seismicbearingcapacityofstripfootingsoncohesivesoil...

Research ArticleSeismic Bearing Capacity of Strip Footings on Cohesive SoilSlopes by Using Adaptive Finite Element Limit Analysis

Weihua Luo12 Minghua Zhao3 Yao Xiao4 Rui Zhang 5 and Wenzhe Peng 4

1Institute of Geotechnical Engineering Hunan University Changsha 410082 China2Construction Development Corporation in Hunan Province Changsha 410000 China3Professor Institute of Geotechnical Engineering Hunan University Changsha 410082 China4PhD Institute of Geotechnical Engineering Hunan University Changsha 410082 China5Assistant Professor PhD Institute of Geotechnical Engineering Hunan University Changsha 410082 China

Correspondence should be addressed to Wenzhe Peng wzpenghnueducn

Received 17 April 2019 Revised 29 June 2019 Accepted 28 July 2019 Published 28 August 2019

Academic Editor Sanjay Nimbalkar

Copyright copy 2019 Weihua Luo et al -is 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

By employing adaptive finite element limit analysis (AFELA) the seismic bearing capacity of strip footing on cohesive soil slopesare investigated To consider the earthquake effects the pseudostatic method is used -e upper and lower bounds for the seismicbearing capacity factor (Nce) are calculated and the relative errors between them are found within 3 or better by adopting theadaptive mesh strategy Based on the obtained results design tables and charts are provided to facilitate engineers use and theeffects of footing position undrained shear strength slope angle slope height and pseudostatic acceleration coefficient are studiedin detail -e collapse mechanisms are also discussed including overall slope failure and foundation failure

1 Introduction

-e stability problem of strip footing on slopes often occursin engineering practice which has been investigated bymany researchers To determine the bearing capacity of stripfooting on slopes different methods are used ranging fromsemiempirical methods [1 2] limit equilibrium techniques[3ndash6] slip-line solutions [7 8] limit analysis [9ndash11] finiteelement methods [12 13] finite element limit analysis [14]and discontinuity layout optimization (DLO) approaches[15ndash17] However the effect of seismicity is not consideredfor all the studies mentioned previously which should betreated carefully in earthquake areas due to the devastatinginfluence of the footing under seismic conditions

Considering the pseudostatic seismic forces a variety ofstudies have been performed to calculate the bearing ca-pacity of strip footing on slopes -eoretical solutionsprovide an efficient way to analyze the problem such aslimit equilibrium approaches [18ndash21] lower bound [22]and upper bound [23ndash27] solutions and stress charac-teristic method [28] Compared to the analytical methods

mentioned above numerical modeling does not requireassumptions of failure mechanisms to be made which mayprovide good accuracy results and consider a wider range ofparameters [29] By using finite element limit analysis(FELA) Shiau et al [30] and Raj et al [31] examined theseismic bearing capacity of footings placed on slopes andthe upper and lower bounds have been presented Kumarand Chakraborty [32] used the lower bound FELA tocompute the bearing capacity factor Nc for a rough stripfooting on cohesionless slopes under earthquake condi-tions Later using the same methods the seismic bearingcapacity of strip footings on a sloping ground surface andembankments has been investigated by Chakraborty andKumar [33] and Chakraborty and Mahesh [34] re-spectively More recently Zhou et al [35] determined theultimate seismic bearing capacity and the collapse mech-anisms for strip footings located adjacent to cohesive-frictional soil slopes by using the DLO method Underundrained conditions Cinicioglu and Erkli [36] appliedfinite element software PLAXIS to investigate the seismicbearing capacity of strip footings resting on or near slopes

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 4548202 16 pageshttpsdoiorg10115520194548202

In this work an AFELA program developed by theauthors is employed to examine the seismic bearing capacityof strip footings on cohesive soil slopes -e pseudostaticapproach is applied to consider the earthquake loadingswhich has been incorporated into the current AFELAprogram Based on the AFELA program the upper bound(UB) and lower bound (LB) results obtained by the presentsolution are first compared to those conducted by theprevious studies -en the effects of seismic accelerationcoefficient soil properties and geometrical parameters onthe seismic bearing capacity factor are explored To facilitateengineers use design tables and charts are provided and thefailure modes are also discussed

2 Statement of the Problem

Under plane strain conditions Figure 1 illustrates a generallayout of the problem analyzed A weightless rough stripfooting of width B is constructed near a slope with a slopeangle β and a slope height H -e normalized footing dis-tance to the crest is given as λ (footing distancefootingwidth) -e soil with unit weight c is assumed as a ho-mogeneous media with undrained shear strength cu fol-lowing the Tresca yield criterion with an associated flow ruleBased on the pseudostatic approach a same horizontalseismic acceleration coefficient kh is applied to the footingand the slope [17 36] -e vertical seismic accelerationcoefficient is ignored owing to its insignificant influence onthe seismic bearing capacity [36 37]

-e seismic bearing capacity of strip footings on cohesiveslopes can be represented by the dimensionless seismicbearing capacity factor Nce which is defined as

Nce qu

cu f

cu

cBH

B λ β kh1113888 1113889 (1)

where qu is the ultimate seismic bearing capacityIn this paper the magnitude of cu(cB) ranges from 0625

to 75 and HB ranges from 1 to 4 [12 36] -e value of λvaries from 0 to 4 corresponding to five different values of βnamely 15deg 30deg 45deg 60deg and 75deg Four different values of thehorizontal seismic acceleration coefficient kh namely 0102 03 and 035 are considered -e problem parametersand adopted values are summarized in Table 1

3 Adaptive Finite Element Limit Analysis

In the last two decades the FELAmethods are routinely usedto analyze a variety of geotechnical engineering stabilityproblems which combine the limit theorems of classicalplasticity with finite elements to give the UB and the LB onthe collapse load [38] -ese techniques do not require theload-settlement curve and assumptions about the collapsemode to determine the limit load which have been widelyused in stability analysis of strip footings [39ndash41] tunnels[42ndash46] slopes [14 47ndash49] and anchors [50 51]

-ere are two distinct solutions produced (1) the UBsolution and (2) the LB solution -e proper FE techniquescan be used to construct kinematically admissible velocityfields for UB solution and statically admissible stress fields

for LB solution In this paper the discontinuous FE for-mulations proposed by Krabbenhoft et al [52] and Lyaminet al [53] are introduced for the discrete UB and LBproblems respectively -is will produce two nonlinearoptimization models which can be expressed as a same form[54 55]

maximize α

subject to bTσ αp + p0

F σi( 1113857le 0 i 1 n

(2)

where α is a load multiplier bT is the discrete equilibrium-type operator σ is the vector of discrete stresses σi n is thetotal number of discrete stresses p and p0 are the vectors ofunknown and prescribed nodal forces respectively and F isthe yield function

