Finite Analytic Method for One-Dimensional Nonlinear Consolidation … · 2019. 7. 30. · ResearchArticle Finite Analytic Method for One-Dimensional Nonlinear Consolidation under

Research ArticleFinite Analytic Method for One-Dimensional NonlinearConsolidation under Time-Dependent Loading

Dawei Cheng12 WenkeWang12 Xi Chen3 and Zaiyong Zhang12

1School of Environmental Science and Engineering Changrsquoan University Xirsquoan China2Key Laboratory of Subsurface Hydrology and Ecological Effect in Arid Region Changrsquoan UniversityMinistry of Education China3School of Civil Engineering and Architecture Shaanxi Sci-Tech University Hanzhong China

Correspondence should be addressed to Dawei Cheng cheng198493qqcom

Received 23 October 2016 Accepted 2 March 2017 Published 30 March 2017

Academic Editor Longjun Dong

Copyright copy 2017 Dawei Cheng 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

For one-dimensional (1D) nonlinear consolidation the governing partial differential equation is nonlinear This paper developsthe finite analytic method (FAM) to simulate 1D nonlinear consolidation under different time-dependent loading and initialconditions To achieve this the assumption of constant initial effective stress is not considered and the governing partial differentialequation is transformed into the diffusion equation Then the finite analytic implicit scheme is established The convergence andstability of finite analytic numerical scheme are proven by a rigorous mathematical analysis In addition the paper obtains threecorrected semianalytical solutions undergoing suddenly imposed constant loading single ramp loading and trapezoidal cyclicloading respectively Comparisons of the results of FAMwith the three semianalytical solutions and the result of FDM respectivelyshow that the FAM can obtain stable and accurate numerical solutions and ensure the convergence of spatial discretization for 1Dnonlinear consolidation

1 Introduction

In geotechnical engineering subsoil is subjected to com-plicated time-dependent loading paths which are inducedby different vibration sources such as wave action ground-water level cyclic variation filling and discharging in silosand tanks vehicular traffic blast and earthquake In thesevibration sources such as waves changes in water tableloading and unloading in the granary and water towersand vehicle traffic can cause regular time-dependent loadingpaths [1 2] However time-dependent loading paths inducedby blast earthquake and so on are irregular [1 3] Manydiscriminant methods just like the waveform spectrumanalysis multivariate statistical methods and soft computingtechniques can identify these vibration sources [4 5]

Based on the cyclic loading classification discriminantcriteria proposed by Zienkiewicz and Bettess [6] theserelatively regular time-dependent loading paths inducedby wave action groundwater level cyclic variation and soforth are considered as low frequency cyclic loadings Three

effects including the soil movement inertia force and relativemovement inertia force between pore water and soil skeletoncan be ignored in soil consolidation behavior under lowfrequency loadings Many methods can be used to simu-late these low frequency cyclic loadings such as suddenlyimposed constant loading single ramp loading rectangu-lar cyclic loading triangle cyclic loading and trapezoidalcyclic loading [7ndash9] A period of trapezoidal cyclic loadingsincludes four phases loading phase the maximum loadphase unloading phase and rest phase respectively It is wellknown that when the duration time of the maximum loadingphase is 0 the trapezoidal cyclic loading will be transformedinto the triangle cyclic loading However when the durationtime of loading and unloading phase is 0 the trapezoidalcyclic loading will be transformed into the rectangular cyclicloading [10]Therefore this paper will discuss the trapezoidalcyclic loading as a result of its flexibility In addition thesuddenly imposed constant loading is a basic load formand the single ramp loading is the simplest time-dependentloading path Thus both of them are also discussed

HindawiShock and VibrationVolume 2017 Article ID 4071268 12 pageshttpsdoiorg10115520174071268

2 Shock and Vibration

The problem of consolidation under time-dependentloading has received attention by various authors [11ndash13]Based on the linear relationship of the soilrsquos constitutiverelation Terzaghi and Frohlich [11] extended Terzaghirsquos linearconsolidation theory to various cases of time-dependentloading following a single ramp loading Olson [12] presentedcharts for 1D consolidation for the case of simple ramp load-ing assuming a constant consolidation coefficient Favarettiand Soranzo [13] derived solutions for common types ofcyclic loadings However soilrsquos constitutive relationships areactually nonlinear [10 14] The coefficient of compressibilitydecreases with increasing effective stress In addition thepermeability coefficient decreases with void ratio decrease

Davis and Raymond [14] developed a nonlinear theory ofconsolidation assuming the relationship between void ratioand the logarithm of effective stress to conform to linear lawand the decrease in permeability coefficient is proportionalto the compressibility Xie et al [10] derived a semianalyticalsolution for trapezoidal cyclic loading by assuming the valid-ity of Davisrsquos nonlinear theory of consolidation Razouki etal [15] derived an analytical solution for consolidation underhaversine cyclic loading based on the same assumption Genget al [7] developed a semianalytical method to solve 1Dconsolidation behavior taking into account the relationshipbetween void ratio and the logarithm of linear effectivestress responses Research has revealed that a stress-strainrelation curve is more aligned with the hyperbolic modelfor certain types of soil such as soft clay [16 17] Shi et al[18] derived a semianalytical solution for consolidation undersuddenly imposed constant loading taking into account thestress-strain relation curve with a hyperbolic model and thedecrease in permeability coefficient being proportional tothe decrease in compressibility Zhang et al [19] adoptedthe same assumption deriving a semianalytical solutionfor consolidation under trapezoidal cyclic loading But forsimplifying nonlinear consolidation model the assumptionsof constant initial effective stress are often used such asDavisrsquos model and Zhangrsquos model which does not conform topractical condition and limits the application of thosemodelsin various initial conditions

It is worth noting that the governing partial differentialequation is nonlinear partial differential equation It is diffi-cult to obtain analytical solution except under specific condi-tions So developing numerical solutions for solving complexrealistic problems is necessary As a methodology the finiteanalytic method (FAM) was first introduced to mainly solveheat conduction and NavierndashStrokes equations [20 21] Thecombination under FAM of the numerical method andanalytical method gives higher precision good numericalstability and fast convergence and is widely employed in fluidmechanics and groundwater dynamics [22ndash25]

In this paper FAM is first developed to solve the govern-ing partial differential equation of 1D nonlinear consolida-tion taking into account the stress-strain behavior expressedby a hyperbolic model and where the permeability coefficientis proportional to compressibility Then three correctedsemianalytical solutions undergoing suddenly imposed con-stant loading single ramp loading and trapezoidal cyclicloading are respectively obtained without the assumption

of constant initial effective stress Finally the numericalsolution of FAM is compared with these three semianalyticalsolutions and the numerical solution of finite differencemethod (FDM) respectively

2 The Governing Partial Differential Equationof 1D Nonlinear Consolidation

Modify Terzaghirsquos hypothesis as follows [19](a) According to results of oedometer tests the stress-

strain relation curve is fitted with a hyperbolic model [16 17]

1205901015840120576 = 119864 + 1198991205901015840 (1)

where 1205901015840 is the effective stress 120576 is the vertical strain 119864 isthe initial elastic modulus 119899 = 1120576119891 is the reciprocal ofthe final vertical strain When 119899 equals zero the stress-strainrelationship is linear

(b) According to results of oedometer tests the coefficientof consolidation 119862V varies much less than the compressibilitycoefficient 120572V and may be taken as constant that is thedecrease in permeability coefficient 119896 is proportional to thedecrease in compressibility coefficient 120572V [14] The coefficientof consolidation 119862V is constant that is

119862V = 119896 (1 + 1198900)120572V = const (2)

where 1198900 is the initial void ratio(c) The initial effective stress is constant that is

12059010158400 = int11987101205741015840119911 119889119911119871 = 121205741015840119871 (3)

where 1205741015840 is the soil effective gravity and 119871 is the depth of thecalculation

(d) Imposed loadings change with timeOther assumptions are the same as those in Terzaghirsquos

theoryBased on the abovementioned assumptions except

hypothesis (c) the governing partial differential equationof 1D consolidation for time-dependent loading [19] can beestablished as follows

minus 119862V119864[ 1(119864 + 1198991205901015840)2

12059721199061205971199112 minus 2119899(119864 + 1198991205901015840)3

1205971205901015840120597119911 120597119906120597119911]

= 119864(119864 + 1198991205901015840)2

1205971205901015840120597119905 (4)

where 119906 is excess pore-water pressure and 119911 is the heightbeneath the upper boundary

Note that hypothesis (c) means that the initial effectivestress is the same for every point in depth which does notconform to practical condition and limits the application of(4) in various initial conditions Thus hypothesis (c) will notbe considered in the paper

Shock and Vibration 3

0

0

j minus 1

j

j + 1

L

Wn+1 On+1 En+1

Wn On EnminusΔz Δz

t

Local element ej

Figure 1 Domain and local element of finite analytical method

By the principle of effective stress this becomes

1205901015840 = 119902 (119905) minus 119906 (5)

where q(t) is cyclic loadingBy defining a new parameter 120596

120596 = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (6)

(4) can be simplified to the following form

119862V12059721205961205971199112 = 120597120596120597119905 + 119877 (119905) (7)

where 119877(119905) = (119864(1 + 1198900)[119864 + 119899119902(119905)]2)(119889119902(119905)119889119905)3 Finite Analytic Scheme

