extended ‘‘mononobe-okabe’’ method for seismic design of

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 136132 10 pageshttpdxdoiorg1011552013136132

Research ArticleExtended lsquolsquoMononobe-Okabersquorsquo Method for Seismic Design ofRetaining Walls

Mahmoud Yazdani1 Ali Azad1 Abol hasan Farshi12 and Siamak Talatahari3

1 Department of Civil and Environmental Engineering Tarbiat Modares University Tehran 1411713116 Iran2 Civil Engineering Faculty Islamic Azad University Central Tehran Branch Tehran Iran3Department of Civil Engineering University of Tabriz Tabriz Iran

Correspondence should be addressed to Siamak Talatahari siamaktalatgmailcom

Received 7 May 2013 Revised 11 July 2013 Accepted 21 August 2013

Academic Editor Ga Zhang

Copyright copy 2013 Mahmoud Yazdani et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Mononobe-Okabe (M-O) method is still employed as the first option to estimate lateral earth pressures during earthquakes bygeotechnical engineers Considering some simple assumptions and using a closed form method M-O solves the equations ofequilibrium and suggests seismic active and passive lateral earth pressures Therefore the results are true in its assumption rangeonly and inmany other practical casesM-Omethod is not applicable Noncontinues backfill slopes cohesive soils and rising waterbehind the wall are some well-known examples in which the M-O theory is irrelevant Using the fundamental framework of M-Omethod this study proposes an iterative method to overcome the limits of the M-O method Based on trial and error process theproposed method is able to cover many of the defects which regularly occur in civil engineering when M-O has no direct answer

1 Introduction

Retaining walls are those structures which are usually con-structed to form roads stabilize trenches and soil slopes andsupport unstable structures Figure 1 shows one of the com-mon configurations of retaining structures schematically

Lateral earth pressure model is belonging to the firstgroup of theories in classical soil mechanics Coulomb [1]and Rankine [2] proposed their theories to estimate activeand passive lateral earth pressures These kinds of theoriespropose a coefficient which is a ratio between horizontaland vertical stress behind retaining walls Using the ratiolateral pressure is simply calculated by the horizontal stressintegration

Mononobe-Okabe method (M-O) a seismic versionof coulomb theory was proposed based on pseudostaticearthquake loading for granular soils This method appliesearthquake force components using two coefficients calledseismic horizontal and vertical coefficients Beside othercomplex theoretical models and numerical methods M-Otheory is one of the best initial estimates

Although M-O is the first choice for engineers to designretaining walls some limitations make it incapable to model

most civil engineering projects This problem rises accordingto the simplifier assumptions in M-O method to solve theequations in a closed form fashion The contribution ofthis paper is primarily to remove these limits and to coverother problems that M-O has no answer for them On theother hand reports on retaining wall failures during majoror minor earthquakes confirm the necessity of immunedesign of retaining structures Since the stability of retainingwalls plays an important role during and right after anearthquake this study strives to provide a reliable tool forquick engineering designs The methodology given in thispaper can also be used as a model to study the effect ofearthquake parameters on retaining structures with a specificgeometry or can be reshaped for any other unusual retainingstructures

2 Mononobe-Okabe Method

Mononobe andMatsuo [3] andOkabe [4] proposed amethodto determine lateral earth pressure of granular cohesionlesssoils during earthquake [5] The method was a modifiedversion of Coulomb theory [1] in which earthquake forces are

2 Journal of Applied Mathematics

Retainingstructure

Figure 1 Application of retaining walls in civil engineering

KW

KW

K

KWh

hW

120573p

120573a

120575p

120575a

Pp

Pa

120588p 120588a

120572p 120572a

Active zonePassive zone

Lateral displacement

Figure 2 Geometry and parameters of M-O method

applied to the failure mass by pseudostatic method To get afinal simple formulation like other closed form solutions ingeotechnical engineeringM-O uses exact form solution withsimple assumption such as simplicity in geometry materialbehavior or dynamic loading to make the equations solvable

Because of the old age of M-O method tens of studieshave been focused on this area (eg [6ndash8]) An importantstudy onM-Owas carried out by Seed andWhitman [9]Theyconfirmed M-O active pressure after long laboratory runsHowever they recommended more studies on passive theoryof M-O They also proposed a method to find the location ofresultant force which acts on 13 of height in M-O methodM-O had been studied by others such as Fang and Chen [10]on the direction of seismic force components on the failuremass

Figure 2 shows the parameters and characteristics of M-O method In M-O static force equilibrium is satisfied for arigid wedge placed on a failure plane with elastic-perfectlyplastic behavior based on Mohr-Coulomb failure criteria