To solve the discrete models presented in equation (2)the authors develop a general nonlinear optimization al-gorithm which is a modified version of the feasible arcinterior point algorithm (FAIPA) proposed by Herskovitset al [56] A distinct feature of the FAIPA algorithm used inthis paper has been introduced by Zhang et al in [57] whichis that a feasible arc is constructed for each iteration of thealgorithm to avoid the violation of inequality constraints insearch of a step length t which is shown in Figure 2 FromFigure 2 it can be observed that a second-order feasible arc xis constructed at the current iterate point x with a feasibledirection d and a restoring direction 1113957d which takes thefollowing form

x x + td + t21113957d (3)

Since it has a curvature close to the active inequalityconstraint f a large allowable step length t can be achievedproviding the ldquoline-searchingrdquo proceeds along this feasiblearc

Moreover the adaptive remeshing procedure proposedby Zhang et al [58] is incorporated into the aforementionedFELA program to reduce the calculation errors which isapplied in both the UB and LB solutions For simplicitydetails of the adaptive remeshing procedure are not givenhere but can be found in Zhang et al [58] Because the

H

β

cu

B

qu

γ

khγ

qh = khqu

λB

Figure 1 Schematic diagram of the model

2 Advances in Civil Engineering

adaptive mesh strategy is used for all our calculations therelative errors (REs) between the UB and LB measured byequation (4) are found to be within 3 or better

relative error plusmn100 times(UB minus LB)

(UB + LB)() (4)

Figure 3 shows a typical adaptive finite element meshemployed in the upper and lower bound for the case with cu(cB) 5 λ 1 HB 2 β 45deg and kh 01 -is figure alsoillustrates the stress boundary conditions for the LB analyses(the normal stress σn and the shear stress τ are labelled) andthe velocity boundary conditions for the UB analyses (thehorizontal velocity u and the vertical velocity v are prescribed)-is study adopts three iterations of adaptive mesh re-finements for all numerical simulations and similar meshesare applied for the UB and LB analyses A typical final adaptivemesh has 9365 triangular elements and 14145 stress-velocitydiscontinuities as shown in Figure 3 -e size of the calcu-lation domain is chosen to be so large that all the plastic-zonesat failure develop within the domain

4 Comparisons with Previous Studies

41 Comparison of Static Bearing Capacity Factor Nc -estatic bearing capacity factor Nc obtained from the presentstudy for rough strip footings on cohesive slopes is com-pared with the solutions of (1) Vesic [2] by using thesemiempirical solution (2) Kusakabe et al [11] on the basisof the upper bound method (3) Georgiadis [12] by usingfinite element method (FEM) and (4) Leshchinsky [15] onthe basis of the DLO approach A comparison of these resultsis given in Figure 4 for the case with kh 0 λ 0 and cu(cB) 1 As seen the AFELA results presented in this study

and the results of Kusakabe et al [11] Georgiadis [12] andLeshchinsky [15] are in excellent agreement while Vesicrsquossolution provides smaller values of Nc

For the case of a footing placed at a distance from theslope a further comparison is made in Figure 5 All resultspresented in Figure 5 are for kh 0 cu(cB) 5 and β 30deg-e values of Nc obtained by Georgiadis [12] based on theFEM and Leshchinsky and Xie [16] by using the DLO ap-proach lie between the present upper and lower boundsolutions -e present results agree reasonably well with the

Table 1 -e problems parameters and adopted values

Parameters Descriptions Values consideredcu(cB) Dimensionless strength ratio 0625ndash75HB Normalized slope height 1ndash4Λ Normalized footing distance 0ndash4Β Slope angle 15degndash75degkh Horizontal seismic acceleration coefficient 01ndash035

d1 Tangent to f

Feasible region

Allowable step t

Feasible directionInitial direction

f(x) = 0

d0ρd1

d

Feasible arc x

d

Figure 2 Illustration of the feasible arc (from Zhang et al [57])

u = v = 0

σn = τ = 0σn = τ = 0

u

v

u =

v = 0

u =

v = 0

6

4

24 6 8 10 12

Figure 3 Finite element mesh showing boundary conditions fornumerical limit analysis (cu(cB) 5 λ 1 HB 2 β 45deg andkh 01)

0 10 20 30 40 5025

30

35

40

45

50

55

Nc

Vesic [2]Kusakabe et al [11]Georgiadis [12]

Leshchinsky [15]Present study (UB)Present study (LB)

β (deg)

Figure 4 Comparison of variation inNc with slope angle β for λ 0and cu(cB) 1

Advances in Civil Engineering 3

UB solution of Kusakabe et al [11] -e limit equilibriummethod reported by Meyerhof [5] and Castelli andMotta [4]provides a greater value of Nc since assumed failuremechanisms are used

42 Comparison of Seismic Bearing Capacity Factor Nce-e present results for seismic bearing capacity of stripfootings placed adjacent to cohesive slopes are comparedwith the FEM conducted by Cinicioglu and Erkli [36]Figure 6 shows the variations of the seismic bearing capacityfactorNce with the horizontal seismic acceleration coefficientkh for the case of λ 0 and cu(cB) 5 It can be observed thatthe present results agree reasonably well with the predictionsof Cinicioglu and Erkli [36]

A final comparison of the results of this study is made forthe case of β 30deg kh 02 and cu(cB) 5 as shown inFigure 7 As seen the present values for Nce lie in betweenthe results obtained by the other researchers -e limitequilibrium method of Castelli and Motta [4] and the FEMof Cinicioglu and Erkli [36] are found to be greater than thepresent upper bound solution and the values ofNce obtainedfrom the lower bound limit analysis of Farzaneh et al [22]become slightly smaller than the present lower boundsolution

5 Results and Discussion

-e upper and lower bound results for the seismic bearingcapacity factor Nce have been computed Since the adaptivemesh strategy is employed for all the numerical simulationsin general the relative errors are found to within 3showing a good accuracy for the present results -ereforeaverage value of the upper and lower bound Nce will beemployed in the following discussions-e results calculatedby AFELA for all the problem parameters considered aresummarized in Tables 1ndash3 and depicted graphically in

Figures 8ndash11 Note that the cases with an infeasible solutionare indicated by ldquomdashrdquo in Tables 1ndash3 indicating that the slopefailure due to gravity occurs In the following sections theinfluences of the normalized footing distance λ the di-mensionless strength ratio cu(cB) the slope angle β theslope height HB and the horizontal seismic accelerationcoefficient kh on Nce will be discussed in detail

Figure 8 shows the variation of the seismic bearingcapacity factor Nce with λ for different combinations of khand β where the values ofNce are for the cases with cu(cB)

1 andHB 1 As seen in Figure 8 it can be found that for allmagnitudes of kh and β the value of Nce increases with theincrease of λ up to a certain critical value of λcr beyond which

Nce

ndash01 00 01 02 03 04

20

25

30

35

40

45

50CE Cinicioglu and Erkli

kh

CE (2018) β = 15degCE (2018) β = 30degCE (2018) β = 45deg

Present study (UB)Present study (LB)

β = 15deg

β = 30deg

β = 45deg

Figure 6 Comparison of variation in Nce with kh for λ 0 and cu(cB) 5

Nc

00 05 10λ

15 20

40

44

48

52

56

Meyerhof [5]Castelli and Motta [4]Kusakabe et al [11]Georgiadis [12]

Leshchinsky andXie [16]Present study (UB)Present study (LB)

