researcharticle a new calculation method for the soil...

Research ArticleA New Calculation Method for the Soil Slope Safety Factor

Fu Zhu12 Wanxi Zhang1 andMingzhi Sun1

1College of Material Science and Engineering Jilin University Changchun 130022 China2School of Transportation Science and Engineering Jilin Jianzhu University Changchun 130118 China

Correspondence should be addressed to Mingzhi Sun mzsun14mailsjlueducn

Received 27 February 2017 Revised 25 July 2017 Accepted 6 September 2017 Published 9 October 2017

Academic Editor Dane Quinn

Copyright copy 2017 Fu Zhu 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

Based on the unified strength theory a new method to calculate the plane soil slope safety factor was derived that considers theeffect of intermediate principal stress 1205902 and at-rest lateral pressure coefficient 1198700 Calculation examples from the literature wereused to compare the new calculationmethod and the current slice method the results showed that both provided good consistencyThe new method can provide a reference for slope stability evaluation The new method was used to calculate the soil slope safetyfactors for different values of intermediate principal stress parameter 119887 double shear stress parameters 1199061015840120591 and static lateral pressurecoefficient 1198700 The results showed that the safety factor 119865119904 increased when 119887 was increased 119865119904 first increased and then decreasedwhen 1199061015840120591 was increased and 119865119904 increased when 1198700 was increased These results show that the intermediate principal stress as wellas the stress state and its changes cannot be ignored during soil slope stability analysisThe slope soil characteristics and stress stateshould be considered to determine the unified strength theoretical parameters and static lateral pressure coefficient maximize thepotential of slope soil strength and effectively reduce the costs of soil slope engineering

1 Introduction

In roads bridges and construction projects slope stabilityproblems are often encountered during cutting or foundationpit excavation Slope instability is due to the destruction ofthe original soil stress state of equilibrium caused by externalforcesmdashsuch as cutting or foundation pit excavationmdashandthe reduction of soil antishear strength by the influence ofvarious external factorsmdashsuch as rainwater intrusion andsoil freeze-thaw In practical engineering slope stability isanalyzed to test the reasonableness of the soil slope sectiondesign If the slope is too steep it will slump easily if the slopeis too gentle it will increase the amount of earthwork needed

The characteristics of the Ordinary Method of Slices [1]Bishoprsquos Modified Method [2] Force Equilibrium Methods(eg Lowe and Karafiath [3]) Janbursquos Genralized Procedureof Slices [4] Morgenstern and Pricersquos Method [5] andSpencerrsquos Method [6] were summarized in most textbooksFall et al [7] have conducted study on the stability analysisof landslides by the finite element method Cheng and Yip[8] have shown that rigorous method is necessary to give a

reliable estimation of stability of landslides in 3D analysesZhu and Lee [9] conducted study on the factor of safety basedon Bellrsquos assumption Bellrsquos method was improved by Zhengand Tham [10] Zheng and Thamrsquo method can be regardedsubsequently as the enhancement of Felleniusrsquo method

The slope stability safety factor 119865119904 refers to the ratio ofthe soil shear strength to the shear stress of a possible slidingsurface in the slope The soil stress state and its changesare prerequisites of slope stability the existing slope circularslipping method (Petterson 1916) and slice method (Fellenius1927) ignore the impact of the stress state In reality the slopestability changes with changes in the stress state Researchers[11ndash14] are currently looking for the sliding center and slipsurface supplementing andmodifying the basic assumptionsof the slice method and providing a fundamental basisin engineering applications for the slice method Howeverdefects of slice method and statically indeterminate problemof this method [15] have produced challenges in practicalengineering applications

Based on multislip mechanism and the model of mul-tishear element Yu established the unified strength theory

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 3569826 6 pageshttpsdoiorg10115520173569826

2 Mathematical Problems in Engineering

which takes into consideration the different contribution ofall stress components on the yield of failure of materials[16 17] The unified strength theory encompasses the twinshear strength theory [18ndash20] and single strength theoryThe excellent agreement between the predicted results by theunified strength theory and the experiment results indicatesthat the unified strength theory is applicable for a wide rangeof stress states in many materials (Ma et al 1985 [21])