Active and passive forces119875119886and119875119901 are then calculated using

the following equations

119875119886

119875119901

=1

21205741198672

(1 minus 119870V) 119870119886

119870119901

119870119886

119870119901

= (cos2 (120593 ∓ 120572 minus 120579))

times ( cos 120579 times cos2120572 times cos (120575 plusmn 120572 + 120579)

times[1 plusmn (sin (120593 + 120575) times sin (120593 ∓ 120573 minus 120579)cos (120575 plusmn 120572 + 120579) times cos (120573 minus 120572)

)

12

]

2

)

minus1

120579 = tanminus1 (119870ℎ

1 minus 119870V

)

(1)

Journal of Applied Mathematics 3

KW998400

W

h

H

F

B A

cL

998400

KhW998400

120593 120575

120572

120573

120588

Pp

c998400L998400

Z0

(a)

h

H

F

BA

cL

KW998400

W998400

KhW998400

120593120575

120572

120573

120588Pa

c998400L998400

Z0

(b)

Figure 3 Proposed model for passive (a) and active (b) pressure

It should be noticed that the above formulations werederived assuming planar failure surfaces This assumptionhas significantly simplified the resultant equations that canpractically capture the contrast between different designscenarios and parameters However in general planar failuresurfaces usually overestimate passive pressures and mightunderestimate active pressure Moreover in reality curvedsurfaces are more often useable Therefore the results of thecurrent study can be used as upper limits while it shouldbe considered with caution for active pressure The worksby Morrison and Ebeling [11] Kumar [12] Subba Rao andChoudhury [13] and Yazdani and Azad [14] explain how theplanar models can be extended

3 Mononobe-Okabe MethodDefects and Limitations

Some of the limits of M-O method that cause the methodnot to cover many of the usual engineering problems are asfollows

(a) M-O method is applicable for cohesionless soils only

(b) Effect of water table behind the wall has not beenconsidered directly in the formula

(c) M-O method has no answer when 120593 minus 120573 minus 120579 le 0

(d) The conventional problems in civil engineering arenot always wall with continues backfill Sometimesone has to use equivalent forms of M-O method tomodel a real problem

4 Overcome Mononobe-Okabe Method Limits

To overcome M-O limits the fundamental basis of limitequilibrium analysis can be used The difference is thesolution type which is carried out using an iterative processfor various values of 120588

119886and 120588

119887to find the minimum and

maximum values of the function instead of using closed formsolutions of differential equations

41 Problem Framework According to Figure 3 wall dis-placement and also its direction produce effective forceson failure mass in both sides of a retaining wall Variableparameters have been listed in Table 1 It can be seen that themain differences between current method and M-O methodare as follows

(a) Geometry of backfill soil has been modeled in anengineering configuration Two real cases with thesame geometry have been illustrated in Figures 4 and5 Both examples are in north side of Tehran Iranwhich are located in a high seismic risk zone Anothersimilar geometry has been plotted in Figure 6 Thefigure also shows the failure mechanism of the wallin Chi-Chi Earthquake Taiwan [15]

(b) In addition to soil cohesion virtual cohesion betweensoil and wall material (adhesion) is included in themodelSeismic active earth pressure considering 119888minus120593 backfillhas been already evaluated by Prakash and Saran [16]as well as Saran and Prakash [17] In their methodsadhesion was considered identical to cohesion Das


Table 1 Parameters of the proposed method

Parameter SymbolWall height HWater depth hWall angle 120572

Soil-wall friction angle 120575

Backfill angle 120573

Soil special weight 120574

Saturated soil special weight 120574sat

Ref to Figure 3 ARef to Figure 3 BSoil internal friction angle 120593

Soil cohesion cSoil-wall cohesion c1015840

Horizontal earthquake coefficient 119870ℎ

Vertical earthquake coefficient 119870V

and Puri [18] improved the analysis by consideringdifferent value of cohesion and adhesion Shukla etal [19] presented an idea to extendMononobe-Okabeconcept for 119888 minus 120593 backfill in such a way to get singlecritical wedge surface Ghosh and Sengupta [20] pre-sented a formulation to evaluate seismic active earthpressure including the influence of both adhesion andcohesion for a nonvertical retaining wall

(c) Static water table is included in themodel to affect theearth pressure directly

(d) The effect of tension crack has been considered Thiseffect is quite important in active earth pressure onretaining wall for cohesive soil backfill [21] Shuklaet al [19] showed that for soil backfill with tensioncracks the total active earth pressure in static con-dition will increase up to 20ndash40 over the casewithout tension cracksTherefore the effect of tensioncracks in cohesive soil backfill should not be neglectedin the calculation of active earth pressure Ghoshand Sharma [22] used the following equation in theiranalysis to compute the depth of tension cracked zonein seismic condition