In the FAM the modeling domain of (7) can be discretizedinto small elements by using the spatial discretization Δ119911(Figure 1) A typical FAM element 119890119895 with the space step 2Δ119911in a given time interval Δ119905 = 119905119899 minus 119905119899minus1 is shown in Figure 1The origin of local coordinates is symbolized by119874Thereforethe local element is

119890119895 = minusΔ119911 le 119911 le Δ119911 0 le 119905 le Δ119905 (8)

The space term is treated as the implicit scheme of thetime term in local element 119890119895 The unsteady term 120597120596120597119905 takesthe values at point 119874 at the 119899th time step 119877(119905) also takes thevalues at the 119899th time step At the (119899 + 1)th time step (7) canbe written as follows

119862V (12059721205961205971199112 )119899+1 = 120597120596120597119905

10038161003816100381610038161003816100381610038161003816119899

119874

+ 119877 (119905)|119899 (9)

where (12059721205961205971199112)119899+1 indicates the values of the element at (119899+1)th time step (120597120596120597119905)|119899119874 indicates the unsteady term takingthe values at point119874 at the 119899th time step 119877(119905)|119899 indicates thatR(t) takes the values at the 119899th time step

The unsteady term 120597120596120597119905 is approximated by (120596119899+1119874 minus120596119899119874)Δ119905 and (9) can be normalized as follows

(12059721205961205971199112 )119899+1 asymp 1119862V

(120596119899+1119874 minus 120596119899119874Δ119905 + 119877 (119905)|119905=119899Δ119905) (10)

where 120596119899+1119874 and 120596119899119874 indicate the unsteady term taking thevalues at point 119874 at the (119899 + 1)th and 119899th time steprespectively 119877(119905)|119905=119899Δ119905 indicates that R(t) takes the values atthe 119899th time step

Here

119891 = 1119862V120596119899+1119874 minus 120596119899119874Δ119905 + 119877 (119899Δ119905) (11)

At the (119899 + 1)th time step (11) can be written as follows

(12059721205961205971199112 )119899+1 = 119891 (12)

The general solution of (12) is

120596119899+1 (119911) = 11989111991122 + 1198881119911 + 1198882 (13)

where the constants 1198881 and 1198882 can be specified by two nodalvalues of 120596 at the 119899th time step (119882119899 and 119864119899 in Figure 1) thatis

120596119882 = 119891Δ11991122 minus 1198881Δ119911 + 1198882 119911 = minusΔ119911120596119864 = 119891Δ11991122 + 1198881Δ119911 + 1198882 119911 = Δ119911

(14)

Solving a system of equations composed of (14) theconstants 1198881 and 1198882 are

1198881 = 120596119864 minus 1205961198822Δ119911 1198882 = 120596119864 + 1205961198822 minus 1198912 Δ1199112

(15)

Substituting (15) into (13) and letting the value of 119911 bezero

120596119899+1119874 = 120596119864 + 1205961198822 minus 1198912 Δ1199112 (16)

Substituting (11) into (16)

120596119899+1119874 = 12 + Δ1199112119862VΔ119905120596119864 +12 + Δ1199112119862VΔ119905120596119882

+ [ Δ1199112119862VΔ1199052 + Δ1199112119862VΔ119905]120596119899119874

minus Δ1199112119862V2 + Δ1199112119862VΔ119905119877 (119899Δ119905)

(17)


Supposing 120582 = Δ1199112119862VΔ119905 the local analytical solution forthe element 119890119895 is then written as follows

120596119899+1119874 = 12 + 120582120596119864 + 12 + 120582120596119882 + ( 1205822 + 120582)120596119899119874minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(18)

This local analytical solution evaluated at the interiornode 119895 of the element 119890119895 at the 119899th time step is

120596119899+1119895 = 12 + 120582120596119899+1119895minus1 + 12 + 120582120596119899+1119895+1 + ( 1205822 + 120582)120596119899119895minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(19)

Equation (19) is the FAM implicit scheme of the govern-ing partial differential equation of 1D nonlinear consolida-tion A set of algebraic equations can be formed from (19)which can be solved by the method of forward eliminationand backward substitution Associated with boundary con-ditions the algebraic equations are solved iteratively for thevariable 120596119899119895 (119899 = 1 2 3 119873 119895 = 1 2 3 119872) Here Nis the number of total time steps and M is the number oftotal nodes in the domain The excess pore-water at the node119895 and the 119899th time step is obtained by applying an inversiontransformation for (5) and (6)

4 The Convergence and Stability ofthe FAM Scheme

41 Convergence of Finite Analytical Numerical SchemeErrors of the FAM implicit scheme (19) are found by using(10) To demonstrate the convergence of FAM the followingequation is used to replace 120597120596120597119905 in (9)

12059712059612059711990510038161003816100381610038161003816100381610038161003816119899

119895

= 120596119899119895 minus 120596119899minus1119895Δ119905 + 119874 (Δ119905) (20)

Here 119874(Δ119905) is truncation error 120596119899119895 represents the exactsolution and 119899119895 represents finite analytic numerical solutionat the node 119895 and the 119899th time step respectively The errorbetween the exact solution 120596119899119895 and finite analytic numericalsolution 119899119895 can be written as 120576119899119895 thus

120576119899119895 = 120596119899119895 minus 119899119895 (21)

120596119899119895 should satisfy the following

120596119899119895 = 12 + 120582120596119899119895minus1 + 12 + 120582120596119899119895+1 + ( 1205822 + 120582)120596119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) + 119874 (Δ119905) (22)

And 119899119895 should satisfy the following

119899119895 = 12 + 120582119899119895minus1 + 12 + 120582119899119895+1 + ( 1205822 + 120582) 119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) (23)

Substituting (22) and (23) into (21)

120576119899119895 = 12 + 120582120576119899119895minus1 + 12 + 120582120576119899119895+1 + ( 1205822 + 120582) 120576119899minus1119895 + 119874 (Δ119905) (24)

Because the coefficients in (24) are positive

10038161003816100381610038161003816120576119899119895 10038161003816100381610038161003816 le 12 + 120582 10038161003816100381610038161003816120576119899119895minus110038161003816100381610038161003816 + 12 + 120582 10038161003816100381610038161003816120576119899119895+110038161003816100381610038161003816 + ( 1205822 + 120582) 10038161003816100381610038161003816120576119899minus1119895

10038161003816100381610038161003816+ 10038161003816100381610038161003816119885119899minus1

119895

10038161003816100381610038161003816 (25)

where 119885119899minus1119895 = 119874(Δ119905)

It can be shown that

120576119899max le 12 + 120582120576119899minus1max + 12 + 120582120576119899minus1max + ( 1205822 + 120582) 120576119899minus1max

+ 119885119899minus1max

(26)

where

120576119899minus1max = max 10038161003816100381610038161003816120576119899minus1119895minus1

10038161003816100381610038161003816 1 le 119895 le 119873 minus 1119885119899minus1max = max 10038161003816100381610038161003816119885119899minus1

119895

10038161003816100381610038161003816 1 le 119895 le 119873 minus 1 (27)

and that

120576119899max le 120576119899minus1max + 119885119899minus1max (119899 = 1 2 119872) (28)

Because the finite analytical equation and the partial differ-ential equation have the same initial condition namely 119899 =1 the finite analytical solution is consistent with the exactsolution That is 1205760max = 0

From (27) the following expressions can be obtained

1205761max le 1205760max + 1198850max = 1198850

max1205762max le 1205761max + 1198851

max le 1198850max + 1198851

max

120576119872max le 120576119872minus1max + 119885119872minus1

max le 1198850max + 1198851

max + sdot sdot sdot + 119885119872minus1max

(29)

Therefore

120576119872max le 119872 sdot 119885max = 119872 sdotmax |119874 (Δ119905)| (30)

where

119885max = max 119885119899max 1 le 119899 le 119872 (31)


When Δ119905 rarr 0 120576119872max rarr 0 thus 119899119895 rarr 120596119899119895 Thus it revealedthat the finite analytical scheme has uniform convergence

42 Stability of the Finite Analytical Numerical Scheme Thestability of finite analytical scheme (19) can be proven asfollows Let 120596119899 = (1205961198990 1205961198991 sdot sdot sdot 120596119899119898)119879 taking account ofcalculation error1205960 becomes1205960lowast and120596119899 becomes120596119899lowastThusthe error of any term is

119890119899119895 = 120596119899119895 minus 120596119899lowast119895 (32)

From (19) the error 119890119899119895 fits withminus1120582119890119899+1119895minus1 + 2 + 120582120582 119890119899+1119895 minus 1120582119890119899+1119895+1 = 119890119899119895 (33)

It can be proven that the error at (119899 + 1)th time step 119890119899+1119895

is between the error at the node 0 and the (119899 + 1)th time step119890119899+10 the error at the node119898 and the (119899 + 1)th time step 119890119899+1119898 and the extreme of the error in 119899th time step 119890119899119895 that is

119898119899 le 119890119899+1119895 le 119872119899 (34)

where

119898119899 = min 119890119899+10 119890119899119895 119890119899+1119898 119872119899 = max 119890119899+10 119890119899119895 119890119899+1119898 (35)

