determination of mohr–coulomb parameters from nonlinear

Research ArticleDetermination of MohrndashCoulomb Parameters from NonlinearStrength Criteria for 3D Slopes

DiWu 1 YukeWang 2 Yue Qiu3 Juan Zhang1 and Yukuai Wan 4

1College of Architectural Engineering Qingdao Binhai University No 425 West Jialingjiang Road Qingdao 266555 China2College of Water Conservancy and Environmental Engineering Zhengzhou University No 100 Science AvenueZhengzhou 450001 China3State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science andTechnology Shandong University of Science and Technology No 579 Qianwangang Road Qingdao 266590 China4Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering Hohai UniversityNo 1 Xikang Road Nanjing 210098 China

Correspondence should be addressed to Di Wu wudiqdbhueducn

Received 21 March 2019 Revised 23 May 2019 Accepted 9 June 2019 Published 11 July 2019

Academic Editor Federico Guarracino

Copyright copy 2019 Di Wu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Many experimental data have illustrated that the strength envelops for soils are not linear Nevertheless the linear MohrndashCoulomb(MC) strength parameters are widely applied for the conventional method software codes and engineering standards in the slopedesign practice Hence this paper developed the 3D limit analysis for the stability of soil slopes with the nonlinear strength criterionBased on a numerical optimization procedure written in Matlab software codes the equivalent MC parameters (the equivalentfriction angle and the equivalent cohesion) from the nonlinear strength envelopes were derived with respect to the least upper-bound solutions Further investigations were made to assess the influences of nonlinear strength parameters and slope geometrieson the equivalent MC parameters The presented results indicate that the equivalent MC parameters are closely related to thenonlinear strength parameters As the inclination angle increases the equivalent friction angle becomes bigger but the equivalentcohesion becomes smaller Besides 3D effects on the equivalent MC parameters were found to be slight The presented approachfor the determination of MC strength parameters is analytical and rigorous and the approximate MC strength parameters in theprovided design tables can be alternative references for practical use

1 Introduction

Strength criterion is critical for all types of materials in thearea of slope stability analysisThe strengths of soils and rocksare universally presented by the linearMohrndashCoulomb (MC)failure envelope which represents the shear strength by twoMC strength parameters the friction angle and the cohesionTheMC strength parameters have been widely applied in theconventional limit equilibrium methods for the calculationsof slope safety factors Besides the computer codes andengineering standards for slope design are commonly onthe base of the MC strength criterion However accordingto the experimental data many studies have illustrated thatthe strength envelops for soils and rocks are not linear [1ndash7] Hence many researchers then utilized some presented

nonlinear strength criteria to conduct slope stability analysis(eg Charles amp Soares [8] Zhang amp Chen [9] Dawson etal [10] Yang amp Yin [11] Li et al [12] Shen amp Karakus [13]Gao et al [14 15] Zhao et al [16] Xu amp Yang [17]) Howeverthese nonlinear failure criteria are not presented in forms ofMC strength parameters and they cannot be directly used inpractice for slope design

To solve this problem many attempts have been madeto derive the equivalent MC parameters from nonlinearstrength criteria Hoek and his partners [18ndash21] have suc-cessively devoted themselves to settling this problem forseveral decades and proposed the widely analytical solutionsfor average MC parameters from the Hoek-Brown strengthenvelope Meanwhile other researchers also developed theanalytical methods to obtain the MC parameters for rock

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 6927654 12 pageshttpsdoiorg10115520196927654

2 Mathematical Problems in Engineering

masses satisfying the Hoek-Brown strength criteria [22ndash26]Besides Shen et al [27] presented an approximate analyticalmethod to determine the MC parameters for slope stabilityassessment based on the Hoek-Brown strength criterionYang amp Yin [28] employed the tangential technique intothe limit analysis method to evaluate the equivalent MCparameters for rock slopes with the Hoek-Brown strengthenvelope Reviewing the literature these presented investi-gations have been made for slopes in rock masses satisfyingthe Hoek-Brown criteria and the slope stability analysis havebeen generally conducted in the condition of plain strainTherefore it is necessary to carry out the estimation of theMC strength parameters of soils satisfying the nonlinearcriteria Further studies should be done to consider thethree dimensional (3D) effects on the determination of MCparameters

For slopes in soils a number of researchers have useda Power-Law (PL) type of nonlinear strength envelope forthe evaluation of slope safety (eg Charles amp Soares [8]Zhang amp Chen [9] Yang amp Yin [11] Gao et al [14 15] Zhaoet al [16] Xu amp Yang [17]) To determine the MC strengthparameters of soil slopes with the PL failure criterion thisstudy adopted the tangential method to carry out the 3Dlimit analysis method for slope stability assessment Theapproximate MC strength parameters could be derived withrespect to the least upper-bound solutions Moreover theinfluences of nonlinear parameters and slope geometries(the slope inclination and 3D effect) on the equivalent MCparameters have been further investigated in this paper

2 Estimation of MC Parameters for3D Soil Slopes

21 PL Strength Criterion and Tangential Method Since thenonlinear PL failure criterion was firstly proposed by Zhangamp Chen [9] to express the failure envelopes of cohesive soilsnumerous researchers have applied this nonlinear criterioninto the slope stability analysis [11 14ndash17] For the PL failurecriterion the shear stress 120591 on the slope slip surface isexpressed in the form of normal stress 120590119899 as follows

120591 = 1198880 (1 + 120590n1205900 )1119898

(1)

where the parameters c0 1205900 andm are the nonlinear strengthconstants of PL failure criterion As presented in Figure 1 theparameter 1198880 is the initial cohesion as120590n is zero the parameter1205900 is the tensile stress as 120591 is zero and the parameterm is thenonlinearity coefficient