1198850=

2119888

120574radic119870119886

119870119886(Rankine) =

1 minus sin1205931 + sin120593

(2)

This equation is based on the Rankine theory of activeearth pressure for cohesive backfill under static conditionThe effect of seismic acceleration on the depth of tensioncrack is neglected in that analysis Given that the inclinationof the stress characteristics depends on acceleration level aRankine condition is valid for the vast majority of cantileverwall configurations under strong seismic action [23] This isapplicable even to short heel walls with an error of about 5[24 25]

Shao-jun et al [21] made an effort to determine the depthof tension cracked zone under seismic loading and used thepseudodynamic approach to compute seismic active force onretaining wall with cohesive backfills

Figure 4Geometry of natural slope behind a retainingwall TehranIran

Figure 5 Geometry of backfill behind a retainingwall Tehran Iran

Overturning

Backfill

Heel

Foot wall

Toe

Parking lotHeaving

Sliding

Taiwan cinema culture town

Hanging wallAssumed

rupture zone

05 m

263 m

1 035

Chelungpu Fault

Figure 6 Geometry and failure mechanism of a retaining wall inChi-Chi Earthquake Taiwan (after [15])


120572 minus 120579

90 minus 120588 + 120579

F

cL

c998400L998400

120579 + 120601 minus 120579

120579 + 120575 minus 120572

W998400radic(1 minus K)2 + K2

h

Pp

(a)

120572 + 120579

90 minus 120588 minus 120579

F

cL

c998400L998400


h

120588 minus 120601 + 120579

Pa

120572 + 120575 + 120579

(b)

Figure 7 Effective force diagrams on failure mass (wedge)

In the current research (2) is adopted to take into accountthe effect of tension crack By this way the shape of activewedge in original M-O method is changed from triangularto trapezoidal and the resultant active earth pressure will becomputed more realistically However in passive conditionthe failure wedge was considered triangular similar to theconventional M-O method though this assumption causesthe passive earth pressure to be overestimated

42 Free Body Diagrams According to limit analysis on thefailure surfacewhich is bilinear in this study shear stresses aredeveloped based onMohr-Coulomb theory A 2D static set ofequilibriums can simply connect stresses in a force (or stress)diagramThe diagrams for active and passive conditions havebeen shown in Figure 7

43 Solution Methodology As mentioned using a closedform solution is not applicable herein according to thelarge number of parameters and nonlinearity of equationsTherefore semianalytical iterative calculations for searchingthe favorite conditions (120588 119875

119886 and 119875

119901) have been chosen in

this studyThe continuity of the equations is plotted in Figure 8

where the active pressure is the maximum while the passivepressure is the minimum points in relevant curves Since theequations have one unique global maximum or minimumvalue no sophisticated search algorithm is needed A simplescanning search method was coded to pick the critical(extremum) conditions

5 Parametric Study

Because of the large number of parameters in each analysisderiving complete series of calculation and results seems

0 20 40 60 80 100

Typical passive condition curve

Typical active condition curve

Failure angle (deg) 120588

(Pa)

or(P

p)

Figure 8 Typical active and passive condition curves

useless Therefore this section just reflects results of a para-metric study on a 10-meter high wall Backfill soil has specificweight of 2 gcm3 Other variables have been explained inthe following subsections Other undefined parameters likeparameter ldquo119860rdquo have been assumed to be zero

51 Effects of Backfill Soil Geometry To assess the effects ofbackfill soil geometry a backfill soil with various values of 120573and 119861119867 (the ratio of slope width to wall height as shownin Figure 3) and internal friction of soil equal to 30∘ wasconsidered Horizontal component of earthquake was set to119870ℎ= 02 This problem was solved with the standard M-O

and the proposed method It should be noted that the valuesof slope angle in M-O were chosen equal to 120573

Figure 9 shows the results of this analysis It is clearthat for 120573 ge 20

∘ M-O method has no result Diagramsexplain that when 120573 increases the difference between the


047

048

049

05

051

052

0 04 08 12 16 2BH

Ka

120573 = 5∘

Mononobe-OkabeThis study

(a)

26

28

3

32

0 04 08 12 16 2BH

Kp

120573 = 5∘


(b)

046

048

05

052

054

056

058

0 04 08 12 16 2BH

Ka


120573 = 10∘

(c)

25

3

35

4

0 04 08 12 16 2BH

Kp


120573 = 10∘

(d)

044

048

052

056

06

064

068

0 04 08 12 16 2

BH

Ka


120573 = 15∘

(e)

25

33

41

49

0 04 08 12 16 2

BH

Kp


120573 = 15∘

(f)

Figure 9 Continued


045

053

061

069

077

085

0 04 08 12 16 2BH

This studyNote no results for M-O method are available

Ka

120573 = 20∘

(g)