Figure 5 Comparison of variation in Nc with λ for β 30deg

Nce

λ00 05 10 15 20

30

32

34

36

38

40

42

Castelli and Motta[4]Farzaneh et al [22]Cinicioglu andErkli [36]

Present study (UB)

Present study (LB)

Figure 7 Comparison of variation in Nce with λ for β 30degkh 02 and cu(cB) 5


Table 2 Seismic bearing capacity factor Nce for strip footings on cohesive soil slopes with HB 1

β (deg) cu(cB) khλ

khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01

381 413 437 441 441 441 441

03

274 297 298 298 298 298 2981 392 422 443 445 445 445 445 277 299 299 299 299 299 29915 397 426 446 447 447 447 447 279 301 301 301 301 301 30125 401 429 448 448 448 448 448 280 301 301 301 301 301 3015 404 431 449 449 449 449 449 281 302 302 302 302 302 30275 405 432 450 450 450 450 450 281 302 302 302 302 302 3020625

02

320 349 362 362 362 362 362

035

mdash mdash mdash mdash mdash mdash mdash1 330 358 366 367 367 367 367 249 269 269 269 269 269 26915 334 362 369 369 369 369 369 252 270 270 270 270 270 27025 338 365 371 371 371 371 371 254 271 271 271 271 271 2715 340 368 372 372 372 372 372 255 272 272 272 272 272 27275 341 368 372 372 372 372 372 256 272 272 272 272 272 272

30

0625

01

312 346 376 404 428 441 441

03


02

267 292 314 333 350 362 362

035


45

0625

01

246 289 328 364 395 441 441

03


02

213 243 273 299 323 360 360

035


60

0625

01

195 244 292 334 370 428 441

03


02

167 204 241 274 303 347 352

035


75

0625

01

148 202 258 307 350 413 441

03


02

127 168 211 250 283 334 336

035





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01

mdash mdash mdash mdash mdash mdash mdash

03

mdash mdash mdash mdash mdash mdash mdash1 382 422 443 445 445 445 445 mdash mdash mdash mdash mdash mdash mdash15 388 426 446 447 447 447 447 272 299 299 299 299 299 29925 393 429 448 448 448 448 448 275 301 301 301 301 301 3015 396 427 449 450 450 450 450 277 302 302 302 302 302 30275 397 432 450 450 450 450 450 277 302 302 302 302 302 3020625

02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035



the normalized footing distance does not influence theseismic bearing capacity For example considering the casesof β 45deg the magnitude of Nce reaches the constant valuesof 445 at λ 3 for kh 01 367 at λ 2 for kh 02 298 atλ 15 for kh 03 and 269 at λ 15 for kh 035 re-spectively Interestingly the value of λcr is generally smallerfor greater value of kh -e reason for this may be that theareas of the failure zone decrease with an increase in thevalue of kh It should be noted that the rate of increase of Ncewith λ is found to be more extensive for smaller values of khwhen λle λcr

-e obtained values of the seismic bearing capacityfactor Nce with different combinations of cu(cB) and kh areshown in Figure 9 for the cases of λ 0 HB 1 and β 15deg45deg 60deg and 75deg -e results indicate that for all the cases asthe value of cu(cB) increases the seismic bearing capacityfactor increases continuously and the rate of increase in Nce

decreases It should be mentioned that the value of Nce isaffected slightly by the dimensionless strength ratio whencu(cB)ge 5 Considering the cases with kh 02 for exampleby increasing the magnitude of cu(cB) from 5 to 75 leads toincrease in the seismic bearing capacityNce by less than 03for β 15deg and 1 for β 75deg On the other hand the effect ofcu(cB) on Nce is found to be more prominent at smallervalues of kh and higher values of β

-e variations of the seismic bearing capacity factor Ncewith β for different combinations of kh and cu(cB) are il-lustrated in Figures 10 and 11 for λ 0 and 1 respectivelyFor the cases of strip footings at the crest of slopes (λ 0) itcan be seen thatNce decreases linearly with an increase in thevalue of β and this trend is predominant at the smallermagnitudes of kh On the contrary for the case of λ 1shown in Figure 11 the existence of a certain critical value ofβcr can be seen beyond which the value of Nce has a linear

Nce

λ0 1 2 3 4

24

28

32

36

40

44

kh = 01kh = 02

kh = 03kh = 035

β = 15deg

(a)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 4

20

25

30

35

40

45

β = 45deg

(b)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 415

20

25

30

35

40

45

β = 60deg

(c)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 410

15

20

25

30

35

40

45

β = 75deg

(d)

Figure 8 Variation of Nce with λ for cu(cB) 1 and HB 1 (a) β 15deg (b) β 45deg (c) β 60deg (d) β 75deg


trend -e reason for this can be attributed to the fact thatthe value of λcr is smaller than 1 when βle βcr It is note-worthy that the magnitude of βcr is generally smaller forlower value of kh

Figure 12 shows the variation of the seismic bearingcapacity Nce with kh for different combinations of β cu(cB) and HB where the value of Nce are for the case ofλ 0 It can be seen that for all the cases the value of Ncereduces linearly with increasing kh -is trend is similar tothe previous studies conducted by Kumar and Rao [28]Farzaneh et al [22] and Cinicioglu and Erkli [36] On theother hand the effect of HB on Nce is also presented inFigure 12 When βge 45deg (Figure 12(b)) it can be found thatthe value of HB does not influence the seismic bearingcapacity factor -is can be attributed that the failuremodes are not affected by HB However from Tables 1ndash3the value ofHB is found to affect the overall slope stability

especially for the case with higher value of kh and lowervalue of cu(cB)

6 Failure Mechanisms

Figure 13 illustrates the plots of power dissipations showingthe influence of the normalized footing distance λ for thecases of cu(cB) 1 and 75 As shown in Figure 13(a) for cu(cB) 1 and λ 0 a nonplastic triangular wedge below thefooting base can be seen and a slip line extends to the freesurface When the value of λ increases to 1 (Figure 13(b))the slip line tends to extend the toe of the slope For the caseof cu(cB) 1 and λ 4 (Figure 13(c)) the failure mechanismis found to be not affected by the slope which is consistentwith the phenomenon illustrated in Figure 8 On the otherhand as shown in Figures 13(d) 13(e) and 13(f) the caseswith higher value of cu(cB) the depth of failure zone is

00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)

Figure 9 Variation of Nce with cu(cB) for λ 0 and HB 1 (a) β 15deg (b) β 45deg (c) β 60deg (d) β 75deg


smaller than the cases with lower value of cu(cB) -e onlyexception to this observation occurs for the cases with thegreater value of λ 4 (Figures 13(c) and 13(f)) where thefailure modes are not unchanged -is indicates that thevalue of cu(cB) does not affect the seismic bearing capacitywhen the normalized footing distance is larger enough

