth at dierent time(a) qBlt ks (b) qBgt ks


piles of foundation pit In addition decreasing thegroundwater level on both sides of the active and passivezone at the same time improves the safety factor moreobviously compared with the one-sided decrease ofgroundwater level in the active zone However the safetyfactor of former decreases sharply as the time elapses

Figures 6(a) and 6(b) describe the variation of safetyfactor against overturning with embedded depth and time asthe unified strength theoretical parameter b is 00 or 10respectively It can be seen that the stability of foundation pitwill be greatly improved with the increase of embeddeddepth In Figure 6(a) the foundation pit may change fromsafe state to unstable state with the increase of rainfallduration as embedded depth is not large (f ) is shows thatthe minimum embedded depth determined by the stabilitychecking calculation in the foundation pit code should retainenough safety reserves to prevent adverse effects of rainfallfrom the stability of foundation pits Parameter b reflects the

effect of medium principal stress σ2 on soil strength andfailure it also represents different strength criteria Com-paring Figures 6(a) and 6(b) it can be seen that the curveshape of the safety factor are basically the same under two b

values e safety factor at b 1 is nearly 50ndash80 higherthan that at b 0 It means that the design of supportingstructure based on MohrndashCoulomb strength criterion isconservative It is necessary to consider the effect of in-termediate principal stress to achieve better economicbenefits

Figures 7 and 8 show the relationship between horizontaldisplacement of pile top and time under different rainfallintensities and saturated permeability coefficient re-spectively a0 001 and a0 01 represent silt and sandrespectively From Figure 7 it can be seen that there is a lagfor the displacement response as the time elapses e

0 20 40 60 80 100 12013

135

14

145

15

155

16

165

t (h)

K q

qB = 50mmdqB = 100mmdqB = 200mmd

ks = 10 times 10ndash6ms

Figure 3 Effect of rainfall intensity on safety coefficient

0 20 40 60 80 100 120115

125

135

145

155

165

t (h)

K q

ks = 10 times 10ndash7msks = 10 times 10ndash6msks = 20 times 10ndash6ms

Figure 4 Effect of saturated permeability coefficient on safetycoefficient

0 20 40 60 80 100 12013

15

17

19

21

23

t (h)

La = 8mLa = 10mLa = 12m

Lp = La ndash H

K q

(a)

La = 8mLa = 10mLa = 12m

0 20 40 60 80 100 12013

14

15

16

17

18

19

2

t (h)

K q

Lp = 2m

(b)

Figure 5 Effect of water table of active area on safety coefficient(a) e same water table on both sides (b)e different water tableon both sides


displacement of pile top is basically constant in the rst halfof the rainfall period However the displacement of pile topincreases gradually later in the rainfall period e lag timemainly depends on the permeability of the soil itself and islittle aected by the rainfall intensity e reason for dis-placement response hysteresis is shown as follows Rain-water needs to go through the critical depth area whereactive earth pressure equals zero then it enters the areawhere the active earth pressure works For silt or clay withhigh suction stress critical depth can be several meters deep(about 4m in this case) Furthermore the displacement ofpile top in silt foundation increases signicantly especiallyin the case of heavy rainfall However for sandy soilfoundation with low suction stress the change of soilpressure caused by rainfall is limitedus the displacementof pile top remains constant with time under dierentrainfall intensity e maximum increase of displacement ofpile top is only about 2mm From Figure 8 it can be seen

that the increase of saturated permeability coecient ksleads to a signicant increase in the displacement of pile topand a great decrease of the time that displacement responsebegins e displacement of pile top at the end of rainfallalmost doubles that at the beginning of rainfall while thesaturated permeability coecient is 20times10minus6mmiddotsminus1 but thedisplacement of pile top remains constant as the saturatedpermeability coecient is small

In general the characteristic of supporting structure isnot very sensitive to short-term rainfall especially for lowpermeability soil Rainfall only aects the distribution ofsuction stress in the soil surface layer However for con-tinuous rainy weather (such as qB 2mm middothminus1) it is foundthat the displacement of pile top is 47mm after 14 dayswhich should cause attention Moreover for expansive soilsor soft soils there may be more tension cracks in foundationpit wall under the condition of wet-dry cycling or excessivedeformation of foundation pit e cracks provide a rapid