To implement the use of nonlinear strength criteria forslope engineering the tangential method was originally pro-posed by Drescher amp Christopoulos [29] to conduct the limitanalysis of slope stability Then many researchers employedthe proposed tangential method to evaluate the slope safetyin the 2D or 3D conditions [11 14ndash17 28]Their studies coulddemonstrate the validity of stability results obtained from

Power-Law criterionTangent line

c0c

0n

T

Figure 1 PL strength envelope and tangent line

the tangential method for slope engineering applicationsTherefore the tangential method was also utilized in thisstudy

As illustrated in Figure 1 the nonlinear strength envelopefor certain stress range could be replaced by a tangent line inthe form of the equivalent MC strength parameters At somepoint T the expression of the tangent line will be given by thefollowing equation

120591 = 119888119890 + 120590n tan120601119890 (2)

where the parameters 120601e and 119888e are the equivalent MCstrength parameters Here the parameter 120601e represents theequivalent friction angle and the parameter 119888e represents theequivalent cohesion

For the PL strength criterion the gradient of the tangentline at some point T can be derived from the deviation ofthe expression of (1) with respect to the normal stress 120590n asshown in (3)

tan120601e = 120597120591120597120590n = 11988801198981205900 (1 +120590n1198880 )((1minus119898)119898)

(3)

By transforming (3) the normal stress 120590n can be given in thefunction of the equivalent friction angle 120601119890 as follows

120590n = 1205900 (1198981205900 tan120601e1198880 )(119898(1minus119898)) minus 1205900 (4)

Combining this expression with (1) the shear stress 120591 can bederived in the form of the equivalent friction angle 120601119890 that is

120591 = 1198880 (1198981205900 tan120601e1198880 )(1(1minus119898)) (5)

Mathematical Problems in Engineering 3

After taking (4) and (5) into (2) the equivalent cohesion c119890can be expressed as follows

119888e1198880 =119898 minus 1119898 (12059001198880 119898 tan120601e)(1(1minus119898)) + 12059001198880 tan120601e (6)

From (6) it can be seen that the equivalent cohesion 119888119890 is afunction of the equivalent friction angle 120601119890 To make indexesbeing dimensionless the parameter ratio of 1198881198901198880 is used asthe equivalent cohesion in this study

22 3D Limit Analysis To establish the 3D limit analysismethod Michalowski amp Drescher [30] and Gao et al [31]have conducted some researches on the 3D rotational failuremechanisms for soil slopes considering toe failure facefailure and base failure Afterwards Gao et al [31] andGao etal [14 15] adopted the 3D failure mechanisms for face failureand base failure to present the 3D limit analysis of slopestability based on the MC strength criterion and nonlinearPL strength criterion respectively Hence this study utilizedthe 3D limit analysis method of Gao et al [14] to derive theequivalentMCparameters for soil slopes with the PL strengthcriterion

As presented by Gao et al [31] Figures 2(a) and 2(b) givethe 3D face failure mechanism and 3D base failure mecha-nism respectively The curvilinear cone can be obtained byrotating a circle with the radius R about an axis The distancefrom the axis to the rotation centre O is defined as the radius119903119898 The expressions of the radiuses 119877 and 119903m are presented asfollows

119877 = 119903 minus 11990310158402 (7)

119903m = 119903 + 11990310158402 (8)

The parameters r and 1199031015840 represent two log-spirals PAD andPA1015840D1015840 passing through the rotation centre O which can beexpressed as

119903 = 1199030 exp [(120579 minus 1205790) tan120601e] (9)

and

1199031015840 = 11990310158400 exp [minus (120579 minus 1205790) tan120601e] (10)

where the parameters r0 and 11990301015840 represent OA and OA1015840 inFigure 2 and 120601e represents the apex angle of curvilinear coneas well as the equivalent friction angle from the PL strengthcriterion Figure 3 shows themodified 3D failuremechanismscomposed by a curvilinear cone with the width 21198871015840 and aninsertosomewith thewidth bThe ratio of the slopewidthB tothe slope heightH namely the relative width BH is adoptedhere to represent the 3D effect of slopes

On the basis of the above 3D failure mechanisms theenergy-balance equation can be established by equating thesoil weight work rate 119882120574 to the internal energy dissipationrate D as shown in the following expression

119882curve120574 +119882plane

120574 = 119863curve + 119863plane (11)

where the parameters 119882curve120574 and 119863curve relate to the work

rates for the curvilinear cone The parameters 119882plane120574 and

119863plane represent the work rates for the insertosome whichcan be seen in the reference of Yang amp Yin [11] For thecurvilinear cone of face failure and base failure the param-eters 119882curve

120574 and 119863curve will be presented in the followinginterpretations

For 3D face failure mechanism (Figure 2(a)) the heightof the rotating block is expressed by the parameter 1198671015840 Bycombining the equivalent strength parameters (119888119890 and 120601119890) theparameters119882curve

120574 and119863curve can be derived by the followingexpressions

119882curve120574

= 2120596120574[int120579B1205790

intradic1198772minus11988620

intradic1198772minus1199092119886

(119903m

+ 119910)2 cos 120579d119910d119909d120579 + int120579h120579B


intradic1198772minus1199092119889

(119903m

+ 119910)2 cos 120579d119910d119909d120579]

(12)

119863curve

= 2119888e12059611990320tan120601e [(minussin21205790

sdot int120579B1205790

cos 120579sin3120579radic1198772 minus 1198862d120579) + (minussin2 (120573 + 120579h)

sdot 1198902(120579hminus1205790) tan120601e int120579h120579B

cos (120579 + 120573)sin3 (120579 + 120573)radic1198772 minus 1198892d120579)]