The coefficient of the earth pressure at rest (1198700) is definedas the ratio of the in situ horizontal effective stress to the insitu vertical effective stress The parameter 1198700 is needed inthe interpretation of laboratory and in situ tests and design ofretaining structures and excavation support systems Schnaidand Yu [22] believe that 1198700 is an important input parameterfor the numerical analyses of geotechnical boundary valueproblems

In this study the perspective of a total stress state wasconsidered to derive a new calculation method for the soilslope safety factor based on the unified strength theory Theslope safety factor was defined by considering the effects ofthe intermediate principal stress and at-rest lateral pressurecoefficient The method was compared and verified with thecurrent slice method and can provide a reference for stabilityevaluation in soil slope engineering

2 Basic Theory and Formula Derivation

21 Unified Strength Theory The MohrndashCoulomb strengththeory is simple and practical It is conducive to engineeringapplications but does not reflect the effect of intermediateprincipal stress and the calculated results are relatively con-servative In 1991 Yu suggested the unified strength theoryto compensate for the shortcomings of the MohrndashCoulombstrength theoryThe unified strength theory can consider theintermediate principal stress effect of the material and cansimulate nearly all materials on the partial plane to developthe material strength potentials There are two equationswith a conditional formula for both the mathematical modeland the theoretical expression of the unified strength theorywhich considers the different contributions of various stresscomponents to the material yield and destruction reducesthe number of material parameters and makes the limitingsurface to reach the outer boundary of the convex criteriathese cannot be achieved by other criteria The mathematicalmodeling methods from the two equations can also be usedto solve issues with determining the intermediate principalshear stress Yu derived a mathematical expression of theunified strength theory using the unified twin shear modeland a new mathematical model [17]

119865 = 1205901 minus 1205721 + 119887 (1198871205902 + 1205903) = 1205901199051205902 le 1205901 + 12057212059031 + 120572

(1a)

1198651015840 = 11 + 119887 (1198871205901 + 1205902) 1205901 minus 1205721205903 = 1205901199051205902 ge 1205901 + 12057212059031 + 120572

(1b)

where 119865 and 1198651015840 are yield functions 120572 = 120590119905120590119888 = (1 minussin1205930)(1 + sin1205930) is the ratio of the tensile to compressionstrengths of the material 120590119905 = 21198880 cos1205930(1 + sin1205930)is the tensile strength 1198880 and 1205930 are the cohesion andinternal friction angle respectively of rock and soil and119887 is the selected failure criterion introduced in the unifiedstrength theory that also reflects the destructive effects of theintermediate principal shear stress and normal stress of thecorresponding surface on the material

The unified strength theory was converted into a formulasimilar to the MohrndashCoulomb strength theory in order toobtain the friction angle 120593uni and unified cohesive force 119888unithese are expressed by the internal friction angle 1205930 andcohesion 1198880 as follows [23]

When 1199061015840120591 le (1 minus sin1205930)2120593uni = arc sin( (1 + 119887) sin12059301 + 119887 (1 minus 1199061015840120591) minus 1198871199061015840120591 sin1205930)

119888uni = (1 + 119887) 1198880 cos1205930 cot (45∘ + 120593uni2)1 + 119887 (1 minus 1199061015840120591) + (1 + 119887 + 1198871199061015840119905) sin1205930

(2a)

When 1199061015840120591 ge (1 minus sin1205930)2120593uni = arc sin( (1 + 119887) sin12059301 + 1198871199061015840120591 + 119887 (1 minus 1199061015840120591) sin1205930)

119888uni = (1 + 119887) 1198880 cos1205930 cot (45∘ + 120593uni2)

(1 + 1198871199061015840120591) (1 minus sin1205930) (2b)

where 1199061015840120591 = 1205912312059113 = (1205902 minus 1205903)(1205901 minus 1205903) is the twin shearstress parameter