2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)

Figure 6 Variation of safety factor against overturning with embedded depth and time (a) b 00 (b) b 10

52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01

qB = 2mmhqB = 5mmhqB = 10mmh

Figure 7 Eect of rainfall intensity on horizontal displacement ofpile top

20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001


Figure 8 Eect of saturated permeability coecient on horizontaldisplacement of pile top


migration channel for rainwater inltration into deep soilsConsequently the permeability of surface soils will increasesignicantly It is found that the wetting front arrives atcritical depth within 15 hours when saturated permeabilitycoecient is 5times10minus5mmiddotsminus1 and rainfall intensity is30mm middothminus1 e existence of soil cracks on the foundationpit wall greatly shortens the time of water inltration intodeep soil which is extremely unfavorable to the stability offoundation pit

When the embedding depth of soldier piles is 8m andrainfall duration is 10 hours the distribution of displace-ment and bending moment of pile body are shown inFigure 9 considering three groundwater levels in activeareas It can be seen that the displacement and bendingmoment of pile body increase gradually with the decrease ofwater table the critical depth of earth pressure moves upaccordingly e displacement of pile top for La 0m al-most doubles that for La 8m Compared to La 8m themaximum bending moment of the pile increases by nearly15 times for La 0m However the locations of themaximum bending moment on both water tables are 037Dbelow the excavation surface Compared with the rise ofshallow groundwater level the rise of deep groundwaterlevel has a more signicant eect on the properties of soldierpiles is is because the soil suction stress is small under thehigh groundwater level and the critical depth is close to thesoil surface

5 Conclusion

Based on the Laplace integral transformation and nite-dierence method approximate solution of permeabilitycoecient of unsaturated soil has been derived Moreoverthe stability and deformation characteristics of soldierpiles are analyzed e conclusions can be obtained asfollows

(1) In the process of rainfall inltration embeddingstability of soldier piles declines obviously withrainfall intensity and saturated permeability co-ecient of soil

(2) ere is hysteresis for displacement response ofretaining piles with rainfall e displacement of piletop increases gradually after rainfall the lag timedecreases with the increase of soil saturated per-meability coecient e rise of groundwater levelwill lead to a signicant increase of displacement andbending moment of pile especially the rise of deepwater level

(3) e inuence of rainfall on the supporting structureof foundation pit in low-suction or low-permeabilitysoil is relatively limited in terms of the moistureabsorption of homogeneous soil

(4) e suction head distribution is taken as the initialcondition for permeability coecient calculation inthe second stage e errors are linearly distributede calculation results have high accuracy whichavoids the complexity caused by the initial condi-tion in the form of series used in the analyticalderivation

Data Availability

e data used to support the ndings of this study are in-cluded within the article e readers can access the datathrough the gures within the article

Conflicts of Interest

e authors declare that there are no conicts of interestregarding the publication of this paper

ndash2

0

2

4

6

8

10

12

14

z (m

)

2 6 10 14 18 22Displacement of pile body (mm)

t = 10hD = 8ma0 = 001

La = 8mLa = 4mLa = 0m

(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500

Bending moment of the pile body (kNmiddotm)

z (m

)

t = 10hD = 8ma0 = 001


(b)

Figure 9 Eects of water table of active area on horizontal displacement and bending moment of pile (a) Horizontal displacement of pilebody (b) Bending moment of pile body


Acknowledgments

is research was funded by specialized research fundfor the Doctoral Program of China (grant number20120162110023)

References

[1] J-M Wu G-X Li and C-H Wang ldquoEffect of matric suctionof unsaturated soil on the estimation of the load internal forceon supporting structure for deep excavationrdquo IndustrialConstruction vol 33 no 7 pp 6ndash10 2003

[2] U A Aqtash and P Bandini ldquoPrediction of unsaturated shearstrength of an adobe soil from the soilndashwater characteristiccurverdquo Construction amp Building Materials vol 98 pp 892ndash899 2015