Equation (34) can be proven as follows Assume 119890119899+1119896 =min119895 119890119899+1119895 When 119896 = 0 or 119896 = 119898 there is 119890119899+1119895 ge 119898119899 clearlyWhen 0 lt 119896 lt 119898 taking into account (33)

119890119899+1119896 = 119890119899119896 + 1120582 (119890119899+1119896minus1 + 119890119899+1119896+1 minus 2119890119899+1119896 )ge 119890119899119896 + 1120582 (119890119899+1119896 + 119890119899+1119896 minus 2119890119899+1119896 ) = 119890119899119896 ge 119898119899

(36)

Evidenced by the same token 119890119899+1119896 le 119872119899The finite analytical equation and the partial differential

equation have the same boundary condition namely 1198901198990 =119890119899119898 = 0 and taking into account (34)

min119895119890119899119895 le 119890119899+1119895 le max

119895119890119899119895 (37)

Therefore

10038171003817100381710038171003817e119899+110038171003817100381710038171003817infin = max119895

10038161003816100381610038161003816119890119899+1119895

10038161003816100381610038161003816 le max119895

10038161003816100381610038161003816119890119899119895 10038161003816100381610038161003816 = 1003817100381710038171003817e1198991003817100381710038171003817infin (38)

The following expression can be obtained

10038171003817100381710038171003817e119899+110038171003817100381710038171003817infin le 1003817100381710038171003817e1198991003817100381710038171003817infin le sdot sdot sdot le 10038171003817100381710038171003817e010038171003817100381710038171003817infin (39)

2L

z

Pervious

Pervious

q(t)

Figure 2 1D foundation model

Equation (39) reveals that the finite analytical scheme (19) isunconditionally stable

5 Semianalytical Solution of GoverningDifferential Equation

The analytical model fitting with all assumptions excepthypothesis (c) mentioned in Section 2 is shown in Figure 2The depth of the soft clay with permeable top and bottom is2119871 Time-dependent loading 119902(119905) is imposed on top of the softclay

According to definition in Figure 2 the initial andboundary conditions of the problem are

119906 (119911 0) = 119902 (0) if 0 le 119911 le 2119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0119906 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(40)

By using (6) the transform of (40) is

120596 (119911 0) = 119876 (0) if 0 le 119911 le 2119871 119905 = 0120596 (0 119905) = 0 if 119911 = 0 119905 gt 0120596 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(41)

where

119876 (119905) = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (42)

When hypothesis (c) is not considered the generalsolution of the problem is modified as

119906 = 120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900) (43)


The average degrees of consolidation of the soil defined inpore-water pressure 119880119901 can be given by

119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906119889119911

= 119902 (119905)119902119906minus 12119871119902119906 int

2119871

0

120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900)119889119911

(44)

where

120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905 sin(1198981205871199112119871 )

sdot [int119905

0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591 minus 119876 (0)]119898 = 1 3 5

(45)

The average degrees of consolidation 119880119901 can be approxi-mately written as follows

119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906 119889119911

= 119902 (119905) minus (12119871)sum119895=2119871Δ119911119895=1 119906119895Δ119911119902119906

(46)

According to (43) and (44) the exact expression of120596(119911 119905) is the key to obtaining excess pore-water pressureand average degrees of consolidation for different time-dependent loadings The following sections will discuss eachof 120596(119911 119905) under suddenly imposed constant loading singleramp loading and trapezoidal cyclic loading

(a) Suddenly Imposed Constant Loading

119902 (119905) = 119902119906 (47)

Therefore

119876 (0) = minus(1 + 1198900) 119902119906119864 + 119899119902119906 (48)


120596 (119911 119905) = infinsum119898=1

4119898120587 sin(1198981205871199112119871 )

sdot 119890minus119862V(1198981205872119871)2119905 (1 + 1198900) 119902119906119864 + 119899119902119906 119898 = 1 3 5

(49)

Substituting (49) into (43) and (44) or (46) the excesspore-water pressure and average degrees of consolidation canbe obtained for suddenly imposed constant loading

(b) Single Ramp Loading

119902 (119905) = 1199021199061199050 119905 if 119905 le 1199050119902119906 if 119905 gt 1199050 (50)

Therefore

119876 (0) = 0 (51)

From (7) and (50)

119877 (119905) = 119864 (1 + 1198900) 1199021199061199050[119864 + 119899 (1199021199061199050) 119905]2 119905 le 11990500 119905 gt 1199050

(52)

The semianalytical solution to the problem is

120596 (119911 119905)

=

minus infinsum119898=1

4119898120587 sin(1198981205871199112119871 )((119864119894 (1 minus119887119889119888 ) minus 119864119894 (1 minus119889 (119905119888 + 119887)119888 )) sdot (119905119887119888119889 + 1198872119889) minus 119890119889(119905119888+119887)119888119887119888 + 119890119887119889119888 (1199051198882 + 119887119888)) 119886119890minus119889(119905119888+119887)1198881198871198882 (119905119888 + 119887) if 119905 le 1199050minus infinsum119898=1

4119898120587 sin(1198981205871199112119871 )((119864119894 (1 minus119887119889119888 ) minus 119864119894 (1 minus119889 (1199050119888 + 119887)119888 )) sdot (1199050119887119888119889 + 1198872119889) minus 119890119889(1199050119888+119887)119888119887119888 + 119890119887119889119888 (11990501198882 + 119887119888)) 119886119890minus119889((119905+1199050)119888+119887)1198881198871198882 (1199050119888 + 119887) if 119905 gt 1199050

119898 = 1 3 5

(53)

where

119886 = 119864 (1 + 1198900) 1199021199061199050 119887 = 119864119888 = 1198991199021199061199050

119889 = 119862V (1198981205872119871 )2

119864119894 (1198861 1198871) = intinfin

1119890minus1199091198871119909minus1198861119889119909

(54)

1198861 and 1198871 are nonzero constantsSubstituting (53) into (43) and (44) or (46) the excess

pore-water pressure and average degrees of consolidation canbe obtained for single ramp loading


(c) Trapezoidal Cyclic Loading (Figure 3)

119902 (119905) =

1199021199061205721199050 [119905 minus (119873 minus 1) 1205731199050] if (119873 minus 1) 1205731199050 le 119905 le [(119873 minus 1) 120573 + 120572] 1199050119902119906 if [(119873 minus 1) 120573 + 120572] 1199050 le 119905 le [(119873 minus 1) 120573 + (1 minus 120572)] 1199050minus 1199021199061205721199050 [119905 minus (119873 minus 1) 1205731199050 minus 1199050] if [(119873 minus 1) 120573 + (1 minus 120572)] 1199050 le 119905 le [(119873 minus 1) 120573 + 1] 11990500 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(55)

where 119902119906 is the maximum value of loading during a cycleperiod 1205731199050 is the period of the loading 120572 120573 are loadingfactors N is the cycle number

Therefore119876 (0) = 0 (56)

From (7) and (50)

119877 (119905) =

119864 (1 + 1198900) 119902119906 (1205721199050)[119864 + 119899 (1199021199061205721199050) (119905 minus (119873 minus 1) 1205731199050)]2 if (119873 minus 1) 1205731199050 le 119905 le [(119873 minus 1) 120573 + 120572] 11990500 if [(119873 minus 1) 120573 + 120572] 1199050 le 119905 le [(119873 minus 1) 120573 + (1 minus 120572)] 1199050minus 119864 (1 + 1198900) 119902119906 (1205721199050)[119864 minus 119899 (1199021199061205721199050) (119905 minus ((119873 minus 1) 120573 + 1) 1199050)]2 if [(119873 minus 1) 120573 + (1 minus 120572)] 1199050 le 119905 le [(119873 minus 1) 120573 + 1] 1199050

0 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(57)

The semianalytical solution for consolidation under theloading case is shown in theAppendix It is clear that the formof the solution is complex Compared with the semianalyticalsolution (A1) the general solution obtained by using (45)and (57) can be easier to program

6 Comparison between Results from FAM andSemianalytical Solutions

The numerical solution of FAM is compared with the semi-analytical solution for different loading cases The value ofconstant parameters from [26] are 119862V = 5789 times 10minus3m2day119864 = 16878 times 106 Pa 119899 = 33 1198900 = 11167 2119871 = 64m and119905 = 800 days In the FAM the time step Δ119905 is 01 days(a) Suddenly Imposed Constant Loading Case For the sud-denly imposed constant loading case 119902119906 = 2 times 105 PaThe duration of loading is 800 days The average degreesof consolidation 119880119901 is calculated by using (46) adoptingthe same space step Δ119911 for FAM and the semianalyticalsolution 120596(119911 119905) is calculated by using (49) The number ofterms in the series is 20 Curves of the average degrees ofconsolidation 119880119901 versus time factor 119879V (shown in (58)) fordifferent methods with different space steps Δ119911 are plotted inFigure 4 (ASM indicates ldquosemianalytical solution methodrdquo inthe figure) Comparison between the result of FAM and thesemianalytical solution shows a negligible gap indicating thatthe method proposed in this paper has sufficient precision

119879V = 119862V119905119871minus2 (58)