The unified internal friction angle 120593uni and unified cohe-sion 119888uni can be used to express the MohrndashCoulomb strengththeory

120591119891 = 119888uni + 120590 tan120593uni (3)

22 Basic Assumptions

(1) The excavated soil is simplified as plane soil slopes

(2) Soil is homogeneous

(3) The stress state can be represented by formula (4)

(4) The soil antishear strength satisfies formula (3)

(5) The horizontal stress of inner points along the depthcauses soil slope instability

(6) During excavation of the soil the static lateral pres-sure coefficient remains unchanged

(7) The impact of pore water and groundwater is notconsidered

23 Formula Derivation The extending direction of the soilslope is taken as the plane stress state and elastic half-spaceplane stress analysis is performed for soil in a steady state

Mathematical Problems in Engineering 3

z

Slope bottom

Slope surfaceOriginal ground

x

Slope height ℎ

Slope angle

1 = z

3 = x

Figure 1 Simplified soil slope excavation

under gravity stress the main stress expression of any pointis as follows

1205901 = 120590119911 = 1199031199111205903 = 120590119909 = 1198700119903119911 (4)

where 1205901 is themaximum principal stress 1205903 is theminimumprincipal stress 119903 is the soil gravity 1198700 is the static lateralpressure coefficient of the soil 119911 is the distance from theground surface to any point 120590119909 is the horizontal stress at anypoint and 120590119911 is the vertical stress at any point

As shown in Figure 1 assuming that the cutting orfoundation soil is not excavated when slope angle 120573 = 0∘then 120590119909 = 1198700119903119911 In vertical excavation of the soil when slopeangle 120573 = 90∘ then 120590119909 = 0 The horizontal stress of the soilexcavation slope angle satisfies the following formula

120590119909 = 1198700119903119911 (1 minus sin120573) (5)

According to the conventional definition of the factor ofsafety for some point within a soil mass the safety factor isthe ratio between the shear strength and shear stress at thatpoint [24 25]

119865119904 = 120591119891120591 =(119888 + 120590 tan120593)120591 (6)

In line with the MohrndashCoulomb criterion for the shearstrength of soils for stresses at some point within a soil massdifferences in the magnitude of shear stress in an arbitrarydirection will result in shear strength differences In otherwords the factor of safety at a point in a soil mass definedin (6) will vary with the direction This leads to complexitiesand difficulties in the methods of slope stability analysisand to a variety of assumptions in calculational theories Toensure the uniqueness of safety factors computed at eachpoint within a soil mass the factor of safety was defined asdescribed below

119865119904 = 120591119891120591max= (119888 + ((1205901 + 1205903) 2) tan120593)120591max

(7)

Given a point with a determined stress state within somesoil mass its margin of safety is the ratio between the shearstrength corresponding to the maximum shear strength at

0

c

max

3 1

f

3 = K0rz 1 = rz

c + ＮＨ

Figure 2 Schematic diagram of accumulated safety in soil

that point and the total maximum shear strength as describedin Figure 2

Then the safety margin of a slope is the ratio betweenthe cumulative shear strength and the cumulative maximumshear stress within the slopersquos height thus

119865119904 = int119911

0120591119891119889119911int1199110120591max119889119911 (8)

since

120591max = 12 (1205901 minus 1205903) =12119903119911 (1 minus 1198700 (1 minus sin120573))

120591119891 = 119888uni + 12 (1205901 + 1205903) tan120593uni= 119888uni + 12119903119911 (1 + 1198700 (1 minus sin120573)) tan120593uni

(9)

Therefore (9) are substituted into (8) to obtain

119865119904 = 4119888uni (1 + 1198700 minus 1198700 sin120573)119903119911 (1 minus 11987020 (1 minus sin120573)2)+ 119903119911 (1 + 1198700 minus 1198700 sin120573)

2 tan120593uni119903119911 (1 minus 11987020 (1 minus sin120573)2)

(10)

3 Safety Factor Calculation and Analysis

The new approach was derived from cutting or foundationpit soil excavation Slope engineering calculation examples intextbooks [26 27] and literature [11] were used to verify thegeneral application of the calculation method with formula(10)