[3] E A Khaboushan H Emami M R Mosaddeghi andA R Astaraei ldquoEstimation of unsaturated shear strengthparameters using easily-available soil propertiesrdquo Soil andTillage Research vol 184 pp 118ndash127 2018

[4] E-C Leong and H Abuel-Naga ldquoContribution of osmoticsuction to shear strength of unsaturated high plasticity siltysoilrdquo Geomechanics for Energy and the Environment vol 15pp 65ndash73 2018

[5] J Kim W Hwang and Y Kim ldquoEffects of hysteresis onhydro-mechanical behavior of unsaturated soilrdquo EngineeringGeology vol 245 pp 1ndash9 2018

[6] L-Z Wu and R-Q Huang ldquoAnalytical analysis of coupledseepage in unsaturated soils considering varying surface fluxrdquoChinese Journal of Geotechnical Engineering vol 33 no 9pp 1370ndash1375 2011

[7] L Z Wu L M Zhang and X Li ldquoOne-dimensional coupledinfiltration and deformation in unsaturated soils subjected tovarying rainfallrdquo International Journal of Geomechanicsvol 16 no 2 article 06015004 2015

[8] S Oh and N Lu ldquoSlope stability analysis under unsaturatedconditions case studies of rainfall-induced failure of cutslopesrdquo Engineering Geology vol 184 pp 96ndash103 2015

[9] H-Q Dou T-C Han X-N Gong and J Zhang ldquoProba-bilistic slope stability analysis considering the variability ofhydraulic conductivity under rainfall infiltration-re-distribution conditionsrdquo Engineering Geology vol 183pp 1ndash13 2014

[10] Y-S Song B-G Chae and J Lee ldquoA method for evaluatingthe stability of an unsaturated slope in natural terrain duringrainfallrdquo Engineering Geology vol 210 pp 84ndash92 2016

[11] D-M Sun X-M Li P Feng and Y-G Zang ldquoStabilityanalysis of unsaturated soil slope during rainfall infiltrationusing coupled liquid-gas-solid three-phase modelrdquo WaterScience and Engineering vol 9 no 3 pp 183ndash194 2016

[12] S E Cho ldquoStability analysis of unsaturated soil slopes con-sidering water-air flow caused by rainfall infiltrationrdquo Engi-neering Geology vol 211 pp 184ndash197 2016

[13] T L T Zhan and C W W Ng ldquoAnalytical analysis of rainfallinfiltration mechanism in unsaturated soilsrdquo InternationalJournal of Geomechanics vol 4 no 4 pp 273ndash284 2004

[14] T L T Zhan G W Jia Y-M Chen D G Fredlund andH Li ldquoAn analytical solution for rainfall infiltration into anunsaturated infinite slope and its application to slope stabilityanalysisrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 12 pp 1737ndash1760 2013

[15] C W W Ng L T Zhan C G Bao D G Fredlund andB W Gong ldquoPerformance of an unsaturated expansive soil

slope subjected to artificial rainfall infiltrationrdquo Geotechniquevol 53 no 2 pp 143ndash157 2003

[16] N Lu and J Godt ldquoInfinite slope stability under steadyunsaturated seepage conditionsrdquo Water Resources Researchvol 44 no 11 pp 1ndash13 2008

[17] C-H Zhang Q-H Zhang and J-H Zhao ldquoDevelopmentsfor unified solution of shear strength and ultimate bearingcapacity of foundation in unsaturated soilsrdquo Rock and SoilMechanics vol 31 no 6 pp 1871ndash1876 2010

[18] N Lu and W J Likos Unsaturated Soil Mechanics WileyHoboken NJ USA 2004

[19] W R Gardner ldquoSome steady-state solutions of the un-saturated moisture flow equation with application to evap-oration from a water tablerdquo Soil Science vol 85 no 4pp 228ndash232 1958

[20] R Srivastava and T-C J Yeh ldquoAnalytical solutions for one-dimensional transient infiltration toward the water table inhomogeneous and layered soilsrdquo Water Resources Researchvol 27 no 5 pp 753ndash762 1991