(13)

For 3D base failure mechanism (Figure 2(b)) an addi-tional angle 1205731015840 is considered to determine the slip surfacegeometry By applying the equivalent strength parameters 119888119890and 120601119890 the parameters 119882curve

120574 and 119863curve can be derived as


R

Rx

x

y

y

a

D

A

P

O

B

H

C

d1

＄

H

A

2

0B

h

rＧ

(a)

R

R

x

x

x

e

y

y

ya

D

A

P

O

B

H

C

d2

＄

A

2

0B

ch

rＧ

(b)

Figure 2 Modified 3D failure mechanisms (a) face failure mechanism and (b) base failure mechanism [31]

H

Bb

b

b

Plane insert

(a)

H

Bb

b

bPlane insert

(b)

Figure 3 Modified 3D failure mechanisms with the insertosome (a) face failure mechanism and (b) base failure mechanism [31]

follows

119882curve120574 = 2120596120574[int120579B

1205790


intradic1198772minus1199092119886

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579C120579B


intradic1198772minus1199092119889

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579h120579C


intradic1198772minus1199092119890

(119903m + 119910)2 cos 120579d119910d119909d120579]

(14)

119863cure = 2119888e12059611990320tan120601e [(minussin21205790 int120579B

1205790

cos 120579sin3120579radic1198772 minus 1198862d120579)

+ (minus sin2 (120573 + 120579C) sin2120579hsin2120579C exp [2 (120579h minus 1205790) tan120601e]

sdot int120579C120579B

cos (120579 + 120573)sin3 (120579 + 120573)radic1198772 minus 1198892d120579)

+ (minussin2120579h exp [2 (120579h minus 1205790) tan120601e] int120579h120579C

cos 120579sin3120579radic1198772 minus 1198902d120579) ]

(15)


where the parameter 119888e is the equivalent cohesion which canbe expressed by the function of the equivalent friction angle

120601e (Equation (6)) The parameters a d e 120579B and 120579C areobtained by the following expressions

119886 = sin 1205790sin 120579 1199030 minus 119903m (16)

119889 = 1199030 sin (120579C + 120573) sin 120579hsin (120579 + 120573) sin 120579C exp [(120579h minus 1205790) tan120601e] minus 119903m (17)

119890 = 1199030 sin 120579hsin 120579 exp [(120579h minus 1205790) tan120601e] minus 119903m (18)

120579B = arctansin 1205790

cos 1205790 minus 1198601015840 (19)

120579C = arctansin 120579h exp [(120579h minus 1205790) tan120601e]

cos 1205790 minus 1198601015840 minus sin 120579h exp [(120579h minus 1205790) tan120601e] minus sin 1205790 tan120573 (20)

1198601015840 = sin (120579h minus 1205790)sin 120579h minus sin (120579h + 120573)

sin 120579h sin120573 sin 120579h exp [(120579h minus 1205790) tan120601e] minus sin 1205790 (21)

On the base of the energy-balance equation the upper-bound solutions (ie the critical height119867cr) would be derivedfor a soil slope with given parameters (ie slope inclinationangle 120573 nonlinear parameters 119898 1198880 1205900 and relative widthBH) To obtain the least upper bound on the criticalheight this study adopted a numerical optimization methodpresented by Chen [32] The optimization procedure wasperformed by using a computer code of Matlab softwareTheleast upper-bound solutions can be calculated with respect toseveral independent variables angles 1205790 and 120579h ratio of 119903010158401199030relative width of the plane insert bH ratio n =1198671015840119867 for the3D face-failuremechanism or angle 1205731015840 for the 3D base-failuremechanism and one additional variable 120601e The variables 1205790120579h 119903010158401199030 bH n or 1205731015840 determine the failure mechanismand the variable 120601e determines the location of tangent lineof PL strength criterion More details for the interpretationsand notations of 3D limit analysis method can be foundin the references of Michalowski amp Drescher [30] andGao et al [31]

23 Determination of Approximate MC Parameters As pre-sented in the 3D limit analysismethod the equivalent frictionangle 120601e represents the apex angle of the curvilinear coneHence the parameter 120601e is a significant variable in theenergy-balance equation The variable 120601e can be obtainedonce the least upper-bound solutions are determined inthe optimization procedure Correspondingly the equivalentcohesion 119888e then can be derived with respect to the equivalentfriction angle120601e as illustrated in (6) Since the shear strengthsof tangent line are equal to or larger than those of thePL strength envelop in the same normal stress range thecalculated solution will be an upper bound of the actuallimit load Here the equivalent MC strength parameters (theequivalent friction angle 120601e and the equivalent cohesion 119888e)are not the conventional strength parameters to reflect thesoil nature But they can represent the approximate shear

strengths of the relevant stress distribution acting on the slopecritical slip surface Therefore the obtained values of 120601e and119888e can be used as the approximate MC strength parameters inslope engineering

3 Numerical Results and Analyses

31 Effect of 11988801205900 on Equivalent MC Parameters Selectingtwo 3D slopes (BH = 20) with 120573 = 30∘ and 120573 = 60∘as examples Figures 4 and 5 present the equivalent MCstrength parameters (the equivalent friction angle 120601119890 and theequivalent cohesion 119888e1198880) as the x-coordinate is the strengthparameter ratio of 11988801205900 Considering different nonlinearitycoefficients m (12 16 and 20) three changing lines werepresented in each figure It should be noted that the strengthparameter ratio of 11988801205900 is adopted as dimensionless param-eter which is consistent with the equivalent cohesion 119888e1198880