Example 1 Slope height ℎ = 6m slope angle 120573 = 55∘ soilgravity 119903 = 186 kNm3 angle of soil internal friction 1205930 =12∘ and cohesion 1198880 = 167 kPa were known The Felleniusrsquoslice method and Bishop formula were used to calculate theslope safety factor the results were 118 and 119 respectively[9]

Using 119887 = 0 and 1198700 = 1 minus sin1205930 = 079 which werecalculated using Jakyrsquos formula formula (10) was used tocalculate the slope safety factor which was 098


Table 1 Material properties

Soil number Gravity (kNm3) Cohesion (kNm2) Internal friction angle (∘)1 soil 195 00 3802 soil 195 53 2303 soil 195 72 200

m

3 soil

2 soil

1 soil(54 31) (70 31)

(70 35)(50 35)

(70 24)

(50 29)

(52 24)(40 27)

(30 25)(20 25)

40

30

2030 40 50 60 7020

m

Figure 3 Contour map of ACADS assessment question EX1 (c)

Example 2 Question EX1 (c) from Australian Associationof Computer Applications (ACADS) assessment in 1987 aheterogeneous soil slope of material properties was shown inTable 1 and slope shape was shown in Figure 3 The problemwas simplified into one with a homogeneous soil slope with aheight of ℎ = 10m slope gradient tan120573 = 1 2 soil gravity119903 = 195 kNm3 soil internal friction angle 1205930 = 38∘and cohesion 1198880 = 53 kPa The Bishop formula and geneticalgorithm were used in the calculation and the ten safetyfactors of the slide facing were in the range of 1398sim140 [11]Using 119887 = 0 and1198700 = 1minussin1205930 = 038 which were calculatedusing Jakyrsquos formula the slope safety factor was calculatedusing formula (10) to be 134 The reference value was 139

The examples above show that the newmethod is simplerthan the current slice method has a clearer theoretical basisand concept does not require programming and effectivelyreduces the computational workload It can be used as ageneral basis for safety evaluation of slope stability

In Example 1 the data substituted into formulas (2a)and (2b) were used to analyze the relationship among theunified internal friction angle 120593uni cohesion 119888uni 119887 and 1199061015840120591The calculation results are shown in Figures 4 and 5

Figures 4 and 5 show that the unified internal frictionangle 120593uni and cohesion 119888uni increased when 119887 was increasedThey firstly increased and then decreased when 1199061015840120591 wasincreased and reached their maximum when 1199061015840120591 = (1 minussin1205930)2 = 04120593uni and 119888uni were obtained with different 119887 and 1199061015840120591 valuesand substituted into formula (10) to calculate various soilslope safety factors 119865119904 as shown in Figure 6 Figure 6 showsthat when 1199061015840120591 was a fixed value the safety factor 119865119904 increasedwith increasing 119887 When 119887 was a fixed value 1199061015840120591 changedfrom 0 to 1 and the safety factor 119865119904 first increased and thendecreased When 1199061015840120591 = (1 minus sin1205930)2 = 04 the safety factor119865119904 was at its maximum when 1199061015840120591 = 0 or 1199061015840120591 = 1 the safetyfactor 119865119904 was at its minimum valueWhen 119887 = 0 formula (10)

b = 0

b = 025

b = 05

b = 075

b = 10

01 04 07 1000u

110

120

130

140

150

160

ＯＨＣ

(∘)

Figure 4 Relationship among 120593uni 1199061015840120591 and 119887

01 04 07 1000u

b = 0

b = 025

b = 075

b = 05

b = 10

160

170

180

190

200

210

220

230

c ＯＨＣ

(kPa

)


01 04 07 1000u

09

10

11

12

13

14

FS

b = 0

b = 025

b = 05

b = 075

b = 10

Figure 6 Relationship among 119865119904 1199061015840120591 and 119887


b = 0

b = 025

b = 05

b = 075

b = 10

059 079039K0

08

09

10

11

12

13

14

FS


degrades to a formula under the MohrndashCoulomb strengththeory and the obtained safety factor 119865119904 will be the smallestfixed value This indicates that the potential of the soil is farfrom being developed and may result in significant waste