25

35

45

55

0 04 08 12 16 2BH

Kp


120573 = 20∘

(h)

045

055

065

075

085

095

105

0 04 08 12 16 2BH

Ka


120573 = 25∘

(i)

25

4

55

7

0 04 08 12 16 2BH

Kp


120573 = 25∘

(j)

Figure 9 Lateral pressure coefficient for various backfill soil geometry

proposed method and M-O results is significant Howeverthis difference decreases when 119861119867 decreases It means thatif variation of 120573 (0sim120573) occurs near the wall (approximatelyfor 119861119867 lt 1) between the results of these methods a largedifference appears However the proposed method is moreaccurate As a result when 120573 is near the wall M-Omethod isnot economical for active conditions and is not accurate forpassive cases

52 Effects of Water Table behind Wall In the original M-O method water table is not considered directly in themodel and the earth pressure is given only for the drycondition To overcome this deficiency either a correctnessfactor recommended by some design codes should be utilizedor the following relationship must be applied in which the

lateral earth pressure ratio in each dry or saturated region isimposed on the relevant unit weight

Total earth pressure (M-O)

= 1

2120574(119867 minus ℎ)

2

+ 120574ℎ (119867 minus ℎ) +1

21205741015840

sdot ℎ2

times 119870119886

(3)

However in the proposed method the total earth pres-sure can be simply obtained by the following equation

Total earth pressure (this study) = 1

2120574 sdot 1198672

times 119870119886 (4)

It should be noted that 119870119886in the above equations has

no similar value The reason is hidden in the philosophy


25

3

35

4

45

0 02 04 06 08 1Level of waterheight of wall

Kp

(a)

045

05

055

06

065

07


Ka

(b)

Figure 10 Effects of water table on lateral earth pressure coefficients

200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)

This studyMononobe-Okabe

30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)

Figure 11 Effects of water table on total lateral earth pressure compared to M-O

of calculations In fact in (4) the weight of the submergedpart of the sliding wedge is considered through the effectiveunit weight of soil which is reflected in the force equilibriumdiagram and then119870

119886is calculatedThe two equations (3) and

(4) can be regenerated in the same way for passive conditionas well

To investigate the ability of the proposed method inaccounting for water effect on lateral earth pressures a simplecase has first been solved with the following parameters withno water table 120573 = 15

∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=

02 The results are plotted in Figure 9 that shows perfectagreement between M-O and the proposed model for thegiven geometry In the second step the same model was

used with water table varying and 120574sat = 120574 = 2 gcm3The sensitivity analysis results are illustrated in Figure 10This figure shows that increasing water depth will decreaselateral pressure in both active and passive conditions Thecomparison between M-O based on (3) and this study basedon (4) for passive case is illustrated in Figure 11 Thesediagrams reveal that the M-O method outputs are not in thesafe side of design Since the proposed model honors thephysics of the problemwithmore details it offers a better toolfor design purposes

53 Effects of Cohesion and Surface Tension Crack Almost allsoils have naturally a very small amount of cohesion Also in


1

2

3

4

5

0 1 2 3 4 5Cohesion (tonm2)

Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)

Figure 12 Effects of cohesion on lateral earth pressure

many projects there is no access to a clean granular materialor it is not economical to use such soils Therefore usage ofcohesive soil is inevitable

To better evaluate the effects of cohesion factor on lateralearth pressure a wall with horizontal backfill and 120601 = 30

∘

was analyzed Analyses have been repeated in the cases ofwith and without considering tension cracks for two differentcategories of 119870

ℎ= 0 and 119870

ℎ= 02 Results are reported

in Figure 12 which shows the significant difference in activecoefficient between cohesionless (119862 = 0) and cohesive (119862 gt

0) soilsWhere119870ℎ= 02 forminimum2 tm2 of cohesion the

active pressure changes 50 and the passive pressure changes27 These two values for 119870

ℎ= 0 are 70 for the active

pressure and 23 for the passive pressure Also it seems thattension cracks have small effects on the passive pressure whilethey are more appropriate for the active conditions

6 Conclusion

In this paper the Mononobe-Okabe (M-O) method wasrevised and a new approach with more general picture ofproblems in civil engineering was proposed Based on thelimit equilibrium analysis and a semianalytical procedurethe proposed model can go over the limitations of closedform solutions of M-O method The modified version is alsocapable of considering different backfill geometry cohesionof backfill soil soil-wall interaction and water table behindthe wall Using the method explained in this paper seismicactive and passive earth pressure could be calculated in manyusual engineering problems without any approximation