For gentle slopes with 120573 = 30∘ (Figure 4) the equivalentfriction angle 120601e appears to be bigger as the ratio of 11988801205900increases However the increasing trend becomes weakerwhen the ratio of 11988801205900 is relatively bigger From Figure 4(a)it can be seen that the changing lines tend to be horizontalin the big range of 11988801205900 Correspondingly the equivalentcohesion 119888e1198880 becomes larger gradually as the ratio of 11988801205900increases By comparing the changing lines with respect todifferent parametersm it can be found that the influences ofthe ratio of 11988801205900 on the equivalent MC parameters becomemore remarkable with the decreasing value ofm

Nevertheless for steep slopes (Figure 5) the equivalentfriction angle 120601e and the equivalent cohesion 119888e1198880 becomebigger gradually with the increasing 11988801205900 Meanwhile as thestrength parameterm decreases the effect of the ratio of 11988801205900on the equivalent friction angle 120601e was found to be moresignificant But the effect of 11988801205900 on the equivalent cohesion119888e1198880 appears to be slight


0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)

Figure 4 Effect of 11988801205900 on equivalent MC parameters (120573 = 30∘)

0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

01

1

10

c c

0c00 = 04

c00 = 10

c00 = 20

(b)

Figure 6 Effect ofm on equivalent MC parameters (120573 = 30∘)

32 Effect of 119898 on Equivalent MC Parameters Figures 6 and7 illustrate the influences of the nonlinearity coefficient mon the equivalent MC strength parameters (120601e and 119888e1198880) for3D slopes with 120573 = 30∘ and 120573 = 60∘ Here the relative widthfor each slope was assumed as BH = 20 From Figures 6and 7 it can be found that the equivalent friction angle 120601eand the equivalent cohesion 119888e1198880 both become smaller asthe parameterm increases whether for gentle slopes or steepslopes The influences of the parameter m on the equivalentstrength parameters tend to be less pronounced with thedecreasing ratio of 11988801205900 especially for steep slopes with thesmall ratio of 11988801205900 As illustrated in Figure 7(b) for slopeswith 120573 = 60∘ and 11988801205900 =20 the equivalent cohesion 119888e1198880would change slightly as the parameterm increases

33 Effect of 120573 on Equivalent MC Parameters To explore theeffects of the slope angle 120573 on the equivalent MC strengthparameters (120601e and 119888e1198880) Figures 8(a) and 8(b) present thedifferent values of 120601e and 119888e1198880 by taking the inclinationangle 120573 as the x-coordinate Four kinds of conditions wereconsidered in this section 11988801205900 = 04 m = 12 11988801205900 =04 m = 20 11988801205900 = 20 m = 12 11988801205900 = 20 m = 20In each condition the slope relative width BH = 20 wasadopted

It is obvious that the equivalent friction angle 120601e becomeslarger as the inclination angle 120573 increases However theequivalent cohesion 119888e1198880 becomes smaller with the increas-ing angle 120573 Comparing these four conditions of 11988801205900 andmthe influences of angle 120573 on the equivalent MC parameters

appear to be more significant for soil slopes with the larger11988801205900 and the smallerm

34 Effect of BH on Equivalent MC Parameters Figure 9gives the values of the equivalent friction angle 120601e for twoslopes (120573 = 30∘ and 120573 = 60∘) with respect to different relativewidths BH Similarly four kinds of combinations of 11988801205900and m were presented in these figures For gentle and steepslopes (Figures 9(a) and 9(b)) the equivalent friction angle120601e was found to be almost constant as the ratio of BHincreases Since the equivalent cohesion 119888e1198880 is a functionof the equivalent friction angle 120601e (as presented in Equation(6)) the equivalent cohesion 119888e1198880 would also change slightlywith the increasingBHThephenomenonmay reveal that 3Deffects nearly have no influences on equivalent MC strengthparameters although 3Deffects have significant influences onthe slope stability [14 15]

35 Charts of Approximate MC Parameters Based on theabove results and analyses it can be concluded that thenonlinear strength parameters and the slope inclination havesignificant influences on the equivalent MC parameters (120601eand 119888e1198880) Nevertheless 3D effects on the equivalent MCparameters can be ignored (but 3D effects on the slope stabil-ity are significant) Hence this study derived the equivalentMC parameters with respect to various nonlinear strengthparameters and common slope inclinations for 2D soil slopesas presented in Tables 1 and 2 The approximate MC strengthparameters in these charts can be alternative references in the


0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)

Figure 8 Effect of 120573 on equivalent MC parameters


0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)

Figure 9 Effect of BH on equivalent MC parameters (a) 120573 = 30∘ (b) 120573 = 60∘

software codes and engineering standards for slope designpractice

4 Example Problems

To verify the accuracy of the presented method and theapplicability of the given approximate MC parameters thissection provides two examples of uniform dry soil slopesin plain-strain conditions Since the limit analysis methodfocuses on the critical state of slope failure the safety factorsfor slopes are assumed as F = 10 in the above studies andthe critical height119867cr are used as the upper-bound solutionsfor slope stability For comparisons with the other resultsrepresented by F the shear strength can be reduced by thesafety factor F and theminimum safety factors will be derivedby using the presented limit analysis method

41 Example 1 For the slope in example 1 the geometryparameters are given asH = 12 m and 120573 = 282∘ This examplewas utilized by Eid [33] based on the test results of shearstrengths given by Chandler [34] for Upper Lias clay Thenonlinear PL strength function can be obtained by curve fit-ting to the test data using the LevenbergndashMarquardt methodThe nonlinear strength parameters have the following valuesc0 =098 kPa1205900 =033 kPa andm= 138The total unitweight120574 is adopted as 20 kNm3