Using 1199061015840120591 = 04 120593uni and 119888uni obtained with different 119887and 1198700 were substituted into formula (10) and various soilslope safety factors 119865119904 were obtained as shown in Figure 7Figure 7 shows that when 1198700 was a fixed value 119865119904 increasedwith 119887 which is consistent with the above conclusions When119887was a fixed value1198700 varied from 039 to 079 and the safetyfactor 119865119904 increased gradually This indicates that 119887 and1198700 arecombined factors that determine slope stability Formula (10)considers the effects of both and can provide a reference forsafety evaluation of a soil slope

4 Conclusions

The new method was verified by comparing results with theexisting slice method using examples in the literature Thenew method can be combined with slope engineering toobtain parameters of the unified strength theory and the staticlateral pressure coefficientThe slope stability and safety wereevaluated with the new method

The effects of different factors on soil slope safety andstability were analyzed including the intermediate principalstress parameter twin shear stress state parameters andstatic lateral pressure coefficient These results indicate thatthe intermediate principal stress and static lateral pressurecoefficient cannot be ignored in slope stability analysis

This study only examined the effects of unified strengththeory parameters and the static lateral pressure coefficienton the slope safety factor In order to determine the param-eters and practical applications of the new method furtherresearch and verification are needed

The theoretical formula was derived calculated andanalyzed from the perspective of the total stress state Theeffects on pore water pressure and groundwater should befurther examined from the perspective of effective stress

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is financially supported by the Project ofEducation Department of Jilin Province (JJKH20170260KJ)the Project ofMinistry ofHousing andUrban-RuralDevelop-ment (2017-K4-004) and the Plan Projects of TransportationScience and Technology in Jilin Province of China (2011103)

References

[1] W Fellenius ldquoCalculation of the stability of earth damsrdquo in InProceedings of the 2nd Congress on Large Dams vol 4 pp 445ndash463 Wash USA 1936

[2] A W Bishop ldquoThe use of the slip circle in the stability analysisof slopesrdquo Geotechnique vol 5 pp 7ndash17 1955

[3] J Lowe and L Karafiath ldquoStability of earth dams upon draw-downrdquo in Proceedings of the 1st Pan-AmConf On SoilMech AndFounf Engrg pp 537ndash552 Mexico City Mexico 1960

[4] N Janbu ldquoSlope stability computationsrdquo in Soil Mechanicsand Foundation Engineering Journal Technical University ofNorway Trondheim Norway 1968

[5] N R Morgenstern and V E Price ldquoThe analysis of the stabilityof general slip surfacesrdquo Geotechnique vol 15 no 1 pp 79ndash931965

[6] E Spencer ldquoA method of analysis of the stability of embank-ments assuming parallel inter-slice forcesrdquoGeotechnique vol 17no 1 pp 11ndash26 1967

[7] M Fall R Azzam and C Noubactep ldquoA multi-methodapproach to study the stability of natural slopes and landslidesusceptibility mappingrdquo Engineering Geology vol 82 no 4 pp241ndash263 2006

[8] Y M Cheng and C J Yip ldquoThree dimensional asymmetri-cal slope stability analysis extension of bishoprsquos Janbursquos andMorgenstern-Pricersquos Techniquesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 133 no 12 pp 1544ndash15552007

[9] D Y Zhu and C F Lee ldquoExplicit limit equilibrium solutionfor slope stabilityrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 26 no 15 pp 1573ndash1590 2002

[10] H Zheng and L G Tham ldquoImproved Bellrsquos method for thestability analysis of slopesrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 33 no 14 pp1673ndash1689 2009

[11] W J Lv X J Li and H H Zhu ldquoGA-based generalizedslope stability analysis methodrdquo Chinese Journal of GeotechnicalEngineering vol 27 no 5 pp 595ndash599 2005