The parametric study on a 10m wall was also performedin the paper to explain the methodology more clearly Theresults reveal this fact that the standard M-O method in


some cases is incapable of offering an answer Because of itssimple assumptionsM-O sometimes stands in an unsafe sideof design or it overestimates and directs the problem into anuneconomical design However the proposed methodologyrelives engineers from some approximations and equivalentmethods This methodology can be easily rederived to anyother simple or sophisticated problems

References

[1] C A Coulomb Essai sur une application desmaximis et minimisa quelques problems de statique relatifs a lrsquoarchitecture vol7 Memoires de lrsquoAcademie Royal Pres Divers Savants ParisFrance 1776

[2] W Rankine ldquoOn the stability of loose earthrdquo PhilosophicalTransactions of the Royal Society of London vol 147 pp 9ndash271857

[3] N Mononobe and H Matsuo ldquoOn the determination of earthpressures during earthquakesrdquo in Proceedings of the WorldEngineering Congress p 9 Tokyo Japan 1929

[4] S Okabe ldquoGeneral theory of earth pressuresrdquo Journal of theJapan Society of Civil Engineering vol 12 no 1 1926

[5] S L Kramer Geotechnical Earthquake Engineering Prentice-Hall New York NY USA 1996

[6] R Richards Jr and X Shi ldquoSeismic lateral pressures in soils withcohesionrdquo Journal of Geotechnical Engineering vol 120 no 7 pp1230ndash1251 1994

[7] H Matsuzawa I Ishibashi and M Kawamura ldquoDynamic soiland water pressures of submerged soilsrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1161ndash1176 1985

[8] A S Veletsos and A H Younan ldquoDynamic modeling andresponse of soil-wall systemsrdquo Journal of Geotechnical Engineer-ing vol 120 no 12 pp 2155ndash2179 1994

[9] H B Seed and R V Whitman ldquoDesign of Earth retainingstructures for dynamic loadsrdquo in Proceedings of the ASCESpecialty Conference Lateral Stresses in the Ground and Designof Earth Retaining Structures pp 103ndash147 June1970

[10] Y-S Fang and T-J Chen ldquoModification of Mononobe-Okabetheoryrdquo Geotechnique vol 45 no 1 pp 165ndash167 1995

[11] E E Morrison and RM Ebeling ldquoLimit equilibrium computa-tion of dynamic passive earth pressurerdquo Canadian GeotechnicalJournal vol 32 pp 481ndash487 1995

[12] J Kumar ldquoSeismic passive earth pressure coefficients for sandsrdquoCanadian Geotechnical Journal vol 38 pp 876ndash881 2001

[13] K S Subba Rao andD Choudhury ldquoSeismic passive earth pres-sures in soilsrdquo Journal of Geotechnical and GeoenvironmentalEngineering vol 131 no 1 pp 131ndash135 2005

[14] M Yazdani and A Azad ldquoEvaluation of Mononobe-Okabemethod and introducing an improved approach to design ofretaining wallsrdquo Transportation Journal no 2 pp 157ndash171 2007(Persian)

[15] Y S Fang Y C Yang and T J Chen ldquoRetaining walls damagedin the Chi-Chi earthquakerdquoCanadian Geotechnical Journal vol40 no 6 pp 1142ndash1153 2003

[16] S Prakash and S Saran ldquoStatic and dynamic earth pressuresbehind retaining wallsrdquo in Proceedings of the 3rd Symposiumon Earthquake Engineering vol 1 pp 277ndash288 University ofRoorkee Roorkee India November 1966

[17] S Saran and S Prakash ldquoDimensionless parameters for staticand dynamic Earth pressure behind retaining wallsrdquo IndianGeotechnical Journal vol 7 no 3 pp 295ndash310 1968

[18] B M Das and V K Puri ldquoStatic and dynamic active earthpressurerdquo Geotechnical and Geological Engineering vol 14 no4 pp 353ndash366 1996

[19] S K Shukla S K Gupta and N Sivakugan ldquoActive earthpressure on retaining wall for c-120593 soil backfill under seismicloading conditionrdquo Journal of Geotechnical andGeoenvironmen-tal Engineering vol 135 no 5 pp 690ndash696 2009

[20] S Ghosh and S Sengupta ldquoExtension of Mononobe-Okabetheory to evaluate seismic active earth pressure supporting c-120593 backfillrdquo Electronic Journal of Geotechnical Engineering vol17 pp 495ndash504 2012

[21] M Shao-jun W Kui-hua and W Wen-bing ldquoPseudo-dynamicactive earth pressure behind retaining wall for cohesive soilbackfillrdquo Journal of Central South University of Technology vol19 pp 3298ndash3304 2012

[22] S Ghosh and R P Sharma ldquoPseudo-dynamic active response ofnon-vertical retaining wall supporting c-120593 backfillrdquo Geotechni-cal and Geological Engineering vol 28 no 5 pp 633ndash641 2010