Several plots of power dissipations showing the effect of βandHB are shown in Figure 14 for the cases with cu(cB) 1λ 1 and kh 01 For the cases with small slope angles(Figures 14(a) 14(d) and 14(g)) it can be seen that thecollapse mechanisms remain unchanged and the failuresurface only extends to the free surface As seen inFigures 14(b) 14(e) and 14(h) for the cases ofmoderate slopeangles similar failure modes showing that the failure surfaceextends to the toe of the slopes can be observed However for

the cases with larger values of β the failure mechanisms aresignificantly affected byHB As illustrated in Figure 14(c) forβ 75deg and HB 1 the collapse mechanism is similar to thecase of β 45deg andHB 1 but the depth of the failure surfaceis smaller For the case of β 75deg and HB 2 (Figure 14(f))the rigid triangular wedge disappeared and only a slip line canbe seen which extends from the right of the footing base tothe toe of the slope For the case of β 75deg and HB 2 asshown in Figure 14(i) a larger slip surface noticeably towardthe ground surface can be observed indicating that the overallslope failure due to the gravity occurs (Table 4)

For comparison Figure 15 shows the plots of powerdissipations for the cases with the greater value of cu(cB)

5 -e failure patterns remain unchanged and become in-dependent of HB which are consistent with those of

Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)

Figure 10 Variation of Nce with β for λ 0 and HB 1 (a) cu(cB) 1 (b) cu(cB) 5

10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)



Figure 12 To show the effect of the horizontal seismic ac-celeration coefficient kh the upper bound failure mecha-nisms are presented in Figure 16 for the cases of cu(cB)

25 λ 1 andHB 2 It clearly can be seen that the depth ofthe failure zone decreases with an increase in the value of kh-is can be used to explain the phenomenon shown in

010 015 020 025 030 035

24

28

32

36

40

kh

cu(γB)=15 HB=1cu(γB)=5 HB=1

cu(γB)=15 HB=2 and 4cu(γB)=5 HB=2 and 4

Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce

cu(γB)=15 HB=1 2 and 4cu(γB)=5 HB=1 2 and 4

β = 45deg 60deg 75deg

(b)

Figure 12 Variation of Nce with kh for λ 0 and HB 1 (a) β 15deg (b) β 45degndash75deg

7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )

Figure 13 Plots of power dissipations for HB 2 β 45deg and kh 01 (a) λ 0 cu(cB) 1 (b) λ 1 cu(cB) 1 (c) λ 4 cu(cB) 1(d) λ 0 cu(cB) 75 (e) λ 1 cu(cB) 75 (f ) λ 4 cu(cB) 75


Figure 8 that the value of λcr is generally smaller for greatervalue of kh

As summarized the failure mechanisms may be classi-fied into four types which are shown in Table 5

7 Conclusions

An adaptive finite element limit analysis (AFELA) pro-gram with incorporation of the pseudostatic approach

has been used to investigate the seismic bearing capacityof strip footings on cohesive soil slopes Based on theAFELA program the upper and lower bounds for theseismic bearing capacity factor Nce have been computedshowing a good accuracy for the results To facilitateengineer use the present results are presented in designtables and charts and the influence of parameters in-cluding λ cu(cB) β kh and HB on Nce has also beenexamined -e following conclusions can be made

2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)

Figure 14 Plots of power dissipations for λ 1 cu(cB) 1and kh 01 (a) HB 1 β 15deg (b) HB 1 β 45deg (c) HB 1 β 75deg(d) HB 2 β 15deg (e) HB 2 β 45deg (f ) HB 2 β 75deg (g) HB 4 β 15deg (h) HB 4 β 45deg (i) HB 4 β 75deg




khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035

mdash mdash mdash mdash mdash mdash mdash1 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash15 334 362 369 369 369 369 369 mdash mdash mdash mdash mdash mdash mdash25 337 365 371 371 371 371 371 254 271 271 271 271 271 2715 340 367 372 372 372 372 372 256 272 272 272 272 272 27275 340 368 372 372 372 372 372 256 272 272 272 272 272 272

30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03

mdash mdash mdash mdash mdash mdash mdash1 161 146 132 124 118 121 121 mdash mdash mdash mdash mdash mdash mdash15 245 297 317 319 326 349 379 mdash mdash mdash mdash mdash mdash mdash25 258 321 363 397 423 448 448 195 252 289 301 301 301 3015 266 333 380 416 446 450 450 200 261 300 302 302 302 30275 268 337 384 421 449 450 450 202 265 302 302 302 302 3020625

02


035


75

0625

01


03


02


035



(1) In general the value of Nce increases with the in-crease of λ up to a certain critical value of λcr beyondwhich the normalized footing distance does notinfluence the seismic bearing capacity and the valueof λcr is smaller for greater value of kh

(2) As the value of cu(cB) increases the value of Nceincreases continuously and the rate of increase inNce

decreases When cu(cB)ge 5 the value of Nce is af-fected slightly by cu(cB) And the effect of cu(cB) onNce is found to be more prominent at smaller valuesof kh and higher values of β

(3) For the cases of strip footings at the crest of slopes(λ 0) the magnitude of Nce decreases linearly withan increase in the value of β and this trend is

7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)

Figure 15 Plots of power dissipations for β 45deg λ 1 cu(cB) 5 and kh 01 (a) HB 1 (b) HB 2 (c) HB 4

Table 5 -e types of slope failure

Failure modes Descriptions Cases

Type 1 Bearing capacity failure for the slip line developedwithin the slope surface

HB 2 β 15deg kh 01 λ 0 and cu(cB) 1(Figure 12(a))

Type 2 Bearing capacity failure for the slip line extend to thetoe of the slope

HB 2 β 45deg kh 01 λ 1 and cu(cB) 1(Figure 13(e))

Type 3 Bearing capacity failure without influence of the slope HB 2 β 45deg kh 01 λ 4 and cu(cB) 75(Figure 13(f ))

Type 4 Overall slope failure HB 4 β 75deg kh 01 λ 1 and cu(cB) 1 (Figure14(i))

4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)

Figure 16 Plots of power dissipations for β 45deg λ 1 cu(cB) 25 andHB 2 (a) kh 0 (b) kh 01 (c) kh 02 (d) kh 03 (e) kh 035


predominant at the smaller magnitudes of kh For thecase of a footing at a distance from the slope (λge 0)the existence of a certain critical value of βcr can beseen beyond which the value ofNce has a linear trend

(4) -e value of Nce reduces linearly with increasing kh-e value of HB does not influence Nce for the casewith higher value of kh and lower value of cu(cB) butaffects the overall slope stability

(5) Two types of failure mechanisms are discussed in-cluding overall slope failure due to gravity andfoundation failure -e depth of the failure zonedecreases with an increase in the value of kh

Note that the soil properties are complex for the naturalslopes this study is limited to slopes for homogeneous soilproperties On the other hand the effect of footingroughness has not been considered in this study which willbe investigated in the future work

Notation

bT Discrete equilibrium-type operatorB Width of the strip footingcu Undrained shear stressF Yield functionH Slope heightkh Horizontal acceleration coefficientNce Seismic bearing capacity factorn Total number of discrete stressesp Vectors of unknownp0 Prescribed nodal forcesu Horizontal velocityv Vertical velocityqu Ultimate seismic bearing capacityα Load multiplierβ Slope anglec Soil unit weightσ Vector of discrete stressesσi Discrete stressesσn Normal stressesτ Shear stressesλ Normalized footing distance to the crest