[12] A I H Malkawi W F Hassan and S K Sarma ldquoGlobal searchmethod for locating general slip surface using Monte Carlotechniquesrdquo Journal of Geotechnical and GeoenvironmentalEngineering vol 127 no 8 pp 688ndash698 2001

[13] G-D Zou ldquoA global optimization method of the slice methodfor slope stability analysisrdquo Chinese Journal of GeotechnicalEngineering vol 24 no 3 pp 309ndash312 2002

[14] G Zou and Y Chen ldquoCoupling algorithm of simulated anneal-ing algorithm and random search method for slope stabilityanalysisrdquo Chinese Journal of Rock Mechanics and Engineeringvol 23 no 12 pp 2032ndash2037 2004

[15] Z Y Chen X GWang Y C Xing et al ldquoTheoretical study andphysical modeling on ldquoPrinciple ofMaximumrdquo in slope stability


analysisrdquo Chinese Journal of Geotechnical Engineering vol 27no 5 pp 495ndash499 2005

[16] M H Yu New System of Strength Theory Xian JiaotongUniversity Press 1992

[17] M-H Yu ldquoUnified strength theory for geomaterials and itsapplicationrdquo Chinese Journal of Geotechnical Engineering vol16 no 2 pp 1ndash9 1994

[18] M-H Yu ldquoTwin shear stress yield criterionrdquo InternationalJournal of Mechanical Sciences vol 25 no 1 pp 71ndash74 1983

[19] M H Yu L He and L Song ldquoTwin shear stress theory and itsgeneralizationrdquo Scientia Sinica Series A-Mathematical PhysicalAstronomical and Technical Sciences vol 28 pp 1174ndash1183 1985

[20] M H Yu and L N He ldquoA new model and theory on yieldand failure of materials under complex stress staterdquoMechanicalBehaviour of Materials vol 6 no 3 pp 841ndash846 1991

[21] M Yu J Li and Y Zhang ldquoUnified characteristics line theory ofspacial axisymmetric plastic problemrdquo Science in China SeriesE Technological Sciences vol 44 no 2 pp 207ndash215 2001

[22] F Schnaid and H S Yu ldquoInterpretation of the seismic cone testin granular soilsrdquoGeotechnique vol 57 no 3 pp 265ndash272 2007

[23] W Fan L S Deng X Y Bai et al ldquoApplication of unifiedstrength theories to the stability of slope analysisrdquo Coal Geologyamp Exploration vol 35 no 1 pp 63ndash66 2007

[24] G T Wang ldquoThe method and formula for stability factorof slope with state of soil original stressrdquo Chinese Journal ofEngineering Science vol 8 no 12 pp 80ndash84 2006

[25] G TWang ldquoIlluminating the newmethod process of slope sta-bility analysis by soil stress staterdquoChinese Journal of EngineeringScience vol 12 no 1 pp 52ndash55 2010

[26] Z Y Chen Soil Slope Stability Analysis-Theory Methods andPrograms China Water Power Press Beijing China 2003

[27] J Y Yuan J G Qian H M Zhang et al Soil Properties and SoilMechanics China Communication Press Beijing china 2009

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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which takes into consideration the different contribution ofall stress components on the yield of failure of materials[16 17] The unified strength theory encompasses the twinshear strength theory [18ndash20] and single strength theoryThe excellent agreement between the predicted results by theunified strength theory and the experiment results indicatesthat the unified strength theory is applicable for a wide rangeof stress states in many materials (Ma et al 1985 [21])

The coefficient of the earth pressure at rest (1198700) is definedas the ratio of the in situ horizontal effective stress to the insitu vertical effective stress The parameter 1198700 is needed inthe interpretation of laboratory and in situ tests and design ofretaining structures and excavation support systems Schnaidand Yu [22] believe that 1198700 is an important input parameterfor the numerical analyses of geotechnical boundary valueproblems

In this study the perspective of a total stress state wasconsidered to derive a new calculation method for the soilslope safety factor based on the unified strength theory Theslope safety factor was defined by considering the effects ofthe intermediate principal stress and at-rest lateral pressurecoefficient The method was compared and verified with thecurrent slice method and can provide a reference for stabilityevaluation in soil slope engineering