Based on the presented method for this slope with non-linear parameters the minimum safety factor is calculated as164 This slope problem has been analyzed by Eid [33] using

the limit equilibrium method and another nonlinear failurecriterion He obtained the safety factor of 150 which is alittle smaller than the result (F = 164) of this study Sincethe limit analysis method adopted in this study derived theupper-bound solutions for slope stability the difference of164 versus 150 between the safety factors is reasonable andthis comparison can confirm the correctness of the presentedresults in this study

For slope design the approximateMCparameters for thisslope example can be obtained from Tables 1 and 2 Givenvalues of m asymp 14 11988801205900 asymp 30 and 120573 = 282∘ we can getthe equivalent friction angle 120601e asymp 1773∘ and the equivalentcohesive 119888e1198880 asymp 3837 by the interpolation calculations ofgiven values Using 1198880 = 098 kPa the approximate cohesive119888e is determined as 3760 kPa In condition of the safety factorF = 10 the presented limit analysis method can derive thecritical height for this slope ie119867cr = 664 m It reveals thatthe design height for this slope should be smaller than 664mto ensure its safety

42 Example 2 The problem considered in this exampleadopts the test data reported by Baker [35] for compactedIsraeli clay The nonlinear strength parameters were derivedas follows c0 = 006 kPa 1205900 = 002 kPa and m = 123 Theslope height H is 6 m and the slope inclination 120573 is 43∘ Thetotal unit weight for Israeli clay is taken as 120574 = 18 kNm3

For such a problem the limit analysis method presentedin this study yielded the safety factor of 114 which is alittle larger than the result of F = 097 derived from the


Table 1 The equivalent friction angle 120601e (∘) for various soils11988801205900 120573 (∘) m

12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766

limit equilibrium method of Baker [35] The small differencecan verify the accuracy of the solutions derived from thepresented method Besides considering the values of m =123 c01205900 = 006002 = 30 and 120573 = 43∘ the approximateMC parameters 120601e asymp 3393∘ and 119888e asymp 319 kPa are determinedfrom Tables 1 and 2 Hence the critical height for this slopecan be calculated as119867cr = 129 m by using the presented limitanalysis method with F = 10

5 Conclusions

On the base of 3D failure mechanisms for soil slopes withtheMC strength criterion this paper employed the tangentialmethod to develop the upper-bound limit analysis of slopestability with the nonlinear PL strength criterion A numer-ical optimization procedure written in a computer code ofMatlab software was applied to calculate the upper-boundsolutions of slope stability The equivalent MC strengthparameters from the PL strength envelope were then derivedwith respect to the least upper-bound solutions Effects ofnonlinear strength parameters and slope geometries on theequivalentMCparameters have beenwell studied and designchats of approximate MC strength parameters have beenprovided for various soil slopes From this study the mainconclusions can be made as follows

(1) The equivalent MC strength parameters 120601e and 119888e1198880both tend to be larger gradually with the increasingratio of 11988801205900 However the effects of the nonlinearitycoefficientm on the equivalent MC strength parame-ters are opposite namely the equivalent friction angle120601e and the equivalent cohesion 119888e1198880 become smallerwith the increasingm

(2) As the inclination angle 120573 increases the equivalentfriction angle 120601e becomes bigger and the equivalentcohesion 119888e1198880 becomes smaller The influences of theinclination on the equivalent MC parameters seem tobe more pronounced for soil slopes with the biggervalue of 11988801205900 or smaller value ofm

(3) Although 3D effect has significant influences on thesafety of soil slopes 3D effect on the equivalent MCstrength parameters seems to be slight

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest


Table 2 The equivalent cohesive 119888e1198880 for various soils11988801205900 120573 (∘) m

12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments

This study was financially supported by National NaturalScience Foundation of China (Grant Nos 51708310 and51809160) Shandong Provincial Natural Science FoundationChina (Grant Nos ZR2017BEE066 and ZR201702160366)a Project of Shandong Province Higher Educational Sci-ence and Technology Program (Grant No J17KB049) andScientific Research Foundation of Shandong University ofScience and Technology for Recruited Talents (Grant No2017RCJJ004)

References

[1] AW Bishop D LWebb and P I Lewin ldquoUndisturbed samplesof london clay from the ashford common shaft Strength-effective stress relationshipsrdquo Geotechnique vol 15 no 1 pp 1ndash31 1965

[2] V M Ponce and J M Bell ldquoShear strength of sand at extremelylow pressuresrdquo Journal of the Soil Mechanics Foundations Divi-sion vol 97 no 4 pp 625ndash638 1971

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] E Hoek and E T Brown ldquoEmpirical strength criterion forrock massesrdquo Journal of Geotechnical and GeoenvironmentalEngineering vol 106 no GT9 pp 1013ndash1035 1980

[5] G Lefebvre ldquoStrength and slope stability in Canadian soft claydepositsrdquo Canadian Geotechnical Journal vol 3 no 2 pp 420ndash442 1981

[6] R Ucar ldquoDetermination of shear failure envelope in rockmassesrdquo Journal of Geotechnical Engineering vol 112 no 3 pp303ndash315 1986

[7] R Baker ldquoNonlinear Mohr envelopes based on triaxial datardquoJournal of Geotechnical and Geoenvironmental Engineering vol130 no 5 pp 498ndash506 2004

[8] J A Charles and M M Soares ldquoThe stability of slopes insoils with nonlinear failure envelopesrdquo Canadian GeotechnicalJournal vol 21 no 3 pp 397ndash406 1984

[9] X J Zhang and W F Chen ldquoStability analysis of slopes withgeneral nonlinear failure criterionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 11 no1 pp 33ndash50 1987

[10] E Dawson K You and Y Park ldquoStrength-reduction stabilityanalysis of rock slopes using the Hoek-Brown failure criterionrdquoin Proceedings of the Sessions of Geo-Denver 2000 - Trends inRock Mechanics GSP 102 pp 65ndash77 2000