[21] N Li and J-C Xu ldquoSimulation of Deformation of infinite soilsloperdquo Chinese Journal of Geotechnical Engineering vol 34no 12 pp 2325ndash2330 2012

[22] M T van Genuchten ldquoA closed-form equation for predictingthe hydraulic conductivity of unsaturated Soils1rdquo Soil ScienceSociety of America Journal vol 44 no 5 pp 892ndash898 1980

[23] L-J Chen Y-J Sun and Z-G Wang ldquoResearch of finitedifference method on retaining structures with internal bracesin deep excavation engineeringrdquo Chinese Journal of RockMechanics and Engineering vol 35 no 1 pp 3316ndash33222016


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

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Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

with the ux and pore pressure boundaries in unsaturatedsoils taking into account the variation of rainfall intensityWu et al [7] studied the seepage problem under the couplingeect of deformation and seepage Oh and Lu [8] analyzedthe transient slope stability under rainfall in a generalizedeective stress framework which increases three un-saturated parameters to the saturated conditions e theoryframework was used to reproduce the slope failure inpractice Dou et al [9] obtained the failure probability ofunsaturated slope over time with rainfall by the Monte Carlomethod Furthermore Song et al [10] investigated theunsaturated slope stability in real time according to thesuction stress attained from monitoring data Sun et al [11]provided a three-phase model to simulate the dynamicchange of safety factor with a given slip surface of un-saturated slope under rainfall inltration Cho [12] studiedthe inuence of pore-air pressure variation on the slopestability due to rainfall A numerical study shows that airow contributes to the silt slope stability but air ow isadverse to the stability of sandy slope In addition otherscholars carried out relevant studies on two-dimensionalrainfall inltration [13] and double-layer foundation [14]

From the literature review the research on the un-saturated soil stability under rainfall inltration mainly fo-cused on slopes or landslides [14ndash16] and less on foundationpits Although they had similarities in stability checking thefoundation pits were dierent from slopes in support forminstability mechanism and analysis methods At presentthere is lack of comprehensive quantitative study on theevolution law of stability and deformation characteristics ofsupporting structures of foundation pits with rainfall which isa worthy theoretical and engineering subject

In this paper based on the Laplace integral trans-formation and nite-dierence method approximate solu-tion of permeability coecient in unsaturated soil duringrainfall inltration is obtained considering the boundarytransition from ux boundary to head boundary A theo-retical model for calculating the stability and deformationcharacteristics of soldier piles of foundation pit is establishede rainfall inltration eect on the supporting structure offoundation pit is discussed through parameter analysis

2 Calculation Model and UnsaturatedSoil Pressure

Figure 1 is the calculation model of soldier piles of foun-dation pit under rainfall e coordinate origin is located onthe soil surface z axis positive direction points to the soilvertically e excavation depth of foundation pit is Hembedded depth is D and pile length is L H +D egroundwater level in active and passive earth pressure areasare La and Lp respectively qA is the initial steady ow intothe soil surface qB is the rainfall intensity

In order to simplify the calculation the assumptions areintroduced as follows (1) the soil is a homogeneous isotropicunsaturated medium and soil surface is a permeableboundary (2) supporting piles are equal in diameter re-gardless of their friction and spatial eects (3) rainfall

intensity is constant neglecting the seepage eect caused bythe groundwater level dierence between the active andpassive zones (4) the bottom of the foundation pit and thesoil behind the pile are at the Rankine limit stress state Inaddition the purpose of this paper is to study the inuenceof matric suction on shear strength and earth pressure ofsoil without considering the soil stiness weakening causedby long-term immersion or periodic soaking-air dryingunder natural environment

According to [17] the shear strength of unsaturated soilscan be obtained on the basis of unied strength theory andBishop eective stress principle