2 Basic Theory and Formula Derivation

21 Unified Strength Theory The MohrndashCoulomb strengththeory is simple and practical It is conducive to engineeringapplications but does not reflect the effect of intermediateprincipal stress and the calculated results are relatively con-servative In 1991 Yu suggested the unified strength theoryto compensate for the shortcomings of the MohrndashCoulombstrength theoryThe unified strength theory can consider theintermediate principal stress effect of the material and cansimulate nearly all materials on the partial plane to developthe material strength potentials There are two equationswith a conditional formula for both the mathematical modeland the theoretical expression of the unified strength theorywhich considers the different contributions of various stresscomponents to the material yield and destruction reducesthe number of material parameters and makes the limitingsurface to reach the outer boundary of the convex criteriathese cannot be achieved by other criteria The mathematicalmodeling methods from the two equations can also be usedto solve issues with determining the intermediate principalshear stress Yu derived a mathematical expression of theunified strength theory using the unified twin shear modeland a new mathematical model [17]

119865 = 1205901 minus 1205721 + 119887 (1198871205902 + 1205903) = 1205901199051205902 le 1205901 + 12057212059031 + 120572

(1a)

1198651015840 = 11 + 119887 (1198871205901 + 1205902) 1205901 minus 1205721205903 = 1205901199051205902 ge 1205901 + 12057212059031 + 120572

(1b)

where 119865 and 1198651015840 are yield functions 120572 = 120590119905120590119888 = (1 minussin1205930)(1 + sin1205930) is the ratio of the tensile to compressionstrengths of the material 120590119905 = 21198880 cos1205930(1 + sin1205930)is the tensile strength 1198880 and 1205930 are the cohesion andinternal friction angle respectively of rock and soil and119887 is the selected failure criterion introduced in the unifiedstrength theory that also reflects the destructive effects of theintermediate principal shear stress and normal stress of thecorresponding surface on the material

The unified strength theory was converted into a formulasimilar to the MohrndashCoulomb strength theory in order toobtain the friction angle 120593uni and unified cohesive force 119888unithese are expressed by the internal friction angle 1205930 andcohesion 1198880 as follows [23]

When 1199061015840120591 le (1 minus sin1205930)2120593uni = arc sin( (1 + 119887) sin12059301 + 119887 (1 minus 1199061015840120591) minus 1198871199061015840120591 sin1205930)

119888uni = (1 + 119887) 1198880 cos1205930 cot (45∘ + 120593uni2)1 + 119887 (1 minus 1199061015840120591) + (1 + 119887 + 1198871199061015840119905) sin1205930

(2a)

When 1199061015840120591 ge (1 minus sin1205930)2120593uni = arc sin( (1 + 119887) sin12059301 + 1198871199061015840120591 + 119887 (1 minus 1199061015840120591) sin1205930)

119888uni = (1 + 119887) 1198880 cos1205930 cot (45∘ + 120593uni2)

(1 + 1198871199061015840120591) (1 minus sin1205930) (2b)

where 1199061015840120591 = 1205912312059113 = (1205902 minus 1205903)(1205901 minus 1205903) is the twin shearstress parameter

The unified internal friction angle 120593uni and unified cohe-sion 119888uni can be used to express the MohrndashCoulomb strengththeory

120591119891 = 119888uni + 120590 tan120593uni (3)

22 Basic Assumptions

(1) The excavated soil is simplified as plane soil slopes

(2) Soil is homogeneous

(3) The stress state can be represented by formula (4)

(4) The soil antishear strength satisfies formula (3)

(5) The horizontal stress of inner points along the depthcauses soil slope instability

(6) During excavation of the soil the static lateral pres-sure coefficient remains unchanged

(7) The impact of pore water and groundwater is notconsidered

23 Formula Derivation The extending direction of the soilslope is taken as the plane stress state and elastic half-spaceplane stress analysis is performed for soil in a steady state


z

Slope bottom


x

Slope height ℎ

Slope angle

1 = z

3 = x



1205901 = 120590119911 = 1199031199111205903 = 120590119909 = 1198700119903119911 (4)