In order to investigate the accuracy of this result furtherthe percentage deviation of 119880119901 between FAM and semian-alytical solutions is evaluated To illustrate the differenceof 119880119901 between the two 21 data points of the percentagedeviation of 119880119901 are chosen at regular intervals for each datagroup plotted in Figure 5The percentage deviation decreaseswith decreasing space step Δ119911 demonstrating that under thesame conditions a smaller space step is helpful in enhancingthe precision of calculation However there is a clear sideeffect the amount of calculation increases Considering theprecision and the amount of calculation Δ119911 = 02m isselected in the next two loading cases single ramp loadingcase and trapezoidal cyclic loading case

(b) Single RampLoadingCase In the single ramp loading case119902119906 = 2 times 105 Pa The value of 119902(119905) increases linearly from 0to 119902119906 during the loading increase phase 1199050 In this sectionthe influence of the loading increase (phase 1199050) is discussedThe loading increasing (phase 1199050) is 1415 283 and 4245 daysrespectively and the value of 119902(119905) remains constant Further120596(119911 119905) is calculated by using (53) in the semianalytical solu-tion In order to avoid overflow the number of terms in theseries is 6 The curves of the average degrees of consolidation119880119901 versus time factor 119879V for different methods with differentloading increase periods are plotted in Figure 6 As shownin Figure 6 119880119901 increases with increasing 119879V but there is aturning point after which the rate of increase slows downFor the different loading increase (period 1199050) the degree ofconsolidation with a short loading increase is higher than thedegree of consolidation with a long loading increase period


qu

0t0 t0 t0 (N minus 1)t0

t

middot middot middot middot middot middot

q(t)

Figure 3 Trapezoidal cyclic loading

0

02

04

06

08

0 02 04 06

Up

T

ASM Δz = 08 mASM Δz = 02 mASM Δz = 005 m

FAM Δz = 08 mFAM Δz = 02 mFAM Δz = 005 m

Figure 4 119880119901 versus 119879V curves for constant loading

Figure 6 also shows that the agreement between the curvesobtained by using the FAM and the semianalytical solutionsis quite remarkable To illustrate the difference of119880119901 betweenthe FAM and the semianalytical solution under single ramploading case 21 data points of the percentage deviation of119880119901are chosen at regular intervals for each data group plottedin Figure 7 Figure 7 shows that the percentage deviation of119880119901 between the FAM and the semianalytical solution changeswith increasing119879V Although there are differences in differentloading increase period cases the percentage deviation of119880119901is generally negligible

(c) Trapezoidal Cyclic Loading Case In the trapezoidal cyclicloading case themaximum value of loading 119902119906 during a cycleperiod is 2times 105 PaTheperiod of the loading is written as12057311990501199050 is a loading period of 1415 daysThe loading factors120572 and120573are 03 and 10 03 and 12 03 and 24 and 05 and 12 whichrepresent trapezoidal cyclic loading without a rest periodtrapezoidal cyclic loading with 021199050 rest period trapezoidalcyclic loading with 141199050 rest period and triangularly cyclicloading with a 021199050 rest period respectively Here 120596(119911 119905)is calculated by using (45) in the semianalytical solution Inorder to avoid overflow the number of terms in the series

0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T

Figure 5 Percentage deviation of 119880119901 between the FAM and thesemianalytical solution versus 119879V curves for constant loading

0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM

t0 = 1415 dayst0 = 283 dayst0 = 4245 days


Figure 6 119880119901 versus 119879V curves for single ramp loading

is 10 The curves of the average degrees of consolidation 119880119901versus time factor 119879V with different loading factors 120572 and 120573for different methods are plotted in Figure 8 showing that119880119901 tends to be fluctuating while increasing with increasing119879V To illustrate the difference of119880119901 between the FAMand thesemianalytical solution under trapezoidal cyclic loading case21 data points of the percentage deviation of119880119901 are chosen atregular intervals for each data group plotted in Figure 8 Thecomparison between the result of FAMand the semianalyticalsolution shows a negligible gap in Figure 8 The percentagedeviation of 119880119901 between the FAM and the semianalyticalsolution for trapezoidal cyclic loading is plotted in Figure 9indicating that fluctuation deviation is small in range in termsof percent and is thus negligible


00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T


Figure 7 Percentage deviation of 119880119901 between the FAM and thesemianalytical solution versus 119879V curves for single ramp loading

0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24

Figure 8 119880119901 versus 119879V curves for trapezoidal cyclic loading

7 Convergence of the SpatialDiscretization for FAM

Proving the convergence of the spatial discretization by amathematical approach is difficult To further explore theconvergence of the spatial discretization a 1D nonlinearconsolidation in the trapezoidal cyclic loading case was intro-duced as an exampleThe initial and boundary conditions areas follows

119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24

Figure 9 Percentage deviation of 119880119901 between the FAM andthe semianalytical solution versus 119879V curves for trapezoidal cyclicloading


120596 (119911 0) = minus2116711991116878 minus 33119911 if 0 le 119911 le 119871 119905 = 0120596 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)

In this example a soft clay layer with permeable top andimpermeable bottomunder initial excess pore-water pressuretriangular distribution is introduced where 119902119906 = 2 times 105 Pa1199050 = 1415 days 120572 = 03 and 120573 = 12 The values ofthe constant parameters are the same as those in Section 6However for the case the analytical solution is difficult tobe obtained But it is easy to solve by numerical methodsuch as FAM and FDM For comparison the space step Δ119911is set to 01 02 04 and 08m by FAM respectively Toachieve good solutions as comparison the space step Δ119911 isadopted to 01m by FDM The space step has influence ondegree of consolidation calculated value of 119880119901 To avoid thisthe excess pore-water pressure 119906 calculated by FAM withdifferent space steps is compared with FDM with specificspace step Δ119911 = 01m shown in Figure 10 It reveals that theresults of FAM matches well with the result of FDM even ifΔ119911 is 08m Therefore FAM can ensure the convergence ofspatial discretization

8 Conclusions

This paper develops the finite analytic method (FAM) forsolving a 1D nonlinear consolidation undergoing time-dependent loadingThe study shows that FAM is very efficientand convenient in simulating the nonlinear consolidationThe following conclusions can be drawn from this research

(1) The governing partial differential equation with non-homogeneous character can be transformed into the


0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)

FDM t = 200 daysFDM t = 400 daysFDM t = 600 daysFDM t = 800 days

FAM t = 200 daysFAM t = 400 daysFAM t = 600 daysFAM t = 800 days

(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m

Figure 10 Comparison between the results of FAM and the results of FDM

diffusion equation where the FAM uses an analyticalsolution in small local elements to formulate thealgebraic representation of the 1D nonlinear consol-idation undergoing time-dependent loading

(2) The FAM is a combination of analytical solution andnumerical approach The finite analytical scheme hasuniform convergence and is unconditionally stablefor 1D nonlinear consolidation Therefore FAM canobtain smooth and high-precision solutions

(3) The comparison between the results of FAM and thesemianalytical solution of the 1Dnonlinear consolida-tion undergoing suddenly imposed constant loadingsingle ramp loading and trapezoidal cyclic loadingindicates that FAM can obtain stable and accuratenumerical solutions

(4) The comparison between the results of FAM withdifferent space steps and the results of FDM with asmall space step indicates that FAM can ensure theconvergence of spatial discretization

Appendix

The modified semianalytical solution of 120596(119911 119905) of a trape-zoidal cyclic loading case is as follows

120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)

where 119898 = 1 3 5 119894 = 1 2 3 4 represents time 119905 lying indifferent loading phase during a loading cycle119879119898119873 representsthe integration for variable 120591 from 0 to (119873 minus 1)1205731199050119879119898119873 = int(119873minus1)1205731199050

0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 minus 119899119902119906 [120591 minus (119896 minus 1) 1205731199050 minus 1199050] 1205721199050]2 119889120591 (A2)


When (119873 minus 1)1205731199050 le 119905 le [(119873 minus 1)120573 + 120572]1199050 (Phase 1)1198791198981 (119905)= int119905

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)

When [(119873minus1)120573+120572]1199050 le 119905 le [(119873minus1)120573+(1minus120572)]1199050 (Phase2)

1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)

When [(119873minus1)120573+(1minus120572)]1199050 le 119905 le [(119873minus1)120573+1]1199050 (Phase3)

1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 minus 119899119902119906 [120591 minus (119896 minus 1) 1205731199050 minus 1199050] 1205721199050]2 119889120591(A5)

When [(119873 minus 1)120573 + 1]1199050 le 119905 le 1198731205731199050 (Phase 4)1198791198984 (119905) = 1198791198983 ([(119873 minus 1) 120573 + 1] 1199050) (A6)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This study was supported by National Natural ScienceFoundation of China (nos 41230314 41602237) and theFundamental Research Funds for the Central Universities ofChina (nos 310829151076 310829161011 and 310829163307)

References

[1] D-Y Xie Soil Dynamics Xirsquoan Jiaotong University Press XirsquoanChina 1987 (Chinese)

[2] J-B Xu C-G Yan X Zhao K Du H Li and Y-L Xie ldquoMoni-toring of train-induced vibrations on rock slopesrdquo InternationalJournal of Distributed SensorNetworks vol 13 no 1 pp 1ndash7 2017