[11] X-L Yang and J-H Yin ldquoSlope stability analysis with nonlinearfailure criterionrdquo Journal of Engineering Mechanics vol 130 no3 pp 267ndash273 2004


[12] A J Li R S Merifield and A V Lyamin ldquoStability chartsfor rock slopes based on the Hoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 689ndash700 2008

[13] J Shen andMKarakus ldquoThree-dimensional numerical analysisfor rock slope stability using shear strength reduction methodrdquoCanadian Geotechnical Journal vol 51 no 2 pp 164ndash172 2014

[14] Y Gao D Wu and F Zhang ldquoEffects of nonlinear failurecriterion on the three-dimensional stability analysis of uniformslopesrdquo Engineering Geology vol 198 pp 87ndash93 2015

[15] Y Gao D Wu F Zhang et al ldquoEffects of nonlinear strengthparameters on the stability of 3D soil slopesrdquo Journal of CentralSouth University vol 23 no 9 pp 2354ndash2363 2016

[16] L-H Zhao X Cheng H-C Dan Z-P Tang and Y ZhangldquoEffect of the vertical earthquake component on permanentseismic displacement of soil slopes based on the nonlinearMohrndashCoulomb failure criterionrdquo Soils and Foundations vol57 no 2 pp 237ndash251 2017

[17] J Xu and X Yang ldquoThree-dimensional stability analysis of slopein unsaturated soils considering strength nonlinearity underwater drawdownrdquo Engineering Geology vol 237 pp 102ndash1152018

[18] E Hoek ldquoStrength of jointed rock massesrdquo Geotechnique vol33 no 3 pp 187ndash223 1983

[19] E Hoek ldquoEstimating Mohr-Coulomb friction and cohesionvalues from the Hoek-Brown failure criterionrdquo InternationalJournal of RockMechanics andMining Sciences amp GeomechanicsAbstracts vol 27 no 3 pp 227ndash229 1990

[20] E Hoek and E T Brown ldquoPractical estimates of rock massstrengthrdquo International Journal of Rock Mechanics and MiningSciences vol 34 no 8 pp 1165ndash1186 1997

[21] E Hoek C Carranza-Torres and B Corkum ldquoHoek-Brownfailure criterionrdquo in Proceedings of NARMS-Tac pp 267ndash2732002

[22] P Londe ldquo Discussion of ldquo Determination of the Shear FailureEnvelope in Rock Masses rdquo by Roberto Ucar (March 1986 Vol112 No 3) rdquo Journal of Geotechnical Engineering vol 114 no 3pp 374ndash376 1988

[23] P Kumar ldquoShear failure envelope of Hoek-Brown criterion forrockmassrdquo Tunnelling and Underground Space Technology vol13 no 4 pp 453ndash458 1998

[24] C Carranza-Torres ldquoSome comments on the application of theHoekBrown failure criterion for intact rock and for rockmassesto the solution of tunnel and slope excavationrdquo in Proceedings ofthe Conference on Rock and EngineeringMechanics pp 285ndash326Torino Italy 2004

[25] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[26] W Fu and Y Liao ldquoNon-linear shear strength reduction tech-nique in slope stability calculationrdquo Computers amp Geosciencesvol 37 no 3 pp 288ndash298 2010

[27] J Shen S D Priest and M Karakus ldquoDetermination ofmohrndashcoulomb shear strength parameters from generalizedhoekndashbrown criterion for slope stability analysisrdquoRockMechan-ics and Rock Engineering vol 45 no 1 pp 123ndash129 2012

[28] X-L Yang and J-H Yin ldquoLinear mohr-coulomb strengthparameters from the non-linear hoek-brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[29] A Drescher and C Christopoulos ldquoLimit analysis slope sta-bility with nonlinear yield conditionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 12 no3 pp 341ndash345 1988

[30] R L Michalowski and A Drescher ldquoThree-dimensional stabil-ity of slopes and excavationsrdquo Geotechnique vol 59 no 10 pp839ndash850 2009

[31] Y F Gao F Zhang G H Lei and D Y Li ldquoAn extended limitanalysis of three-dimensional slope stabilityrdquoGeotechnique vol63 no 6 pp 518ndash524 2013

[32] Z Y Chen ldquoRandom trials used in determining global mini-mum factors of safety of slopesrdquoCanadianGeotechnical Journalvol 29 no 2 pp 225ndash233 1992

[33] H T Eid ldquoStability charts for uniform slopes in soils withnonlinear failure envelopesrdquo Engineering Geology vol 168 pp38ndash45 2014

[34] R J Chandler ldquoLias clay the long-term stability of cuttingslopesrdquo Geotechnique vol 24 no 1 pp 21ndash38 1974

[35] R Baker ldquoInter-relations between experimental and computa-tional aspects of slope stability analysisrdquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 27no 5 pp 379ndash401 2003

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


masses satisfying the Hoek-Brown strength criteria [22ndash26]Besides Shen et al [27] presented an approximate analyticalmethod to determine the MC parameters for slope stabilityassessment based on the Hoek-Brown strength criterionYang amp Yin [28] employed the tangential technique intothe limit analysis method to evaluate the equivalent MCparameters for rock slopes with the Hoek-Brown strengthenvelope Reviewing the literature these presented investi-gations have been made for slopes in rock masses satisfyingthe Hoek-Brown criteria and the slope stability analysis havebeen generally conducted in the condition of plain strainTherefore it is necessary to carry out the estimation of theMC strength parameters of soils satisfying the nonlinearcriteria Further studies should be done to consider thethree dimensional (3D) effects on the determination of MCparameters