[23] D S Scotto A PennaA Evangelista et al ldquoExperimental inves-tigation of dynamic behaviour of cantilever retaining wallsrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisbon Portugal 2012

[24] W Huntington Earth Pressures and Retaining Walls JohnWiley New York NY USA 1957

[25] V R Greco ldquoActive earth thrust on cantilever walls with shortheelrdquoCanadian Geotechnical Journal vol 38 no 2 pp 401ndash4092001


Retainingstructure

Figure 1 Application of retaining walls in civil engineering

KW

KW

K

KWh

hW

120573p

120573a

120575p

120575a

Pp

Pa

120588p 120588a

120572p 120572a

Active zonePassive zone

Lateral displacement

Figure 2 Geometry and parameters of M-O method

applied to the failure mass by pseudostatic method To get afinal simple formulation like other closed form solutions ingeotechnical engineeringM-O uses exact form solution withsimple assumption such as simplicity in geometry materialbehavior or dynamic loading to make the equations solvable

Because of the old age of M-O method tens of studieshave been focused on this area (eg [6ndash8]) An importantstudy onM-Owas carried out by Seed andWhitman [9]Theyconfirmed M-O active pressure after long laboratory runsHowever they recommended more studies on passive theoryof M-O They also proposed a method to find the location ofresultant force which acts on 13 of height in M-O methodM-O had been studied by others such as Fang and Chen [10]on the direction of seismic force components on the failuremass

Figure 2 shows the parameters and characteristics of M-O method In M-O static force equilibrium is satisfied for arigid wedge placed on a failure plane with elastic-perfectlyplastic behavior based on Mohr-Coulomb failure criteria

Active and passive forces119875119886and119875119901 are then calculated using

the following equations

119875119886

119875119901

=1

21205741198672

(1 minus 119870V) 119870119886

119870119901

119870119886

119870119901

= (cos2 (120593 ∓ 120572 minus 120579))

times ( cos 120579 times cos2120572 times cos (120575 plusmn 120572 + 120579)

times[1 plusmn (sin (120593 + 120575) times sin (120593 ∓ 120573 minus 120579)cos (120575 plusmn 120572 + 120579) times cos (120573 minus 120572)

)

12

]

2

)

minus1

120579 = tanminus1 (119870ℎ

1 minus 119870V

)

(1)


KW998400

W

h

H

F

B A

cL

998400

KhW998400

120593 120575

120572

120573

120588

Pp

c998400L998400

Z0

(a)

h

H

F

BA

cL

KW998400

W998400

KhW998400

120593120575

120572

120573

120588Pa

c998400L998400

Z0

(b)











119886and 120588




















1198850=

2119888

120574radic119870119886

119870119886(Rankine) =

1 minus sin1205931 + sin120593

(2)





Overturning

Backfill

Heel

Foot wall

Toe

Parking lotHeaving

Sliding


Hanging wallAssumed

rupture zone

05 m

263 m

1 035

Chelungpu Fault



120572 minus 120579

90 minus 120588 + 120579

F

cL

c998400L998400

120579 + 120601 minus 120579

120579 + 120575 minus 120572


h

Pp

(a)

120572 + 120579

90 minus 120588 minus 120579

F

cL

c998400L998400


h

120588 minus 120601 + 120579

Pa

120572 + 120575 + 120579

(b)





119886 and 119875




5 Parametric Study


0 20 40 60 80 100




(Pa)

or(P

p)








047

048

049

05

051

052

0 04 08 12 16 2BH

Ka

120573 = 5∘


(a)

26

28

3

32

0 04 08 12 16 2BH

Kp

120573 = 5∘


(b)

046

048

05

052

054

056

058

0 04 08 12 16 2BH

Ka


120573 = 10∘

(c)

25

3

35

4

0 04 08 12 16 2BH

Kp


120573 = 10∘

(d)

044

048

052

056

06

064

068

0 04 08 12 16 2

BH

Ka


120573 = 15∘

(e)

25

33

41

49

0 04 08 12 16 2

BH

Kp


120573 = 15∘

(f)

Figure 9 Continued


045

053

061

069

077

085

0 04 08 12 16 2BH


Ka

120573 = 20∘

(g)

25

35

45

55

0 04 08 12 16 2BH

Kp


120573 = 20∘

(h)

045

055

065

075

085

095

105

0 04 08 12 16 2BH

Ka


120573 = 25∘

(i)

25

4

55

7

0 04 08 12 16 2BH

Kp


120573 = 25∘

(j)






= 1

2120574(119867 minus ℎ)

2

+ 120574ℎ (119867 minus ℎ) +1

21205741015840

sdot ℎ2

times 119870119886

(3)



2120574 sdot 1198672

times 119870119886 (4)