τfprime ctprime + σ minus ua( ) + χ ua minus uw( )[ ]tanφtprime (1)

where σ is the total normal stress τfprime is the eective shearstrength and χ is the eective stress coecient of un-saturated soil ua and uw are the air pressure and waterpressure respectively ua minus uw is the matric suction ctprime andφtprime are the unied eective cohesion and unied eectiveinternal friction angle respectively which can be calculatedby ctprime (2(1 + b)cprime cosφprime)2 + b(1 + sinφprime)(1cos φtprime) andsin φtprime (2(1 + b)sin φprime2 + b(1 + sinφprime)) cprime and φprime are theeective cohesion and eective internal friction of soilrespectively b is the unied strength theory parameterwhich reects the inuence of intermediate principal stresson the yield or failure of materials e range of value is0le ble 1

ere is a highly nonlinear relationship between χ andmatric suction or saturation e range of χ is from 0 to 1which reects the contribution of suction stress to soilshear strength It also shows that there is a nonlinearfunctional relationship between unsaturated soil shearstrength and matric suction Based on equation (1) un-saturated Rankine active and passive earth pressure can beobtained which considers intermediate principal stress asfollows [18]

σh minus ua σv minus ua( )ka minus 2ctprimeka

radicminus χ ua minus uw( ) 1minus ka( ) (2a)

σh minus ua σv minus ua( )kp + 2ctprimekpradic

+ χ ua minus uw( ) kp minus 1( ) (2b)

qB

L p

(σh ndash ua)p

(σh ndash ua)a

Water table

Zot

0

qB

L a

HD

Z

Figure 1 Computation model for deep excavation under rainfall




3 Model Solution


k hm( 1113857 ks exp βhm( 1113857 (3a)


exp βhm( 1113857 (3b)





eminusT4


n1


1 + La2 + 2λ2nLa1113872 1113873

(4)


tan λLa( 1113857 + 2λ 0 (5)





minuszK

zZ+ K1113890 1113891

Z0 QB (6)


K21113868111386811138681113868Z0 1 (7)



s


minusLa1113876 1113877eZ2

middot F(s)

(8)


s + (14)


s + (14)

1113968La))) p(s)q(s) and



F(T) 1113944n

k1Res F(s)e

sT sk1113960 1113961 1113944

n

k1

p sk( 1113857

qprime sk( 1113857 (9)




F(T) sinh La minusZ( 111385721113858 1113859

sinh La2( 1113857

+ 1113944+infin

n0


4π2n2 + L2a1113872 1113873

eminus π2n2L

2a( 1113857+(14)( 1113857T

(10)


sn + (14)1113968




minus LaminusZ( )


minusLa1113876 1113877eZ2

middot F(T)

(11)






1 minusK2 Z T0( 1113857 (13)







Ki1 1

Kin+1 1(15a)


K1j K1 Zj T01113872 1113873 (15b)



1 + ΔZ (16)




⋮ ⋮ ⋮ ⋮ ⋮ ⋮




⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


Ki2

Ki3

⋮

Kinminus1

Kin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


dKiminus13

⋮

dKiminus1nminus1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)






ka

1113969


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18a)


1113969+


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18b)




Kq Mp

Ma

Mpun + Mpsa

Maun + Masa (19)

where

Maun 1113946La

zot



3

6+ Lminus La( 1113857


6minus ctprimeka

11139681113896 1113897

Mpun 1113946Lp



3

6+ Dminus Lp1113872 1113873


6 + ctprimekp

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭


(20)




EIdy4






0 (22a)


EI (22b)












0

1

2

3

4

5

6

7

8

z (m

)



hm (m)

1h

10h

20h

80h


(a)

z (m

)0

1

2

3

4

5

6

7



hm (m)

0h20h

40h70h


(b)








0 20 40 60 80 100 12013

135

14

145

15

155

16

165

t (h)

K q




0 20 40 60 80 100 120115

125

135

145

155

165

t (h)

K q



0 20 40 60 80 100 12013

15

17

19

21

23

t (h)

La = 8mLa = 10mLa = 12m

Lp = La ndash H

K q

(a)

La = 8mLa = 10mLa = 12m

0 20 40 60 80 100 12013

14

15

16

17

18

19

2

t (h)

K q

Lp = 2m

(b)






2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)