Data Availability

-e data in Tables 1ndash3 used to support the findings of thisstudy are included within the article In addition the dataused in Figures 3ndash6 were from Refs [2 4 511 12 15 16 22 36]and they are cited at relevant placeswithin the text -e remaining data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-e authors would like to acknowledge the financial supportof the China Postdoctoral Science Foundation (140050006)and the National Natural Science Foundation of China (Nos51478178 and 51608540) which made the work presented inthis paper possible

References

[1] B Hansen ldquoA general formula for bearing capacityrdquo DanishGeotechnical Institute Bulletin vol 11 pp 38ndash46 1961

[2] A S Vesic ldquoBearing capacity of shallow foundationsrdquo inFoundation Engineering Handbook H F Winterkorn andH Y Fang Eds Van Nostrand Reinhold New York NYUSA 1975

[3] A S Azzouz and M M Baligh ldquoLoaded areas on cohesiveslopesrdquo Journal of Geotechnical Engineering vol 109 no 5pp 724ndash729 1983

[4] F Castelli and E Motta ldquoBearing capacity of strip footingsnear slopesrdquo Geotechnical and Geological Engineering vol 28no 2 pp 187ndash198 2010

[5] G G Meyerhof ldquo-e ultimate bearing capacity of founda-tions on slopesrdquo in Proceedings of the Fourth InternationalConference on Soil Mechanics and Foundation Engineeringvol 1 pp 384ndash386 London UK August 1957

[6] K Narita and H Yamaguchi ldquoBearing capacity analysis offoundations on slopes by use of log-spiral sliding surfacesrdquoSoils and Foundations vol 30 no 3 pp 144ndash152 1990

[7] J Graham M Andrews and D H Shields ldquoStress charac-teristics for shallow footings in cohesionless slopesrdquo Cana-dian Geotechnical Journal vol 26 no 4 pp 748ndash755 1988

[8] V V Sokolovskii Statics of Granular Media ButterworthScientific Publications London UK 1960

[9] E H Davis and J R Booker ldquoSome adaptations of classicalplasticity theory for soil stability problemsrdquo in Proceedings ofthe Symposium Role of Plasticity in Soil MechanicsA C Palmer Ed pp 24ndash41 Cambridge UK September1973

[10] K Georgiadis ldquoAn upper bound solution for the undrainedbearing capacity of strip footings at the top of a sloperdquoGeotechnique vol 60 no 10 pp 801ndash806 2010

[11] O Kusakabe T Kimura and H Yamaguchi ldquoBearing ca-pacity of slopes under strip loads on the top surfacesrdquo Soilsand Foundations vol 21 no 4 pp 29ndash40 1981

[12] K Georgiadis ldquoUndrained bearing capacity of strip footingson slopesrdquo Journal of Geotechnical and GeoenvironmentalEngineering vol 136 no 5 pp 677ndash685 2010

[13] K Georgiadis ldquo-e influence of load inclination on theundrained bearing capacity of strip footings on slopesrdquoComputers and Geotechnics vol 37 no 3 pp 311ndash322 2010

[14] J S Shiau R S Merifield A V Lyamin and S W SloanldquoUndrained stability of footings on slopesrdquo InternationalJournal of Geomechanics vol 11 no 5 pp 381ndash390 2011

[15] B Leshchinsky ldquoBearing capacity of footings placed adjacentto cprime-ϕprime slopesrdquo Journal of Geotechnical amp GeoenvironmentalEngineering vol 141 no 6 article 04015022 2015

[16] B Leshchinsky and Y Xie ldquoBearing capacity for spreadfootings placed near cprime-ϕprime slopesrdquo Journal of Geotechnical ampGeoenvironmental Engineering vol 143 no 1 article06016020 2017


[17] H Zhou G Zheng X Yang T Li and P Yang ldquoUltimateseismic bearing capacities and failure mechanisms for stripfootings placed adjacent to slopesrdquo Canadian GeotechnicalJournal 2018

[18] M Budhu and A Al-Karni ldquoSeismic bearing capacity ofsoilsrdquo Geotechnique vol 43 no 1 pp 181ndash187 1993

[19] D Choudhury and K S Subba Rao ldquoSeismic bearing capacityof shallow strip footings embedded in sloperdquo InternationalJournal of Geomechanics vol 6 no 3 pp 176ndash184 2006

[20] J Kumar and N Kumar ldquoSeismic bearing capacity of roughfootings on slopes using limit equilibriumrdquo Geotechniquevol 53 no 3 pp 363ndash369 2003

[21] S K Sarma and I S Iossifelis ldquoSeismic bearing capacityfactors of shallow strip footingsrdquo Geotechnique vol 40 no 2pp 265ndash273 1990

[22] O Farzaneh J Mofidi and F Askari ldquoSeismic bearing ca-pacity of strip footings near cohesive slopes using lower boundlimit analysisrdquo in Proceedings of the 18th InternationalConference on Soil Mechanics and Geotechnical EngineeringParis France September 2013

[23] F Askari and O Farzaneh ldquoUpper-bound solution for seismicbearing capacity of shallow foundations near slopesrdquoGeotechnique vol 53 no 8 pp 697ndash702 2003

[24] L Dormieux and A Pecker ldquoSeismic bearing capacity offoundation on cohesionless soilrdquo Journal of GeotechnicalEngineering vol 121 no 3 pp 300ndash303 1995

[25] K Georgiadis and E Chrysouli ldquoSeismic bearing capacity ofstrip footings on clay slopesrdquo in Proceedings of the 15thEuropean Conference on Soil Mechanics and GeotechnicalEngineering pp 723ndash728 Athens Greece September 2011

[26] J Kumar and P Ghosh ldquoSeismic bearing capacity for em-bedded footings on sloping groundrdquo Geotechnique vol 56no 2 pp 133ndash140 2006

[27] K Yamamoto ldquoSeismic bearing capacity of shallow foun-dations near slopes using the upper-bound methodrdquo In-ternational Journal of Geotechnical Engineering vol 4 no 2pp 255ndash267 2010

[28] J Kumar and V B K M Rao ldquoSeismic bearing capacity offoundations on slopesrdquo Geotechnique vol 53 no 3pp 347ndash361 2003

[29] C Jiang J-L He L Liu and B-W Sun ldquoEffect of loadingdirection and slope on laterally loaded pile in sloping groundrdquoAdvances in Civil Engineering vol 2018 Article ID 756957812 pages 2018

[30] J S Shiau A V Lyamin and S W Sloan ldquoApplication ofpseudo-static limit analysis in geotechnical earthquake de-signrdquo in Proceedings of the 6th European Conference onNumerical Methods in Geotechnical Engineering pp 249ndash255Taylor amp Francis Graz Austria September 2006

[31] D Raj Y Singh and S K Shukla ldquoSeismic bearing capacity ofstrip foundation embedded in c-ϕ soil sloperdquo InternationalJournal of Geomechanics vol 18 no 7 article 04018076 2018