For slopes in soils a number of researchers have useda Power-Law (PL) type of nonlinear strength envelope forthe evaluation of slope safety (eg Charles amp Soares [8]Zhang amp Chen [9] Yang amp Yin [11] Gao et al [14 15] Zhaoet al [16] Xu amp Yang [17]) To determine the MC strengthparameters of soil slopes with the PL failure criterion thisstudy adopted the tangential method to carry out the 3Dlimit analysis method for slope stability assessment Theapproximate MC strength parameters could be derived withrespect to the least upper-bound solutions Moreover theinfluences of nonlinear parameters and slope geometries(the slope inclination and 3D effect) on the equivalent MCparameters have been further investigated in this paper

2 Estimation of MC Parameters for3D Soil Slopes

21 PL Strength Criterion and Tangential Method Since thenonlinear PL failure criterion was firstly proposed by Zhangamp Chen [9] to express the failure envelopes of cohesive soilsnumerous researchers have applied this nonlinear criterioninto the slope stability analysis [11 14ndash17] For the PL failurecriterion the shear stress 120591 on the slope slip surface isexpressed in the form of normal stress 120590119899 as follows

120591 = 1198880 (1 + 120590n1205900 )1119898

(1)

where the parameters c0 1205900 andm are the nonlinear strengthconstants of PL failure criterion As presented in Figure 1 theparameter 1198880 is the initial cohesion as120590n is zero the parameter1205900 is the tensile stress as 120591 is zero and the parameterm is thenonlinearity coefficient

To implement the use of nonlinear strength criteria forslope engineering the tangential method was originally pro-posed by Drescher amp Christopoulos [29] to conduct the limitanalysis of slope stability Then many researchers employedthe proposed tangential method to evaluate the slope safetyin the 2D or 3D conditions [11 14ndash17 28]Their studies coulddemonstrate the validity of stability results obtained from

Power-Law criterionTangent line

c0c

0n

T

Figure 1 PL strength envelope and tangent line

the tangential method for slope engineering applicationsTherefore the tangential method was also utilized in thisstudy

As illustrated in Figure 1 the nonlinear strength envelopefor certain stress range could be replaced by a tangent line inthe form of the equivalent MC strength parameters At somepoint T the expression of the tangent line will be given by thefollowing equation

120591 = 119888119890 + 120590n tan120601119890 (2)

where the parameters 120601e and 119888e are the equivalent MCstrength parameters Here the parameter 120601e represents theequivalent friction angle and the parameter 119888e represents theequivalent cohesion

For the PL strength criterion the gradient of the tangentline at some point T can be derived from the deviation ofthe expression of (1) with respect to the normal stress 120590n asshown in (3)

tan120601e = 120597120591120597120590n = 11988801198981205900 (1 +120590n1198880 )((1minus119898)119898)

(3)

By transforming (3) the normal stress 120590n can be given in thefunction of the equivalent friction angle 120601119890 as follows

120590n = 1205900 (1198981205900 tan120601e1198880 )(119898(1minus119898)) minus 1205900 (4)

Combining this expression with (1) the shear stress 120591 can bederived in the form of the equivalent friction angle 120601119890 that is

120591 = 1198880 (1198981205900 tan120601e1198880 )(1(1minus119898)) (5)







119877 = 119903 minus 11990310158402 (7)

119903m = 119903 + 11990310158402 (8)


119903 = 1199030 exp [(120579 minus 1205790) tan120601e] (9)

and




119882curve120574 +119882plane

120574 = 119863curve + 119863plane (11)







119882curve120574

= 2120596120574[int120579B1205790


intradic1198772minus1199092119886

(119903m

+ 119910)2 cos 120579d119910d119909d120579 + int120579h120579B


intradic1198772minus1199092119889

(119903m

+ 119910)2 cos 120579d119910d119909d120579]

(12)

119863curve

= 2119888e12059611990320tan120601e [(minussin21205790

sdot int120579B1205790




(13)




R

Rx

x

y

y

a

D

A

P

O

B

H

C

d1

＄

H

A

2

0B

h

rＧ

(a)

R

R

x

x

x

e

y

y

ya

D

A

P

O

B

H

C

d2

＄

A

2

0B

ch

rＧ

(b)


H

Bb

b

b

Plane insert

(a)

H

Bb

b

bPlane insert

(b)


follows

119882curve120574 = 2120596120574[int120579B

1205790


intradic1198772minus1199092119886

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579C120579B


intradic1198772minus1199092119889

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579h120579C


intradic1198772minus1199092119890

(119903m + 119910)2 cos 120579d119910d119909d120579]

(14)


1205790







(15)




119886 = sin 1205790sin 120579 1199030 minus 119903m (16)



120579B = arctansin 1205790

cos 1205790 minus 1198601015840 (19)













0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

01

1

10

c c

0c00 = 04

c00 = 10

c00 = 20

(b)









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018














119877 = 119903 minus 11990310158402 (7)

119903m = 119903 + 11990310158402 (8)


119903 = 1199030 exp [(120579 minus 1205790) tan120601e] (9)

and




119882curve120574 +119882plane

120574 = 119863curve + 119863plane (11)







119882curve120574

= 2120596120574[int120579B1205790


intradic1198772minus1199092119886

(119903m

+ 119910)2 cos 120579d119910d119909d120579 + int120579h120579B


intradic1198772minus1199092119889

(119903m

+ 119910)2 cos 120579d119910d119909d120579]

(12)

119863curve

= 2119888e12059611990320tan120601e [(minussin21205790

sdot int120579B1205790




(13)




R

Rx

x

y

y

a

D

A

P

O

B

H

C

d1

＄

H

A

2

0B

h

rＧ

(a)