[3] J-B Xu C-G Yan H Yuan et al ldquoThe blasting vibration ofdeep buried soft rock tunnelrdquo Electronic Journal of GeotechnicalEngineering vol 22 no 4 pp 1219ndash1237 2017

[4] L-J Dong J Wesseloo Y Potvin and X-B Li ldquoDiscriminantmodels of blasts and seismic events in mine seismologyrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 86 pp 282ndash291 2016

[5] L-J Dong X-B Li and G-N Xie ldquoNonlinear methodologiesfor identifying seismic event and nuclear explosion usingrandom forest support vector machine and naive bayes clas-sificationrdquo Abstract and Applied Analysis vol 2014 Article ID459137 8 pages 2014

[6] O C Zienkiewicz and P Bettess ldquoSoils and other satu-rated media under transient dynamic conditionsrdquo in SoilMechanicsmdashTransient and Cyclic Loads pp 1ndash16 John Wiley ampSons 1982

[7] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[8] D B McLean Permanent Deformation Characteristics ofAsphalt Concrete University of California Berkeley Calif USA1974

[9] Y H Huang Pavement Analysis and Design Prentice HallEnglewood Cliffs NJ USA 1993

[10] K-H Xie T Qi and Y-Q Dong ldquoNonlinear analytical solutionfor one-dimensional consolidation of soft soil under cyclicloadingrdquo Journal of Zhejiang University Science vol 7 no 8 pp1358ndash1364 2006

[11] K Terzaghi and O K Frohlich Theorie der Setzung vonTonschichten Franz Deutike Leipzig Germany 1936

[12] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977

[13] M Favaretti andM Soranzo ldquoA simplified consolidation theoryin cyclic loading conditionsrdquo in Proceedings of the InternationalSymposium on Compression and Consolidation of Clayey Soilspp 405ndash409 A A Balkema Hiroshima Japan May 1995

[14] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[15] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013

[16] Y-R Zheng and L Kong Fundamentals of Geotechnical PlasticMechanics China Building Industry Press Beijing China 1989(Chinese)

[17] S-M Xu ldquoThe normalization curve method for saturated clayfoundation settlement calculationrdquo Yan Tu Gong Cheng XueBao vol 9 no 4 pp 70ndash77 1987 (Chinese)

[18] J-Y Shi L-A Yang and W-B Zhao ldquoResearch of one-dimensional consolidation theory considering nonlinear char-acteristics of soilrdquo Journal of Hohai University vol 29 no 1 pp1ndash5 2001 (Chinese)

[19] L Zhang S-L Sun X-N Gong and J Zhang ldquoSolutionfor one-dimensional consolidation based on hyperbola modelunder cyclic loadingrdquo Rock and Soil Mechanics vol 31 no 2pp 456ndash460 2010 (Chinese)

[20] C J Chen and P Li ldquoThe finite analytic method for steadyand unsteady heat transfer problemsrdquo in Proceedings of the JointNationalHeat Transfer Conference pp 80ndash86 American Societyof Mechanical Engineers and American Institute of ChemicalEngineers Orlando Fla USA July 1980

[21] C-J Chen and H-C Chen ldquoFinite analytic numerical methodfor unsteady two-dimensional Navier-Stokes equationsrdquo Jour-nal of Computational Physics vol 53 no 2 pp 209ndash226 1984

[22] W-F Tsai C-J Chen and H-C Tien ldquoFinite analytic numer-ical solutions for unsaturated flow with irregular boundariesrdquoJournal of Hydraulic Engineering vol 119 no 11 pp 1274ndash12981993

[23] W-K Wang Z-X Dai J-T Li and L-L Zhou ldquoA hybridLaplace transform finite analytic method for solving transportproblems with large Peclet and Courant numbersrdquo Computersand Geosciences vol 49 pp 182ndash189 2012

[24] Z-Y ZhangW-KWang L Chen et al ldquoFinite analyticmethodfor solving the unsaturated flow equationrdquoVadose Zone Journalvol 14 no 1 2015


[25] Z-Y Zhang W-K Wang T-C J Yeh et al ldquoFinite analyticmethod based onmixed-formRichardsrsquo equation for simulatingwater flow in vadose zonerdquo Journal of Hydrology vol 537 pp146ndash156 2016

[26] L Zhang One-Dimensional Consolidation Theory Modified byHyperbola Model under Generalized Time-Depending LoadingHohai University Nanjing China 2007 (Chinese)

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The problem of consolidation under time-dependentloading has received attention by various authors [11ndash13]Based on the linear relationship of the soilrsquos constitutiverelation Terzaghi and Frohlich [11] extended Terzaghirsquos linearconsolidation theory to various cases of time-dependentloading following a single ramp loading Olson [12] presentedcharts for 1D consolidation for the case of simple ramp load-ing assuming a constant consolidation coefficient Favarettiand Soranzo [13] derived solutions for common types ofcyclic loadings However soilrsquos constitutive relationships areactually nonlinear [10 14] The coefficient of compressibilitydecreases with increasing effective stress In addition thepermeability coefficient decreases with void ratio decrease

Davis and Raymond [14] developed a nonlinear theory ofconsolidation assuming the relationship between void ratioand the logarithm of effective stress to conform to linear lawand the decrease in permeability coefficient is proportionalto the compressibility Xie et al [10] derived a semianalyticalsolution for trapezoidal cyclic loading by assuming the valid-ity of Davisrsquos nonlinear theory of consolidation Razouki etal [15] derived an analytical solution for consolidation underhaversine cyclic loading based on the same assumption Genget al [7] developed a semianalytical method to solve 1Dconsolidation behavior taking into account the relationshipbetween void ratio and the logarithm of linear effectivestress responses Research has revealed that a stress-strainrelation curve is more aligned with the hyperbolic modelfor certain types of soil such as soft clay [16 17] Shi et al[18] derived a semianalytical solution for consolidation undersuddenly imposed constant loading taking into account thestress-strain relation curve with a hyperbolic model and thedecrease in permeability coefficient being proportional tothe decrease in compressibility Zhang et al [19] adoptedthe same assumption deriving a semianalytical solutionfor consolidation under trapezoidal cyclic loading But forsimplifying nonlinear consolidation model the assumptionsof constant initial effective stress are often used such asDavisrsquos model and Zhangrsquos model which does not conform topractical condition and limits the application of thosemodelsin various initial conditions

It is worth noting that the governing partial differentialequation is nonlinear partial differential equation It is diffi-cult to obtain analytical solution except under specific condi-tions So developing numerical solutions for solving complexrealistic problems is necessary As a methodology the finiteanalytic method (FAM) was first introduced to mainly solveheat conduction and NavierndashStrokes equations [20 21] Thecombination under FAM of the numerical method andanalytical method gives higher precision good numericalstability and fast convergence and is widely employed in fluidmechanics and groundwater dynamics [22ndash25]

In this paper FAM is first developed to solve the govern-ing partial differential equation of 1D nonlinear consolida-tion taking into account the stress-strain behavior expressedby a hyperbolic model and where the permeability coefficientis proportional to compressibility Then three correctedsemianalytical solutions undergoing suddenly imposed con-stant loading single ramp loading and trapezoidal cyclicloading are respectively obtained without the assumption

of constant initial effective stress Finally the numericalsolution of FAM is compared with these three semianalyticalsolutions and the numerical solution of finite differencemethod (FDM) respectively

2 The Governing Partial Differential Equationof 1D Nonlinear Consolidation

Modify Terzaghirsquos hypothesis as follows [19](a) According to results of oedometer tests the stress-

strain relation curve is fitted with a hyperbolic model [16 17]

1205901015840120576 = 119864 + 1198991205901015840 (1)

where 1205901015840 is the effective stress 120576 is the vertical strain 119864 isthe initial elastic modulus 119899 = 1120576119891 is the reciprocal ofthe final vertical strain When 119899 equals zero the stress-strainrelationship is linear

(b) According to results of oedometer tests the coefficientof consolidation 119862V varies much less than the compressibilitycoefficient 120572V and may be taken as constant that is thedecrease in permeability coefficient 119896 is proportional to thedecrease in compressibility coefficient 120572V [14] The coefficientof consolidation 119862V is constant that is

119862V = 119896 (1 + 1198900)120572V = const (2)

where 1198900 is the initial void ratio(c) The initial effective stress is constant that is

12059010158400 = int11987101205741015840119911 119889119911119871 = 121205741015840119871 (3)

where 1205741015840 is the soil effective gravity and 119871 is the depth of thecalculation

(d) Imposed loadings change with timeOther assumptions are the same as those in Terzaghirsquos

theoryBased on the abovementioned assumptions except

hypothesis (c) the governing partial differential equationof 1D consolidation for time-dependent loading [19] can beestablished as follows

minus 119862V119864[ 1(119864 + 1198991205901015840)2

12059721199061205971199112 minus 2119899(119864 + 1198991205901015840)3

1205971205901015840120597119911 120597119906120597119911]

= 119864(119864 + 1198991205901015840)2

1205971205901015840120597119905 (4)

where 119906 is excess pore-water pressure and 119911 is the heightbeneath the upper boundary