[32] J Kumar and D Chakraborty ldquoSeismic bearing capacity offoundations on cohesionless slopesrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 139 no 11pp 1986ndash1993 2013

[33] D Chakraborty and J Kumar ldquoSeismic bearing capacity ofshallow embedded foundations on a sloping ground surfacerdquoInternational Journal of Geomechanics vol 15 no 1 article04014035 2015

[34] D Chakraborty and Y Mahesh ldquoSeismic bearing capacityfactors for strip footings on an embankment by using lower-bound limit analysisrdquo International Journal of Geomechanicsvol 16 no 3 article 06015008 2016

[35] H Zhou G Zheng X Yin R Jia and X Yang ldquo-e bearingcapacity and failure mechanism of a vertically loaded stripfooting placed on the top of slopesrdquo Computers and Geo-technics vol 94 pp 12ndash21 2018

[36] O Cinicioglu and A Erkli ldquoSeismic bearing capacity ofsurficial foundations on sloping cohesive groundrdquo Soil Dy-namics and Earthquake Engineering vol 111 pp 53ndash64 2018

[37] R Richards Jr and X Shi ldquoSeismic lateral pressures in soilswith cohesionrdquo Journal of Geotechnical Engineering vol 120no 7 pp 1230ndash1251 1994

[38] S W Sloan ldquoGeotechnical stability analysisrdquo Geotechniquevol 63 no 7 pp 531ndash571 2013

[39] J S Shiau A V Lyamin and SW Sloan ldquoBearing capacity ofa sand layer on clay by finite element limit analysisrdquo CanadianGeotechnical Journal vol 40 no 5 pp 900ndash915 2003

[40] Y Xiao M Zhao and H Zhao ldquoUndrained stability of stripfooting above voids in two-layered clays by finite elementlimit analysisrdquo Computers and Geotechnics vol 97 pp 124ndash133 2018

[41] Y Xiao M Zhao H Zhao and R Zhang ldquoFinite elementlimit analysis of the bearing capacity of strip footing on a rockmass with voidsrdquo International Journal of Geomechanicsvol 18 no 9 article 04018108 2018

[42] J P Sahoo and J Kumar ldquoRequired lining pressure for thestability of twin circular tunnels in soilsrdquo InternationalJournal of Geomechanics vol 18 no 7 article 04018069 2018

[43] D W Wilson A J Abbo S W Sloan and A V LyaminldquoUndrained stability of dual circular tunnelsrdquo InternationalJournal of Geomechanics vol 14 no 1 pp 69ndash79 2014

[44] Y Xiao M Zhao R Zhang H Zhao and W Peng ldquoStabilityof two circular tunnels at different depths in cohesive-fric-tional soils subjected to surcharge loadingrdquo Computers andGeotechnics vol 112 pp 23ndash34 2019

[45] Y Xiao M Zhao R Zhang H Zhao and G Wu ldquoStability ofdual square tunnels in rock masses subjected to surchargeloadingrdquo Tunnelling and Underground Space Technologyvol 92 article 103037 2019

[46] R Zhang Y Xiao M Zhao and H Zhao ldquoStability of dualcircular tunnels in a rock mass subjected to surchargeloadingrdquo Computers and Geotechnics vol 108 pp 257ndash2682019

[47] K Lim M J Cassidy A J Li and A V Lyamin ldquoMeanparametric Monte Carlo study of fill slopesrdquo InternationalJournal of Geomechanics vol 17 no 4 article 04016105 2017

[48] Z G Qian A J Li R S Merifield and A V Lyamin ldquoSlopestability charts for two-layered purely cohesive soils based onfinite-element limit analysis methodsrdquo International Journalof Geomechanics vol 15 no 3 article 06014022 2015

[49] Y Xiao M Zhao R Zhang H Zhao and GWu ldquoUndrainedbearing capacity of strip footings placed adjacent to two-layered slopesrdquo International Journal of Geomechanicsvol 19 no 8 article 06019014 2019

[50] J Kumar and V N Khatri ldquoEffect of footing roughness onlower bound Nc valuesrdquo International Journal of Geo-mechanics vol 8 no 3 pp 176ndash187 2008

[51] J Kumar and J P Sahoo ldquoUpper bound solution for pulloutcapacity of vertical anchors in sand using finite elements andlimit analysisrdquo International Journal of Geomechanics vol 12no 3 pp 333ndash337 2011

[52] K Krabbenhoft A V Lyamin M Hjiaj and S W Sloan ldquoAnew discontinuous upper bound limit analysis formulationrdquoInternational Journal for Numerical Methods in Engineeringvol 63 no 7 pp 1069ndash1088 2005


[53] A V Lyamin K Krabbenhoslashft A J Abbo and S W SloanldquoGeneral approach to modelling discontinuities in limitanalysisrdquo in Proceedings of the 11th International Conferenceof International Association for Computer Methods and Ad-vances in Geomechanics (IACMAG 2005) vol 1 pp 95ndash102Torino Italy June 2005

[54] H Ciria J Peraire and J Bonet ldquoMesh adaptive computationof upper and lower bounds in limit analysisrdquo InternationalJournal for Numerical Methods in Engineering vol 75 no 8pp 899ndash944 2008

[55] J J Muntildeoz J Bonet A Huerta and J Peraire ldquoUpper andlower bounds in limit analysis adaptive meshing strategiesand discontinuous loadingrdquo International Journal for Nu-merical Methods in Engineering vol 77 no 4 pp 471ndash5012009

[56] J Herskovits P Mappa E Goulart and C M Mota SoaresldquoMathematical programming models and algorithms forengineering design optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 194 no 30ndash33pp 3244ndash3268 2005

[57] R Zhang G Chen J Zou L Zhao and C Jiang ldquoStudy onroof collapse of deep circular cavities in jointed rock massesusing adaptive finite element limit analysisrdquo Computers andGeotechnics vol 111 pp 42ndash55 2019

[58] R Zhang L Li L Zhao and G Tang ldquoAn adaptive remeshingprocedure for discontinuous finite element limit analysisrdquoInternational Journal for Numerical Methods in Engineeringvol 116 no 5 pp 1ndash21 2018


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Active and Passive Electronic Components

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Shock and Vibration


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Journal of

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

In this work an AFELA program developed by theauthors is employed to examine the seismic bearing capacityof strip footings on cohesive soil slopes -e pseudostaticapproach is applied to consider the earthquake loadingswhich has been incorporated into the current AFELAprogram Based on the AFELA program the upper bound(UB) and lower bound (LB) results obtained by the presentsolution are first compared to those conducted by theprevious studies -en the effects of seismic accelerationcoefficient soil properties and geometrical parameters onthe seismic bearing capacity factor are explored To facilitateengineers use design tables and charts are provided and thefailure modes are also discussed