25

3

35

4

45


Kp

(a)

045

05

055

06

065

07


Ka

(b)


200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)


30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)






∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=





1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References



























KW998400

W

h

H

F

B A

cL

998400

KhW998400

120593 120575

120572

120573

120588

Pp

c998400L998400

Z0

(a)

h

H

F

BA

cL

KW998400

W998400

KhW998400

120593120575

120572

120573

120588Pa

c998400L998400

Z0

(b)











119886and 120588




















1198850=

2119888

120574radic119870119886

119870119886(Rankine) =

1 minus sin1205931 + sin120593

(2)





Overturning

Backfill

Heel

Foot wall

Toe

Parking lotHeaving

Sliding


Hanging wallAssumed

rupture zone

05 m

263 m

1 035

Chelungpu Fault



120572 minus 120579

90 minus 120588 + 120579

F

cL

c998400L998400

120579 + 120601 minus 120579

120579 + 120575 minus 120572


h

Pp

(a)

120572 + 120579

90 minus 120588 minus 120579

F

cL

c998400L998400


h

120588 minus 120601 + 120579

Pa

120572 + 120575 + 120579

(b)





119886 and 119875




5 Parametric Study


0 20 40 60 80 100




(Pa)

or(P

p)








047

048

049

05

051

052

0 04 08 12 16 2BH

Ka

120573 = 5∘


(a)

26

28

3

32

0 04 08 12 16 2BH

Kp

120573 = 5∘


(b)

046

048

05

052

054

056

058

0 04 08 12 16 2BH

Ka


120573 = 10∘

(c)

25

3

35

4

0 04 08 12 16 2BH

Kp


120573 = 10∘

(d)

044

048

052

056

06

064

068

0 04 08 12 16 2

BH

Ka


120573 = 15∘

(e)

25

33

41

49

0 04 08 12 16 2

BH

Kp


120573 = 15∘

(f)

Figure 9 Continued


045

053

061

069

077

085

0 04 08 12 16 2BH


Ka

120573 = 20∘

(g)

25

35

45

55

0 04 08 12 16 2BH

Kp


120573 = 20∘

(h)

045

055

065

075

085

095

105

0 04 08 12 16 2BH

Ka


120573 = 25∘

(i)

25

4

55

7

0 04 08 12 16 2BH

Kp


120573 = 25∘

(j)






= 1

2120574(119867 minus ℎ)

2

+ 120574ℎ (119867 minus ℎ) +1

21205741015840

sdot ℎ2

times 119870119886

(3)



2120574 sdot 1198672

times 119870119886 (4)




25

3

35

4

45


Kp

(a)

045

05

055

06

065

07


Ka

(b)


200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)


30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)






∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=





1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References








































1198850=

2119888

120574radic119870119886

119870119886(Rankine) =

1 minus sin1205931 + sin120593

(2)





Overturning

Backfill

Heel

Foot wall

Toe

Parking lotHeaving

Sliding


Hanging wallAssumed

rupture zone

05 m

263 m

1 035

Chelungpu Fault



120572 minus 120579

90 minus 120588 + 120579

F

cL

c998400L998400

120579 + 120601 minus 120579

120579 + 120575 minus 120572


h

Pp

(a)

120572 + 120579

90 minus 120588 minus 120579

F

cL

c998400L998400


h

120588 minus 120601 + 120579

Pa

120572 + 120575 + 120579

(b)





119886 and 119875




5 Parametric Study


0 20 40 60 80 100




(Pa)

or(P

p)








047

048

049

05

051

052

0 04 08 12 16 2BH

Ka

120573 = 5∘


(a)

26

28

3

32

0 04 08 12 16 2BH

Kp

120573 = 5∘


(b)

046

048

05

052

054

056

058

0 04 08 12 16 2BH

Ka


120573 = 10∘

(c)

25

3

35

4

0 04 08 12 16 2BH

Kp


120573 = 10∘

(d)

044

048

052

056

06

064

068

0 04 08 12 16 2

BH

Ka


120573 = 15∘

(e)

25

33

41

49

0 04 08 12 16 2

BH

Kp


120573 = 15∘

(f)

Figure 9 Continued


045

053

061

069

077

085

0 04 08 12 16 2BH


Ka

120573 = 20∘

(g)

25

35

45

55

0 04 08 12 16 2BH

Kp


120573 = 20∘

(h)

045

055

065

075

085

095

105

0 04 08 12 16 2BH

Ka


120573 = 25∘

(i)

25

4

55

7

0 04 08 12 16 2BH

Kp


120573 = 25∘

(j)






= 1

2120574(119867 minus ℎ)

2

+ 120574ℎ (119867 minus ℎ) +1

21205741015840

sdot ℎ2

times 119870119886

(3)