Note that hypothesis (c) means that the initial effectivestress is the same for every point in depth which does notconform to practical condition and limits the application of(4) in various initial conditions Thus hypothesis (c) will notbe considered in the paper


0

0

j minus 1

j

j + 1

L

Wn+1 On+1 En+1


t

Local element ej



1205901015840 = 119902 (119905) minus 119906 (5)


120596 = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (6)


119862V12059721205961205971199112 = 120597120596120597119905 + 119877 (119905) (7)





119862V (12059721205961205971199112 )119899+1 = 120597120596120597119905

10038161003816100381610038161003816100381610038161003816119899

119874

+ 119877 (119905)|119899 (9)



(12059721205961205971199112 )119899+1 asymp 1119862V

(120596119899+1119874 minus 120596119899119874Δ119905 + 119877 (119905)|119905=119899Δ119905) (10)


Here

119891 = 1119862V120596119899+1119874 minus 120596119899119874Δ119905 + 119877 (119899Δ119905) (11)


(12059721205961205971199112 )119899+1 = 119891 (12)


120596119899+1 (119911) = 11989111991122 + 1198881119911 + 1198882 (13)



(14)


1198881 = 120596119864 minus 1205961198822Δ119911 1198882 = 120596119864 + 1205961198822 minus 1198912 Δ1199112

(15)


120596119899+1119874 = 120596119864 + 1205961198822 minus 1198912 Δ1199112 (16)


120596119899+1119874 = 12 + Δ1199112119862VΔ119905120596119864 +12 + Δ1199112119862VΔ119905120596119882

+ [ Δ1199112119862VΔ1199052 + Δ1199112119862VΔ119905]120596119899119874

minus Δ1199112119862V2 + Δ1199112119862VΔ119905119877 (119899Δ119905)

(17)



120596119899+1119874 = 12 + 120582120596119864 + 12 + 120582120596119882 + ( 1205822 + 120582)120596119899119874minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(18)


120596119899+1119895 = 12 + 120582120596119899+1119895minus1 + 12 + 120582120596119899+1119895+1 + ( 1205822 + 120582)120596119899119895minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(19)




12059712059612059711990510038161003816100381610038161003816100381610038161003816119899

119895

= 120596119899119895 minus 120596119899minus1119895Δ119905 + 119874 (Δ119905) (20)


120576119899119895 = 120596119899119895 minus 119899119895 (21)


120596119899119895 = 12 + 120582120596119899119895minus1 + 12 + 120582120596119899119895+1 + ( 1205822 + 120582)120596119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) + 119874 (Δ119905) (22)


119899119895 = 12 + 120582119899119895minus1 + 12 + 120582119899119895+1 + ( 1205822 + 120582) 119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) (23)


120576119899119895 = 12 + 120582120576119899119895minus1 + 12 + 120582120576119899119895+1 + ( 1205822 + 120582) 120576119899minus1119895 + 119874 (Δ119905) (24)


10038161003816100381610038161003816120576119899119895 10038161003816100381610038161003816 le 12 + 120582 10038161003816100381610038161003816120576119899119895minus110038161003816100381610038161003816 + 12 + 120582 10038161003816100381610038161003816120576119899119895+110038161003816100381610038161003816 + ( 1205822 + 120582) 10038161003816100381610038161003816120576119899minus1119895

10038161003816100381610038161003816+ 10038161003816100381610038161003816119885119899minus1

119895

10038161003816100381610038161003816 (25)

where 119885119899minus1119895 = 119874(Δ119905)



+ 119885119899minus1max

(26)

where



119895

10038161003816100381610038161003816 1 le 119895 le 119873 minus 1 (27)

and that




1205761max le 1205760max + 1198850max = 1198850

max1205762max le 1205761max + 1198851

max le 1198850max + 1198851

max


max le 1198850max + 1198851


(29)

Therefore


where

119885max = max 119885119899max 1 le 119899 le 119872 (31)




119890119899119895 = 120596119899119895 minus 120596119899lowast119895 (32)




119898119899 le 119890119899+1119895 le 119872119899 (34)

where

119898119899 = min 119890119899+10 119890119899119895 119890119899+1119898 119872119899 = max 119890119899+10 119890119899119895 119890119899+1119898 (35)



(36)



min119895119890119899119895 le 119890119899+1119895 le max

119895119890119899119895 (37)

Therefore

10038171003817100381710038171003817e119899+110038171003817100381710038171003817infin = max119895

10038161003816100381610038161003816119890119899+1119895

10038161003816100381610038161003816 le max119895

10038161003816100381610038161003816119890119899119895 10038161003816100381610038161003816 = 1003817100381710038171003817e1198991003817100381710038171003817infin (38)



2L

z

Pervious

Pervious

q(t)






119906 (119911 0) = 119902 (0) if 0 le 119911 le 2119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0119906 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(40)


120596 (119911 0) = 119876 (0) if 0 le 119911 le 2119871 119905 = 0120596 (0 119905) = 0 if 119911 = 0 119905 gt 0120596 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(41)

where

119876 (119905) = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (42)


119906 = 120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900) (43)



119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906119889119911

= 119902 (119905)119902119906minus 12119871119902119906 int

2119871

0

120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900)119889119911

(44)

where

120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905 sin(1198981205871199112119871 )

sdot [int119905

0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591 minus 119876 (0)]119898 = 1 3 5

(45)


119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906 119889119911

= 119902 (119905) minus (12119871)sum119895=2119871Δ119911119895=1 119906119895Δ119911119902119906

(46)



119902 (119905) = 119902119906 (47)

Therefore

119876 (0) = minus(1 + 1198900) 119902119906119864 + 119899119902119906 (48)


120596 (119911 119905) = infinsum119898=1

4119898120587 sin(1198981205871199112119871 )

sdot 119890minus119862V(1198981205872119871)2119905 (1 + 1198900) 119902119906119864 + 119899119902119906 119898 = 1 3 5

(49)



119902 (119905) = 1199021199061199050 119905 if 119905 le 1199050119902119906 if 119905 gt 1199050 (50)

Therefore

119876 (0) = 0 (51)

From (7) and (50)

119877 (119905) = 119864 (1 + 1198900) 1199021199061199050[119864 + 119899 (1199021199061199050) 119905]2 119905 le 11990500 119905 gt 1199050

(52)


120596 (119911 119905)

=




119898 = 1 3 5

(53)

where

119886 = 119864 (1 + 1198900) 1199021199061199050 119887 = 119864119888 = 1198991199021199061199050

119889 = 119862V (1198981205872119871 )2

119864119894 (1198861 1198871) = intinfin

1119890minus1199091198871119909minus1198861119889119909

(54)





119902 (119905) =


(55)


Therefore119876 (0) = 0 (56)

From (7) and (50)

119877 (119905) =


0 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(57)




119879V = 119862V119905119871minus2 (58)




qu


t


q(t)


0

02

04

06

08

0 02 04 06

Up

T






0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T


0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM






00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0

j minus 1

j

j + 1

L

Wn+1 On+1 En+1


t

Local element ej



1205901015840 = 119902 (119905) minus 119906 (5)


120596 = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (6)


119862V12059721205961205971199112 = 120597120596120597119905 + 119877 (119905) (7)





119862V (12059721205961205971199112 )119899+1 = 120597120596120597119905

10038161003816100381610038161003816100381610038161003816119899

119874

+ 119877 (119905)|119899 (9)



(12059721205961205971199112 )119899+1 asymp 1119862V

(120596119899+1119874 minus 120596119899119874Δ119905 + 119877 (119905)|119905=119899Δ119905) (10)


Here

119891 = 1119862V120596119899+1119874 minus 120596119899119874Δ119905 + 119877 (119899Δ119905) (11)


(12059721205961205971199112 )119899+1 = 119891 (12)


120596119899+1 (119911) = 11989111991122 + 1198881119911 + 1198882 (13)



(14)


1198881 = 120596119864 minus 1205961198822Δ119911 1198882 = 120596119864 + 1205961198822 minus 1198912 Δ1199112

(15)


120596119899+1119874 = 120596119864 + 1205961198822 minus 1198912 Δ1199112 (16)


120596119899+1119874 = 12 + Δ1199112119862VΔ119905120596119864 +12 + Δ1199112119862VΔ119905120596119882

+ [ Δ1199112119862VΔ1199052 + Δ1199112119862VΔ119905]120596119899119874

minus Δ1199112119862V2 + Δ1199112119862VΔ119905119877 (119899Δ119905)

(17)



120596119899+1119874 = 12 + 120582120596119864 + 12 + 120582120596119882 + ( 1205822 + 120582)120596119899119874minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(18)


120596119899+1119895 = 12 + 120582120596119899+1119895minus1 + 12 + 120582120596119899+1119895+1 + ( 1205822 + 120582)120596119899119895minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(19)




12059712059612059711990510038161003816100381610038161003816100381610038161003816119899

119895

= 120596119899119895 minus 120596119899minus1119895Δ119905 + 119874 (Δ119905) (20)


120576119899119895 = 120596119899119895 minus 119899119895 (21)


120596119899119895 = 12 + 120582120596119899119895minus1 + 12 + 120582120596119899119895+1 + ( 1205822 + 120582)120596119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) + 119874 (Δ119905) (22)