2 Statement of the Problem

Under plane strain conditions Figure 1 illustrates a generallayout of the problem analyzed A weightless rough stripfooting of width B is constructed near a slope with a slopeangle β and a slope height H -e normalized footing dis-tance to the crest is given as λ (footing distancefootingwidth) -e soil with unit weight c is assumed as a ho-mogeneous media with undrained shear strength cu fol-lowing the Tresca yield criterion with an associated flow ruleBased on the pseudostatic approach a same horizontalseismic acceleration coefficient kh is applied to the footingand the slope [17 36] -e vertical seismic accelerationcoefficient is ignored owing to its insignificant influence onthe seismic bearing capacity [36 37]

-e seismic bearing capacity of strip footings on cohesiveslopes can be represented by the dimensionless seismicbearing capacity factor Nce which is defined as

Nce qu

cu f

cu

cBH

B λ β kh1113888 1113889 (1)

where qu is the ultimate seismic bearing capacityIn this paper the magnitude of cu(cB) ranges from 0625

to 75 and HB ranges from 1 to 4 [12 36] -e value of λvaries from 0 to 4 corresponding to five different values of βnamely 15deg 30deg 45deg 60deg and 75deg Four different values of thehorizontal seismic acceleration coefficient kh namely 0102 03 and 035 are considered -e problem parametersand adopted values are summarized in Table 1

3 Adaptive Finite Element Limit Analysis

In the last two decades the FELAmethods are routinely usedto analyze a variety of geotechnical engineering stabilityproblems which combine the limit theorems of classicalplasticity with finite elements to give the UB and the LB onthe collapse load [38] -ese techniques do not require theload-settlement curve and assumptions about the collapsemode to determine the limit load which have been widelyused in stability analysis of strip footings [39ndash41] tunnels[42ndash46] slopes [14 47ndash49] and anchors [50 51]

-ere are two distinct solutions produced (1) the UBsolution and (2) the LB solution -e proper FE techniquescan be used to construct kinematically admissible velocityfields for UB solution and statically admissible stress fields

for LB solution In this paper the discontinuous FE for-mulations proposed by Krabbenhoft et al [52] and Lyaminet al [53] are introduced for the discrete UB and LBproblems respectively -is will produce two nonlinearoptimization models which can be expressed as a same form[54 55]

maximize α

subject to bTσ αp + p0

F σi( 1113857le 0 i 1 n

(2)

where α is a load multiplier bT is the discrete equilibrium-type operator σ is the vector of discrete stresses σi n is thetotal number of discrete stresses p and p0 are the vectors ofunknown and prescribed nodal forces respectively and F isthe yield function

To solve the discrete models presented in equation (2)the authors develop a general nonlinear optimization al-gorithm which is a modified version of the feasible arcinterior point algorithm (FAIPA) proposed by Herskovitset al [56] A distinct feature of the FAIPA algorithm used inthis paper has been introduced by Zhang et al in [57] whichis that a feasible arc is constructed for each iteration of thealgorithm to avoid the violation of inequality constraints insearch of a step length t which is shown in Figure 2 FromFigure 2 it can be observed that a second-order feasible arc xis constructed at the current iterate point x with a feasibledirection d and a restoring direction 1113957d which takes thefollowing form

x x + td + t21113957d (3)

Since it has a curvature close to the active inequalityconstraint f a large allowable step length t can be achievedproviding the ldquoline-searchingrdquo proceeds along this feasiblearc

Moreover the adaptive remeshing procedure proposedby Zhang et al [58] is incorporated into the aforementionedFELA program to reduce the calculation errors which isapplied in both the UB and LB solutions For simplicitydetails of the adaptive remeshing procedure are not givenhere but can be found in Zhang et al [58] Because the

H

β

cu

B

qu

γ

khγ

qh = khqu

λB

Figure 1 Schematic diagram of the model




(UB + LB)() (4)








d1 Tangent to f

Feasible region

Allowable step t


f(x) = 0

d0ρd1

d

Feasible arc x

d


u = v = 0

σn = τ = 0σn = τ = 0

u

v

u =

v = 0

u =

v = 0

6

4

24 6 8 10 12


0 10 20 30 40 5025

30

35

40

45

50

55

Nc



β (deg)











Nce

ndash01 00 01 02 03 04

20

25

30

35

40

45


kh



β = 15deg

β = 30deg

β = 45deg


Nc

00 05 10λ

15 20

40

44

48

52

56




Nce

λ00 05 10 15 20

30

32

34

36

38

40

42


Present study (UB)

Present study (LB)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01

381 413 437 441 441 441 441

03

274 297 298 298 298 298 2981 392 422 443 445 445 445 445 277 299 299 299 299 299 29915 397 426 446 447 447 447 447 279 301 301 301 301 301 30125 401 429 448 448 448 448 448 280 301 301 301 301 301 3015 404 431 449 449 449 449 449 281 302 302 302 302 302 30275 405 432 450 450 450 450 450 281 302 302 302 302 302 3020625

02

320 349 362 362 362 362 362

035


30

0625

01

312 346 376 404 428 441 441

03


02

267 292 314 333 350 362 362

035


45

0625

01

246 289 328 364 395 441 441

03


02

213 243 273 299 323 360 360

035


60

0625

01

195 244 292 334 370 428 441

03


02

167 204 241 274 303 347 352

035


75

0625

01

148 202 258 307 350 413 441

03


02

127 168 211 250 283 334 336

035





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







Nce

λ0 1 2 3 4

24

28

32

36

40

44

kh = 01kh = 02

kh = 03kh = 035

β = 15deg

(a)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 4

20

25

30

35

40

45

β = 45deg

(b)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 415

20

25

30

35

40

45

β = 60deg

(c)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 410

15

20

25

30

35

40

45

β = 75deg

(d)








00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)








Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




(UB + LB)() (4)








d1 Tangent to f

Feasible region

Allowable step t


f(x) = 0

d0ρd1

d

Feasible arc x

d


u = v = 0

σn = τ = 0σn = τ = 0

u

v

u =

v = 0

u =

v = 0

6

4

24 6 8 10 12


0 10 20 30 40 5025

30

35

40

45

50

55

Nc



β (deg)











Nce

ndash01 00 01 02 03 04

20

25

30

35

40

45


kh



β = 15deg

β = 30deg

β = 45deg


Nc

00 05 10λ

15 20

40

44

48

52

56




Nce

λ00 05 10 15 20

30

32

34

36

38

40

42


Present study (UB)

Present study (LB)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01

381 413 437 441 441 441 441

03

274 297 298 298 298 298 2981 392 422 443 445 445 445 445 277 299 299 299 299 299 29915 397 426 446 447 447 447 447 279 301 301 301 301 301 30125 401 429 448 448 448 448 448 280 301 301 301 301 301 3015 404 431 449 449 449 449 449 281 302 302 302 302 302 30275 405 432 450 450 450 450 450 281 302 302 302 302 302 3020625

02

320 349 362 362 362 362 362

035


30

0625

01

312 346 376 404 428 441 441

03


02

267 292 314 333 350 362 362

035


45

0625

01

246 289 328 364 395 441 441

03


02

213 243 273 299 323 360 360

035


60

0625

01

195 244 292 334 370 428 441

03


02

167 204 241 274 303 347 352

035


75

0625

01

148 202 258 307 350 413 441

03


02

127 168 211 250 283 334 336

035





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







Nce

λ0 1 2 3 4

24

28

32

36

40

44

kh = 01kh = 02

kh = 03kh = 035

β = 15deg

(a)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 4

20

25

30

35

40

45

β = 45deg

(b)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 415

20

25

30

35

40

45

β = 60deg

(c)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 410

15

20

25

30

35

40

45

β = 75deg

(d)