R

R

x

x

x

e

y

y

ya

D

A

P

O

B

H

C

d2

＄

A

2

0B

ch

rＧ

(b)


H

Bb

b

b

Plane insert

(a)

H

Bb

b

bPlane insert

(b)


follows

119882curve120574 = 2120596120574[int120579B

1205790


intradic1198772minus1199092119886

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579C120579B


intradic1198772minus1199092119889

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579h120579C


intradic1198772minus1199092119890

(119903m + 119910)2 cos 120579d119910d119909d120579]

(14)


1205790







(15)




119886 = sin 1205790sin 120579 1199030 minus 119903m (16)



120579B = arctansin 1205790

cos 1205790 minus 1198601015840 (19)













0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

01

1

10

c c

0c00 = 04

c00 = 10

c00 = 20

(b)









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









R

Rx

x

y

y

a

D

A

P

O

B

H

C

d1

＄

H

A

2

0B

h

rＧ

(a)

R

R

x

x

x

e

y

y

ya

D

A

P

O

B

H

C

d2

＄

A

2

0B

ch

rＧ

(b)


H

Bb

b

b

Plane insert

(a)

H

Bb

b

bPlane insert

(b)


follows

119882curve120574 = 2120596120574[int120579B

1205790


intradic1198772minus1199092119886

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579C120579B


intradic1198772minus1199092119889

(119903m + 119910)2 cos 120579d119910d119909d120579

+ int120579h120579C


intradic1198772minus1199092119890

(119903m + 119910)2 cos 120579d119910d119909d120579]

(14)


1205790







(15)




119886 = sin 1205790sin 120579 1199030 minus 119903m (16)



120579B = arctansin 1205790

cos 1205790 minus 1198601015840 (19)













0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

01

1

10

c c

0c00 = 04

c00 = 10

c00 = 20

(b)









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018











119886 = sin 1205790sin 120579 1199030 minus 119903m (16)



120579B = arctansin 1205790

cos 1205790 minus 1198601015840 (19)













0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

01

1

10

c c

0c00 = 04

c00 = 10

c00 = 20

(b)









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

04 08 12 16 2000c00

m = 12m = 16m = 20

(a)

m = 12m = 16m = 20

04 08 12 16 2000c00

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

01

1

10

c c

0c00 = 04

c00 = 10

c00 = 20

(b)









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

01

1

10

c c

0c00 = 04

c00 = 10

c00 = 20

(b)









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









0

10

20

30

40

50

60

70

80

90

(∘)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

(a)

12 14 16 18 20 22 24 2610m

c00 = 04

c00 = 10

c00 = 20

01

1

10

c c

0

(b)


0

10

20

30

40

50

60

70

80

90

(∘)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

(a)

80 9030 5040 60 70 (∘)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

01

1

10

c c

0

(b)



0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018









0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 30∘

(a)

c00 = 04 m = 12c00 = 04 m = 20c00 = 20 m = 12c00 = 20 m = 20

= 60∘

0

10

20

30

40

50

60

70

80

90

(∘)

2 3 4 5 6 7 8 9 101BH

(b)



4 Example Problems










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018










12 14 16 18 20 22 25

05

20 1430 1057 831 678 569 487 39830 1752 1315 1050 872 744 649 54340 1896 1463 1186 995 856 750 63250 1978 1560 1282 1085 939 827 70160 2033 1630 1353 1154 1005 889 757

10

20 1541 1207 976 803 675 579 47230 2299 1744 1384 1140 966 837 69540 2873 2155 1708 1408 1195 1036 86250 3206 2449 1961 1627 1386 1205 100660 3401 2663 2161 1807 1548 1352 1134

20

20 1545 1235 1013 847 719 619 50830 2354 1877 1528 1274 1086 943 78540 3206 2532 2036 1685 1429 1237 102650 4063 3164 2526 2081 1760 1521 125960 4752 3719 2979 2456 2078 1796 1487

30

20 1546 1238 1020 855 728 629 51730 2355 1891 1552 1303 1117 973 81240 3217 2588 2111 1761 1500 1302 108250 4145 3336 2700 2236 1895 1638 135760 5113 4109 3312 2730 2305 1987 1640

50

20 1546 1240 1023 859 733 635 52330 2355 1894 1562 1318 1134 991 83040 3218 2605 2145 1802 1543 1344 112150 4154 3397 2790 2330 1985 1722 143060 5186 4303 3525 2925 2477 2138 1766


5 Conclusions





Data Availability






12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018










12 14 16 18 20 22 25

05

20 246 182 162 152 145 141 13530 130 129 127 124 122 120 11840 112 115 115 114 113 112 11150 107 109 109 109 109 108 10860 104 106 106 106 106 106 105

10

20 4232 605 339 261 223 201 18030 530 254 202 178 164 154 14540 190 165 153 145 139 135 13050 132 134 131 129 126 124 12160 115 119 120 119 118 117 115

20

20 132929 3119 968 555 403 326 26430 13666 1052 487 337 270 233 20140 2255 476 302 241 209 189 17150 504 265 214 189 173 163 15260 193 177 166 158 151 145 138

30

20 100617 8518 1872 903 591 447 33930 103392 2809 913 532 387 313 25340 16556 1187 535 364 289 247 21150 3059 568 346 269 230 206 18360 595 300 240 210 190 177 163

50

20 129396 30420 4349 1695 974 676 47130 132962 9987 2100 983 627 466 34740 212308 4133 1197 656 458 361 28450 38373 1862 734 465 353 293 24260 6277 837 461 340 280 244 211

Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018








































Journal of












Journal of







Volume 2018






Volume 2018








determination of mohr–coulomb parameters from nonlinear

Documents