119899119895 = 12 + 120582119899119895minus1 + 12 + 120582119899119895+1 + ( 1205822 + 120582) 119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) (23)


120576119899119895 = 12 + 120582120576119899119895minus1 + 12 + 120582120576119899119895+1 + ( 1205822 + 120582) 120576119899minus1119895 + 119874 (Δ119905) (24)


10038161003816100381610038161003816120576119899119895 10038161003816100381610038161003816 le 12 + 120582 10038161003816100381610038161003816120576119899119895minus110038161003816100381610038161003816 + 12 + 120582 10038161003816100381610038161003816120576119899119895+110038161003816100381610038161003816 + ( 1205822 + 120582) 10038161003816100381610038161003816120576119899minus1119895

10038161003816100381610038161003816+ 10038161003816100381610038161003816119885119899minus1

119895

10038161003816100381610038161003816 (25)

where 119885119899minus1119895 = 119874(Δ119905)



+ 119885119899minus1max

(26)

where



119895

10038161003816100381610038161003816 1 le 119895 le 119873 minus 1 (27)

and that




1205761max le 1205760max + 1198850max = 1198850

max1205762max le 1205761max + 1198851

max le 1198850max + 1198851

max


max le 1198850max + 1198851


(29)

Therefore


where

119885max = max 119885119899max 1 le 119899 le 119872 (31)




119890119899119895 = 120596119899119895 minus 120596119899lowast119895 (32)




119898119899 le 119890119899+1119895 le 119872119899 (34)

where

119898119899 = min 119890119899+10 119890119899119895 119890119899+1119898 119872119899 = max 119890119899+10 119890119899119895 119890119899+1119898 (35)



(36)



min119895119890119899119895 le 119890119899+1119895 le max

119895119890119899119895 (37)

Therefore

10038171003817100381710038171003817e119899+110038171003817100381710038171003817infin = max119895

10038161003816100381610038161003816119890119899+1119895

10038161003816100381610038161003816 le max119895

10038161003816100381610038161003816119890119899119895 10038161003816100381610038161003816 = 1003817100381710038171003817e1198991003817100381710038171003817infin (38)



2L

z

Pervious

Pervious

q(t)






119906 (119911 0) = 119902 (0) if 0 le 119911 le 2119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0119906 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(40)


120596 (119911 0) = 119876 (0) if 0 le 119911 le 2119871 119905 = 0120596 (0 119905) = 0 if 119911 = 0 119905 gt 0120596 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(41)

where

119876 (119905) = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (42)


119906 = 120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900) (43)



119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906119889119911

= 119902 (119905)119902119906minus 12119871119902119906 int

2119871

0

120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900)119889119911

(44)

where

120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905 sin(1198981205871199112119871 )

sdot [int119905

0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591 minus 119876 (0)]119898 = 1 3 5

(45)


119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906 119889119911

= 119902 (119905) minus (12119871)sum119895=2119871Δ119911119895=1 119906119895Δ119911119902119906

(46)



119902 (119905) = 119902119906 (47)

Therefore

119876 (0) = minus(1 + 1198900) 119902119906119864 + 119899119902119906 (48)


120596 (119911 119905) = infinsum119898=1

4119898120587 sin(1198981205871199112119871 )

sdot 119890minus119862V(1198981205872119871)2119905 (1 + 1198900) 119902119906119864 + 119899119902119906 119898 = 1 3 5

(49)



119902 (119905) = 1199021199061199050 119905 if 119905 le 1199050119902119906 if 119905 gt 1199050 (50)

Therefore

119876 (0) = 0 (51)

From (7) and (50)

119877 (119905) = 119864 (1 + 1198900) 1199021199061199050[119864 + 119899 (1199021199061199050) 119905]2 119905 le 11990500 119905 gt 1199050

(52)


120596 (119911 119905)

=




119898 = 1 3 5

(53)

where

119886 = 119864 (1 + 1198900) 1199021199061199050 119887 = 119864119888 = 1198991199021199061199050

119889 = 119862V (1198981205872119871 )2

119864119894 (1198861 1198871) = intinfin

1119890minus1199091198871119909minus1198861119889119909

(54)





119902 (119905) =


(55)


Therefore119876 (0) = 0 (56)

From (7) and (50)

119877 (119905) =


0 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(57)




119879V = 119862V119905119871minus2 (58)




qu


t


q(t)


0

02

04

06

08

0 02 04 06

Up

T






0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T


0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM






00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120596119899+1119874 = 12 + 120582120596119864 + 12 + 120582120596119882 + ( 1205822 + 120582)120596119899119874minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(18)


120596119899+1119895 = 12 + 120582120596119899+1119895minus1 + 12 + 120582120596119899+1119895+1 + ( 1205822 + 120582)120596119899119895minus 120582Δ1199052 + 120582119877 (119899Δ119905)

(19)




12059712059612059711990510038161003816100381610038161003816100381610038161003816119899

119895

= 120596119899119895 minus 120596119899minus1119895Δ119905 + 119874 (Δ119905) (20)


120576119899119895 = 120596119899119895 minus 119899119895 (21)


120596119899119895 = 12 + 120582120596119899119895minus1 + 12 + 120582120596119899119895+1 + ( 1205822 + 120582)120596119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) + 119874 (Δ119905) (22)


119899119895 = 12 + 120582119899119895minus1 + 12 + 120582119899119895+1 + ( 1205822 + 120582) 119899minus1119895

minus 120582Δ1199052 + 120582119877 (119899Δ119905) (23)


120576119899119895 = 12 + 120582120576119899119895minus1 + 12 + 120582120576119899119895+1 + ( 1205822 + 120582) 120576119899minus1119895 + 119874 (Δ119905) (24)


10038161003816100381610038161003816120576119899119895 10038161003816100381610038161003816 le 12 + 120582 10038161003816100381610038161003816120576119899119895minus110038161003816100381610038161003816 + 12 + 120582 10038161003816100381610038161003816120576119899119895+110038161003816100381610038161003816 + ( 1205822 + 120582) 10038161003816100381610038161003816120576119899minus1119895

10038161003816100381610038161003816+ 10038161003816100381610038161003816119885119899minus1

119895

10038161003816100381610038161003816 (25)

where 119885119899minus1119895 = 119874(Δ119905)



+ 119885119899minus1max

(26)

where



119895

10038161003816100381610038161003816 1 le 119895 le 119873 minus 1 (27)

and that




1205761max le 1205760max + 1198850max = 1198850

max1205762max le 1205761max + 1198851

max le 1198850max + 1198851

max


max le 1198850max + 1198851


(29)

Therefore


where

119885max = max 119885119899max 1 le 119899 le 119872 (31)




119890119899119895 = 120596119899119895 minus 120596119899lowast119895 (32)




119898119899 le 119890119899+1119895 le 119872119899 (34)

where

119898119899 = min 119890119899+10 119890119899119895 119890119899+1119898 119872119899 = max 119890119899+10 119890119899119895 119890119899+1119898 (35)



(36)



min119895119890119899119895 le 119890119899+1119895 le max

119895119890119899119895 (37)

Therefore

10038171003817100381710038171003817e119899+110038171003817100381710038171003817infin = max119895

10038161003816100381610038161003816119890119899+1119895

10038161003816100381610038161003816 le max119895

10038161003816100381610038161003816119890119899119895 10038161003816100381610038161003816 = 1003817100381710038171003817e1198991003817100381710038171003817infin (38)



2L

z

Pervious

Pervious

q(t)






119906 (119911 0) = 119902 (0) if 0 le 119911 le 2119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0119906 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(40)


120596 (119911 0) = 119876 (0) if 0 le 119911 le 2119871 119905 = 0120596 (0 119905) = 0 if 119911 = 0 119905 gt 0120596 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(41)

where

119876 (119905) = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (42)


119906 = 120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900) (43)



119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906119889119911

= 119902 (119905)119902119906minus 12119871119902119906 int

2119871

0

120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900)119889119911

(44)

where

120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905 sin(1198981205871199112119871 )

sdot [int119905

0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591 minus 119876 (0)]119898 = 1 3 5

(45)


119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906 119889119911

= 119902 (119905) minus (12119871)sum119895=2119871Δ119911119895=1 119906119895Δ119911119902119906

(46)



119902 (119905) = 119902119906 (47)

Therefore

119876 (0) = minus(1 + 1198900) 119902119906119864 + 119899119902119906 (48)


120596 (119911 119905) = infinsum119898=1

4119898120587 sin(1198981205871199112119871 )

sdot 119890minus119862V(1198981205872119871)2119905 (1 + 1198900) 119902119906119864 + 119899119902119906 119898 = 1 3 5

(49)



119902 (119905) = 1199021199061199050 119905 if 119905 le 1199050119902119906 if 119905 gt 1199050 (50)

Therefore

119876 (0) = 0 (51)

From (7) and (50)

119877 (119905) = 119864 (1 + 1198900) 1199021199061199050[119864 + 119899 (1199021199061199050) 119905]2 119905 le 11990500 119905 gt 1199050

(52)


120596 (119911 119905)

=




119898 = 1 3 5

(53)

where

119886 = 119864 (1 + 1198900) 1199021199061199050 119887 = 119864119888 = 1198991199021199061199050