52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01



20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001






5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




3 Model Solution


k hm( 1113857 ks exp βhm( 1113857 (3a)


exp βhm( 1113857 (3b)





eminusT4


n1


1 + La2 + 2λ2nLa1113872 1113873

(4)


tan λLa( 1113857 + 2λ 0 (5)





minuszK

zZ+ K1113890 1113891

Z0 QB (6)


K21113868111386811138681113868Z0 1 (7)



s


minusLa1113876 1113877eZ2

middot F(s)

(8)


s + (14)


s + (14)

1113968La))) p(s)q(s) and



F(T) 1113944n

k1Res F(s)e

sT sk1113960 1113961 1113944

n

k1

p sk( 1113857

qprime sk( 1113857 (9)




F(T) sinh La minusZ( 111385721113858 1113859

sinh La2( 1113857

+ 1113944+infin

n0


4π2n2 + L2a1113872 1113873

eminus π2n2L

2a( 1113857+(14)( 1113857T

(10)


sn + (14)1113968




minus LaminusZ( )


minusLa1113876 1113877eZ2

middot F(T)

(11)






1 minusK2 Z T0( 1113857 (13)







Ki1 1

Kin+1 1(15a)


K1j K1 Zj T01113872 1113873 (15b)



1 + ΔZ (16)




⋮ ⋮ ⋮ ⋮ ⋮ ⋮




⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


Ki2

Ki3

⋮

Kinminus1

Kin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


dKiminus13

⋮

dKiminus1nminus1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)






ka

1113969


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18a)


1113969+


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18b)




Kq Mp

Ma

Mpun + Mpsa

Maun + Masa (19)

where

Maun 1113946La

zot



3

6+ Lminus La( 1113857


6minus ctprimeka

11139681113896 1113897

Mpun 1113946Lp



3

6+ Dminus Lp1113872 1113873


6 + ctprimekp

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭


(20)




EIdy4






0 (22a)


EI (22b)












0

1

2

3

4

5

6

7

8

z (m

)



hm (m)

1h

10h

20h

80h


(a)

z (m

)0

1

2

3

4

5

6

7



hm (m)

0h20h

40h70h


(b)








0 20 40 60 80 100 12013

135

14

145

15

155

16

165

t (h)

K q




0 20 40 60 80 100 120115

125

135

145

155

165

t (h)

K q



0 20 40 60 80 100 12013

15

17

19

21

23

t (h)

La = 8mLa = 10mLa = 12m

Lp = La ndash H

K q

(a)

La = 8mLa = 10mLa = 12m

0 20 40 60 80 100 12013

14

15

16

17

18

19

2

t (h)

K q

Lp = 2m

(b)






2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)


52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01



20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001






5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


F(T) sinh La minusZ( 111385721113858 1113859

sinh La2( 1113857

+ 1113944+infin

n0


4π2n2 + L2a1113872 1113873

eminus π2n2L

2a( 1113857+(14)( 1113857T

(10)


sn + (14)1113968




minus LaminusZ( )


minusLa1113876 1113877eZ2

middot F(T)

(11)






1 minusK2 Z T0( 1113857 (13)







Ki1 1

Kin+1 1(15a)


K1j K1 Zj T01113872 1113873 (15b)



1 + ΔZ (16)




⋮ ⋮ ⋮ ⋮ ⋮ ⋮




⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


Ki2

Ki3

⋮

Kinminus1

Kin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


dKiminus13

⋮

dKiminus1nminus1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)






ka

1113969


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18a)


1113969+


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18b)




Kq Mp

Ma

Mpun + Mpsa

Maun + Masa (19)

where

Maun 1113946La

zot



3

6+ Lminus La( 1113857


6minus ctprimeka

11139681113896 1113897

Mpun 1113946Lp



3

6+ Dminus Lp1113872 1113873


6 + ctprimekp

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭


(20)




EIdy4






0 (22a)


EI (22b)












0

1

2

3

4

5

6

7

8

z (m

)



hm (m)

1h

10h

20h

80h


(a)

z (m

)0

1

2

3

4

5

6

7



hm (m)