00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)








Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










Nce

ndash01 00 01 02 03 04

20

25

30

35

40

45


kh



β = 15deg

β = 30deg

β = 45deg


Nc

00 05 10λ

15 20

40

44

48

52

56




Nce

λ00 05 10 15 20

30

32

34

36

38

40

42


Present study (UB)

Present study (LB)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01

381 413 437 441 441 441 441

03

274 297 298 298 298 298 2981 392 422 443 445 445 445 445 277 299 299 299 299 299 29915 397 426 446 447 447 447 447 279 301 301 301 301 301 30125 401 429 448 448 448 448 448 280 301 301 301 301 301 3015 404 431 449 449 449 449 449 281 302 302 302 302 302 30275 405 432 450 450 450 450 450 281 302 302 302 302 302 3020625

02

320 349 362 362 362 362 362

035


30

0625

01

312 346 376 404 428 441 441

03


02

267 292 314 333 350 362 362

035


45

0625

01

246 289 328 364 395 441 441

03


02

213 243 273 299 323 360 360

035


60

0625

01

195 244 292 334 370 428 441

03


02

167 204 241 274 303 347 352

035


75

0625

01

148 202 258 307 350 413 441

03


02

127 168 211 250 283 334 336

035





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







Nce

λ0 1 2 3 4

24

28

32

36

40

44

kh = 01kh = 02

kh = 03kh = 035

β = 15deg

(a)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 4

20

25

30

35

40

45

β = 45deg

(b)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 415

20

25

30

35

40

45

β = 60deg

(c)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 410

15

20

25

30

35

40

45

β = 75deg

(d)








00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)








Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01

381 413 437 441 441 441 441

03

274 297 298 298 298 298 2981 392 422 443 445 445 445 445 277 299 299 299 299 299 29915 397 426 446 447 447 447 447 279 301 301 301 301 301 30125 401 429 448 448 448 448 448 280 301 301 301 301 301 3015 404 431 449 449 449 449 449 281 302 302 302 302 302 30275 405 432 450 450 450 450 450 281 302 302 302 302 302 3020625

02

320 349 362 362 362 362 362

035


30

0625

01

312 346 376 404 428 441 441

03


02

267 292 314 333 350 362 362

035


45

0625

01

246 289 328 364 395 441 441

03


02

213 243 273 299 323 360 360

035


60

0625

01

195 244 292 334 370 428 441

03


02

167 204 241 274 303 347 352

035


75

0625

01

148 202 258 307 350 413 441

03


02

127 168 211 250 283 334 336

035





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







Nce

λ0 1 2 3 4

24

28

32

36

40

44

kh = 01kh = 02

kh = 03kh = 035

β = 15deg

(a)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 4

20

25

30

35

40

45

β = 45deg

(b)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 415

20

25

30

35

40

45

β = 60deg

(c)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 410

15

20

25

30

35

40

45

β = 75deg

(d)








00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)








Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







Nce

λ0 1 2 3 4

24

28

32

36

40

44

kh = 01kh = 02

kh = 03kh = 035

β = 15deg

(a)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 4

20

25

30

35

40

45

β = 45deg

(b)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 415

20

25

30

35

40

45

β = 60deg

(c)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 410

15

20

25

30

35

40

45

β = 75deg

(d)








00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)








Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Nce

λ0 1 2 3 4

24

28

32

36

40

44

kh = 01kh = 02

kh = 03kh = 035

β = 15deg

(a)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 4

20

25

30

35

40

45

β = 45deg

(b)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 415

20

25

30

35

40

45

β = 60deg

(c)

Nce

λ

kh = 01kh = 02

kh = 03kh = 035

0 1 2 3 410

15

20

25

30

35

40

45

β = 75deg

(d)








00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)








Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







00 15 30 45 60 7524

27

30

33

36

39

Nce

cu(γB)

β = 15deg

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB)00 15 30 45 60 75

20

22

24

26

28

30

32

β = 45deg

kh = 01kh = 02

kh = 03kh = 035

(b)

Nce

cu(γB)00 15 30 45 60 75

16

18

20

22

24

26

28

β = 60deg

kh = 01kh = 02

kh = 03kh = 035

(c)

Nce

cu(γB)00 15 30 45 60 75

12

14

16

18

20

22

β = 75deg

kh = 01kh = 02

kh = 03kh = 035

(d)








Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







Nce

cu(γB) = 1

β (deg)10 20 30 40 50 60 70 80

10

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(a)

Nce

cu(γB) = 5

β (deg)10 20 30 40 50 60 70 80

15

20

25

30

35

40

kh = 01kh = 02

kh = 03kh = 035

(b)


10 20 30 40 50 60 70 8020

25

30

35

40

45

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 1

(a)

10 20 30 40 50 60 70 8024

28

32

36

40

44

Nce

β (deg)

kh = 01kh = 02

kh = 03kh = 035

cu(γB) = 5

(b)





010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




010 015 020 025 030 035

24

28

32

36

40

kh



Nce

β = 15deg

(a)

kh

010 015 020 025 030 035

16

20

24

28

32

36

Nce



(b)


7

6

5

4

3

22 4 6 8 10

(a)

24 6 8 10 12

3

4

5

6

7

(b)

0 2 4 6 8 10 122

4

6

(c)

7

6

5

4

3

22 4 6 8 10

(d)

7

6

5

4

3

24 6 8 10 12

(e)

0 2 4 6 8 10 122

4

6

(f )





7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




7 Conclusions



2 44

6

6 8 10 12

(a)

765

655

545

46 7 8 9 10 11

(b)

46 7 8 9 10 11

455

556

657

(c)

4

2ndash2 0 2 4 6 8 10 12

6

(d)

4

5

24 6 8 10 12

3

6

7

(e)

4 6 8 10 12

4

5

2

3

6

7

(f )

10500

5

ndash5

(g)121086420

0

2

4

6

(h)

40

2

4

6

6 8 10 12 14

(i)





khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




khλ

0 05 1 15 2 3 4 0 05 1 15 2 3 4

15

0625

01


03


02


035


30

0625

01


03


02


035


45

0625

01


03


02


035


60

0625

01


03


02


035


75

0625

01


03


02


035







7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






7

65

655

5

454

6 7 8 9 10 11(a)

4 6 8 10 12

7

6

5

4

3

2

(b)0 2 6 84 10 12

6

4

2

0

(c)










4 6 8 10 12

7

6

5

4

3

2

(a)

4 6 8 10 12

7

6

5

4

3

2

(b)

4 6 8 10 12

7

6

5

4

3

2

(c)

4 6 8 10 12

7

6

5

4

3

2

(d)

4 6 8 10 12

7

6

5

4

3

2

(e)







Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Notation


Data Availability




Acknowledgments


References
































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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