120590119909 = 1198700119903119911 (1 minus sin120573) (5)


119865119904 = 120591119891120591 =(119888 + 120590 tan120593)120591 (6)


119865119904 = 120591119891120591max= (119888 + ((1205901 + 1205903) 2) tan120593)120591max

(7)


0

c

max

3 1

f

3 = K0rz 1 = rz

c + ＮＨ




119865119904 = int119911

0120591119891119889119911int1199110120591max119889119911 (8)

since



(9)




(10)








m

3 soil

2 soil

1 soil(54 31) (70 31)

(70 35)(50 35)

(70 24)

(50 29)

(52 24)(40 27)

(30 25)(20 25)

40

30

2030 40 50 60 7020

m






b = 0

b = 025

b = 05

b = 075

b = 10

01 04 07 1000u

110

120

130

140

150

160

ＯＨＣ

(∘)


01 04 07 1000u

b = 0

b = 025

b = 075

b = 05

b = 10

160

170

180

190

200

210

220

230

c ＯＨＣ

(kPa

)


01 04 07 1000u

09

10

11

12

13

14

FS

b = 0

b = 025

b = 05

b = 075

b = 10



b = 0

b = 025

b = 05

b = 075

b = 10

059 079039K0

08

09

10

11

12

13

14

FS




4 Conclusions







Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





z

Slope bottom


x

Slope height ℎ

Slope angle

1 = z

3 = x



1205901 = 120590119911 = 1199031199111205903 = 120590119909 = 1198700119903119911 (4)



120590119909 = 1198700119903119911 (1 minus sin120573) (5)


119865119904 = 120591119891120591 =(119888 + 120590 tan120593)120591 (6)


119865119904 = 120591119891120591max= (119888 + ((1205901 + 1205903) 2) tan120593)120591max

(7)


0

c

max

3 1

f

3 = K0rz 1 = rz

c + ＮＨ




119865119904 = int119911

0120591119891119889119911int1199110120591max119889119911 (8)

since



(9)




(10)








m

3 soil

2 soil

1 soil(54 31) (70 31)

(70 35)(50 35)

(70 24)

(50 29)

(52 24)(40 27)

(30 25)(20 25)

40

30

2030 40 50 60 7020

m






b = 0

b = 025

b = 05

b = 075

b = 10

01 04 07 1000u

110

120

130

140

150

160

ＯＨＣ

(∘)


01 04 07 1000u

b = 0

b = 025

b = 075

b = 05

b = 10

160

170

180

190

200

210

220

230

c ＯＨＣ

(kPa

)


01 04 07 1000u

09

10

11

12

13

14

FS

b = 0

b = 025

b = 05

b = 075

b = 10



b = 0

b = 025

b = 05

b = 075

b = 10

059 079039K0

08

09

10

11

12

13

14

FS




4 Conclusions







Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







m

3 soil

2 soil

1 soil(54 31) (70 31)

(70 35)(50 35)

(70 24)

(50 29)

(52 24)(40 27)

(30 25)(20 25)

40

30

2030 40 50 60 7020

m






b = 0

b = 025

b = 05

b = 075

b = 10

01 04 07 1000u

110

120

130

140

150

160

ＯＨＣ

(∘)


01 04 07 1000u

b = 0

b = 025

b = 075

b = 05

b = 10

160

170

180

190

200

210

220

230

c ＯＨＣ

(kPa

)


01 04 07 1000u

09

10

11

12

13

14

FS

b = 0

b = 025

b = 05

b = 075

b = 10



b = 0

b = 025

b = 05

b = 075

b = 10

059 079039K0

08

09

10

11

12

13

14

FS




4 Conclusions







Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





b = 0

b = 025

b = 05

b = 075

b = 10

059 079039K0

08

09

10

11

12

13

14

FS




4 Conclusions







Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




researcharticle a new calculation method for the soil...

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