0h20h

40h70h


(b)








0 20 40 60 80 100 12013

135

14

145

15

155

16

165

t (h)

K q




0 20 40 60 80 100 120115

125

135

145

155

165

t (h)

K q



0 20 40 60 80 100 12013

15

17

19

21

23

t (h)

La = 8mLa = 10mLa = 12m

Lp = La ndash H

K q

(a)

La = 8mLa = 10mLa = 12m

0 20 40 60 80 100 12013

14

15

16

17

18

19

2

t (h)

K q

Lp = 2m

(b)






2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)


52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01



20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001






5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ka

1113969


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18a)


1113969+


1 + a0 ua minus uw( 1113857( 1113857d01113960 1113961

1minus 1d0( )

(18b)




Kq Mp

Ma

Mpun + Mpsa

Maun + Masa (19)

where

Maun 1113946La

zot



3

6+ Lminus La( 1113857


6minus ctprimeka

11139681113896 1113897

Mpun 1113946Lp



3

6+ Dminus Lp1113872 1113873


6 + ctprimekp

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭


(20)




EIdy4






0 (22a)


EI (22b)












0

1

2

3

4

5

6

7

8

z (m

)



hm (m)

1h

10h

20h

80h


(a)

z (m

)0

1

2

3

4

5

6

7



hm (m)

0h20h

40h70h


(b)








0 20 40 60 80 100 12013

135

14

145

15

155

16

165

t (h)

K q




0 20 40 60 80 100 120115

125

135

145

155

165

t (h)

K q



0 20 40 60 80 100 12013

15

17

19

21

23

t (h)

La = 8mLa = 10mLa = 12m

Lp = La ndash H

K q

(a)

La = 8mLa = 10mLa = 12m

0 20 40 60 80 100 12013

14

15

16

17

18

19

2

t (h)

K q

Lp = 2m

(b)






2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)


52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01



20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001






5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







0

1

2

3

4

5

6

7

8

z (m

)



hm (m)

1h

10h

20h

80h


(a)

z (m

)0

1

2

3

4

5

6

7



hm (m)

0h20h

40h70h


(b)








0 20 40 60 80 100 12013

135

14

145

15

155

16

165

t (h)

K q




0 20 40 60 80 100 120115

125

135

145

155

165

t (h)

K q



0 20 40 60 80 100 12013

15

17

19

21

23

t (h)

La = 8mLa = 10mLa = 12m

Lp = La ndash H

K q

(a)

La = 8mLa = 10mLa = 12m

0 20 40 60 80 100 12013

14

15

16

17

18

19

2

t (h)

K q

Lp = 2m

(b)






2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)


52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01



20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001






5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







0 20 40 60 80 100 12013

135

14

145

15

155

16

165

t (h)

K q




0 20 40 60 80 100 120115

125

135

145

155

165

t (h)

K q



0 20 40 60 80 100 12013

15

17

19

21

23

t (h)

La = 8mLa = 10mLa = 12m

Lp = La ndash H

K q

(a)

La = 8mLa = 10mLa = 12m

0 20 40 60 80 100 12013

14

15

16

17

18

19

2

t (h)

K q

Lp = 2m

(b)






2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)


52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01



20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001






5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





2 3 4 5 6 7 8 9 10025

5075

100125

05

075

1

125

15

175

2

D (m)t (h)

K q

(a)

D (m)t (h)

K q

2 3 4 5 6 7 8 9 10025

5075

100125

1125

15175

2225

25275

(b)


52

48

44

40

36

32

28

240 20 40 60 80 100

t (h)

Disp

lace

men

t (m

m)

a0 = 0 1

a0 = 0 01



20 40 60 80 1000t (h)

50

45

40

35

30

25

20

Disp

lace

men

t (m

m)

a0 = 001






5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




5 Conclusion






Data Availability




ndash2

0

2

4

6

8

10

12

14

z (m

)


t = 10hD = 8ma0 = 001


(a)

0

2

4

6

8

10

12

140 100 200 300 400 600500


z (m

)

t = 10hD = 8ma0 = 001


(b)



Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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