119889 = 119862V (1198981205872119871 )2

119864119894 (1198861 1198871) = intinfin

1119890minus1199091198871119909minus1198861119889119909

(54)





119902 (119905) =


(55)


Therefore119876 (0) = 0 (56)

From (7) and (50)

119877 (119905) =


0 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(57)




119879V = 119862V119905119871minus2 (58)




qu


t


q(t)


0

02

04

06

08

0 02 04 06

Up

T






0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T


0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM






00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119890119899119895 = 120596119899119895 minus 120596119899lowast119895 (32)




119898119899 le 119890119899+1119895 le 119872119899 (34)

where

119898119899 = min 119890119899+10 119890119899119895 119890119899+1119898 119872119899 = max 119890119899+10 119890119899119895 119890119899+1119898 (35)



(36)



min119895119890119899119895 le 119890119899+1119895 le max

119895119890119899119895 (37)

Therefore

10038171003817100381710038171003817e119899+110038171003817100381710038171003817infin = max119895

10038161003816100381610038161003816119890119899+1119895

10038161003816100381610038161003816 le max119895

10038161003816100381610038161003816119890119899119895 10038161003816100381610038161003816 = 1003817100381710038171003817e1198991003817100381710038171003817infin (38)



2L

z

Pervious

Pervious

q(t)






119906 (119911 0) = 119902 (0) if 0 le 119911 le 2119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0119906 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(40)


120596 (119911 0) = 119876 (0) if 0 le 119911 le 2119871 119905 = 0120596 (0 119905) = 0 if 119911 = 0 119905 gt 0120596 (2119871 119905) = 0 if 119911 = 2119871 119905 gt 0

(41)

where

119876 (119905) = (1 + 1198900) 1205901015840119864 + 1198991205901015840 minus (1 + 1198900) 119902 (119905)119864 + 119899119902 (119905) (42)


119906 = 120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900) (43)



119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906119889119911

= 119902 (119905)119902119906minus 12119871119902119906 int

2119871

0

120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900)119889119911

(44)

where

120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905 sin(1198981205871199112119871 )

sdot [int119905

0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591 minus 119876 (0)]119898 = 1 3 5

(45)


119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906 119889119911

= 119902 (119905) minus (12119871)sum119895=2119871Δ119911119895=1 119906119895Δ119911119902119906

(46)



119902 (119905) = 119902119906 (47)

Therefore

119876 (0) = minus(1 + 1198900) 119902119906119864 + 119899119902119906 (48)


120596 (119911 119905) = infinsum119898=1

4119898120587 sin(1198981205871199112119871 )

sdot 119890minus119862V(1198981205872119871)2119905 (1 + 1198900) 119902119906119864 + 119899119902119906 119898 = 1 3 5

(49)



119902 (119905) = 1199021199061199050 119905 if 119905 le 1199050119902119906 if 119905 gt 1199050 (50)

Therefore

119876 (0) = 0 (51)

From (7) and (50)

119877 (119905) = 119864 (1 + 1198900) 1199021199061199050[119864 + 119899 (1199021199061199050) 119905]2 119905 le 11990500 119905 gt 1199050

(52)


120596 (119911 119905)

=




119898 = 1 3 5

(53)

where

119886 = 119864 (1 + 1198900) 1199021199061199050 119887 = 119864119888 = 1198991199021199061199050

119889 = 119862V (1198981205872119871 )2

119864119894 (1198861 1198871) = intinfin

1119890minus1199091198871119909minus1198861119889119909

(54)





119902 (119905) =


(55)


Therefore119876 (0) = 0 (56)

From (7) and (50)

119877 (119905) =


0 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(57)




119879V = 119862V119905119871minus2 (58)




qu


t


q(t)


0

02

04

06

08

0 02 04 06

Up

T






0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T


0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM






00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906119889119911

= 119902 (119905)119902119906minus 12119871119902119906 int

2119871

0

120596 (119911 119905) [119864 + 119899119902 (119905)]2119899120596 (119911 119905) [119864 + 119899119902 (119905)] minus 119864 (1 + 1198900)119889119911

(44)

where

120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905 sin(1198981205871199112119871 )

sdot [int119905

0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591 minus 119876 (0)]119898 = 1 3 5

(45)


119880119901 = int21198710(1205901015840 minus 12059010158400) 119889119911

int21198710(1205901015840

119891minus 12059010158400) 119889119911 =

int21198710[119902 (119905) minus 119906] 119889119911int21198710119902119906 119889119911

= 119902 (119905) minus (12119871)sum119895=2119871Δ119911119895=1 119906119895Δ119911119902119906

(46)



119902 (119905) = 119902119906 (47)

Therefore

119876 (0) = minus(1 + 1198900) 119902119906119864 + 119899119902119906 (48)


120596 (119911 119905) = infinsum119898=1

4119898120587 sin(1198981205871199112119871 )

sdot 119890minus119862V(1198981205872119871)2119905 (1 + 1198900) 119902119906119864 + 119899119902119906 119898 = 1 3 5

(49)



119902 (119905) = 1199021199061199050 119905 if 119905 le 1199050119902119906 if 119905 gt 1199050 (50)

Therefore

119876 (0) = 0 (51)

From (7) and (50)

119877 (119905) = 119864 (1 + 1198900) 1199021199061199050[119864 + 119899 (1199021199061199050) 119905]2 119905 le 11990500 119905 gt 1199050

(52)


120596 (119911 119905)

=




119898 = 1 3 5

(53)

where

119886 = 119864 (1 + 1198900) 1199021199061199050 119887 = 119864119888 = 1198991199021199061199050

119889 = 119862V (1198981205872119871 )2

119864119894 (1198861 1198871) = intinfin

1119890minus1199091198871119909minus1198861119889119909

(54)





119902 (119905) =


(55)


Therefore119876 (0) = 0 (56)

From (7) and (50)

119877 (119905) =


0 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(57)




119879V = 119862V119905119871minus2 (58)




qu


t


q(t)


0

02

04

06

08

0 02 04 06

Up

T






0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T


0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM






00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119902 (119905) =


(55)


Therefore119876 (0) = 0 (56)

From (7) and (50)

119877 (119905) =


0 if [(119873 minus 1) 120573 + 1] 1199050 le 119905 le 1198731205731199050

(57)




119879V = 119862V119905119871minus2 (58)




qu


t


q(t)


0

02

04

06

08

0 02 04 06

Up

T






0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T


0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM






00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










qu


t


q(t)


0

02

04

06

08

0 02 04 06

Up

T






0

1

2

3

4

5

6

0 01 02 03 04 05

Δz = 08 mΔz = 02 mΔz = 005 m

minus1

Perc

enta

ge d

evia

tion

ofU

p(

)

T


0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMASM

FAMFAMFAM






00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

01

02

03

0 01 02 03 04 05minus01

minus02

minus03

minus04

Perc

enta

ge d

evia

tion

ofU

p(

)

T



0

01

02

03

04

05

06

07

0 01 02 03 04 05

Up

T

ASMASMFAMFAM

ASMASMFAMFAM

= 03 = 12

= 03 = 12

= 05 = 12

= 05 = 12

= 03 = 10

= 03 = 10

= 03 = 24

= 03 = 24




119906 (119911 0) = 104119911 if 0 le 119911 le 119871 119905 = 0119906 (0 119905) = 0 if 119911 = 0 119905 gt 0

120597119906 (119871 119905)120597119911 = 0 if 119911 = 119871 119905 gt 0(59)

00

05

10

0 02 04 06

minus05

minus10

minus15

Perc

enta

ge d

evia

tion

ofU

p(

)

T

= 03 = 12 = 05 = 12

= 03 = 10 = 03 = 24




120597120596 (119871 119905)120597119905 = 0 if 119911 = 119871 119905 gt 0(60)


8 Conclusions




0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4

u(p

a)25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(a) FDM Δ119911 = 01m FAM Δ119911 = 01m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(b) FDM Δ119911 = 01m FAM Δ119911 = 02m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(c) FDM Δ119911 = 01 m FAM Δ119911 = 04m

0 1 2 3 4

u(p

a)

25E + 05

20E + 05

15E + 05

10E + 05

50E + 04

00E + 00

minus50E + 04 Z (m)



(d) FDM Δ119911 = 01m FAM Δ119911 = 08m






Appendix


120596 (119911 119905) = minus infinsum119898=1

4119898120587119890minus119862V(1198981205872119871)2119905

sdot sin(1198981205871199112119871 ) [119879119898119873 + 119879119898119894] (A1)


0119877 (120591) 119890119862V(1198981205872119871)

2120591119889120591= 119873minus1sum

119896=1

int[(119896minus1)120573+120572]1199050

(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591

minus int[(119896minus1)120573+1]1199050

[(119896minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591




(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(119896minus1)1205731199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591

[119864 + 119899119902119906 [120591 minus (119896 minus 1) 1205731199050] 1205721199050]2 119889120591(A3)


1198791198982 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050) (A4)


1198791198983 (119905) = 1198791198981 ([(119873 minus 1) 120573 + 120572] 1199050)minus int120591

[(119873minus1)120573+(1minus120572)]1199050

119864 (1 + 1198900) (1199021199061205721199050) 119890119862V(1198981205872119871)2120591





Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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