2120574 sdot 1198672

times 119870119886 (4)




25

3

35

4

45


Kp

(a)

045

05

055

06

065

07


Ka

(b)


200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)


30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)






∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=





1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References



























120572 minus 120579

90 minus 120588 + 120579

F

cL

c998400L998400

120579 + 120601 minus 120579

120579 + 120575 minus 120572


h

Pp

(a)

120572 + 120579

90 minus 120588 minus 120579

F

cL

c998400L998400


h

120588 minus 120601 + 120579

Pa

120572 + 120575 + 120579

(b)





119886 and 119875




5 Parametric Study


0 20 40 60 80 100




(Pa)

or(P

p)








047

048

049

05

051

052

0 04 08 12 16 2BH

Ka

120573 = 5∘


(a)

26

28

3

32

0 04 08 12 16 2BH

Kp

120573 = 5∘


(b)

046

048

05

052

054

056

058

0 04 08 12 16 2BH

Ka


120573 = 10∘

(c)

25

3

35

4

0 04 08 12 16 2BH

Kp


120573 = 10∘

(d)

044

048

052

056

06

064

068

0 04 08 12 16 2

BH

Ka


120573 = 15∘

(e)

25

33

41

49

0 04 08 12 16 2

BH

Kp


120573 = 15∘

(f)

Figure 9 Continued


045

053

061

069

077

085

0 04 08 12 16 2BH


Ka

120573 = 20∘

(g)

25

35

45

55

0 04 08 12 16 2BH

Kp


120573 = 20∘

(h)

045

055

065

075

085

095

105

0 04 08 12 16 2BH

Ka


120573 = 25∘

(i)

25

4

55

7

0 04 08 12 16 2BH

Kp


120573 = 25∘

(j)






= 1

2120574(119867 minus ℎ)

2

+ 120574ℎ (119867 minus ℎ) +1

21205741015840

sdot ℎ2

times 119870119886

(3)



2120574 sdot 1198672

times 119870119886 (4)




25

3

35

4

45


Kp

(a)

045

05

055

06

065

07


Ka

(b)


200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)


30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)






∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=





1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References



























047

048

049

05

051

052

0 04 08 12 16 2BH

Ka

120573 = 5∘


(a)

26

28

3

32

0 04 08 12 16 2BH

Kp

120573 = 5∘


(b)

046

048

05

052

054

056

058

0 04 08 12 16 2BH

Ka


120573 = 10∘

(c)

25

3

35

4

0 04 08 12 16 2BH

Kp


120573 = 10∘

(d)

044

048

052

056

06

064

068

0 04 08 12 16 2

BH

Ka


120573 = 15∘

(e)

25

33

41

49

0 04 08 12 16 2

BH

Kp


120573 = 15∘

(f)

Figure 9 Continued


045

053

061

069

077

085

0 04 08 12 16 2BH


Ka

120573 = 20∘

(g)

25

35

45

55

0 04 08 12 16 2BH

Kp


120573 = 20∘

(h)

045

055

065

075

085

095

105

0 04 08 12 16 2BH

Ka


120573 = 25∘

(i)

25

4

55

7

0 04 08 12 16 2BH

Kp


120573 = 25∘

(j)






= 1

2120574(119867 minus ℎ)

2

+ 120574ℎ (119867 minus ℎ) +1

21205741015840

sdot ℎ2

times 119870119886

(3)



2120574 sdot 1198672

times 119870119886 (4)




25

3

35

4

45


Kp

(a)

045

05

055

06

065

07


Ka

(b)


200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)


30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)






∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=





1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References



























045

053

061

069

077

085

0 04 08 12 16 2BH


Ka

120573 = 20∘

(g)

25

35

45

55

0 04 08 12 16 2BH

Kp


120573 = 20∘

(h)

045

055

065

075

085

095

105

0 04 08 12 16 2BH

Ka


120573 = 25∘

(i)

25

4

55

7

0 04 08 12 16 2BH

Kp


120573 = 25∘

(j)






= 1

2120574(119867 minus ℎ)

2

+ 120574ℎ (119867 minus ℎ) +1

21205741015840

sdot ℎ2

times 119870119886

(3)



2120574 sdot 1198672

times 119870119886 (4)




25

3

35

4

45


Kp

(a)

045

05

055

06

065

07


Ka

(b)


200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)


30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)






∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=





1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References



























25

3

35

4

45


Kp

(a)

045

05

055

06

065

07


Ka

(b)


200

250

300

350

400

450


Tota

l pas

sive p

ress

ure (

ton

m)


(a)


30

40

50

60

70


Tota

l act

ive p

ress

ure (

ton

m)

(b)






∘ 119861119867 = 2 120601 = 30∘ and 119870

ℎ=





1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References



























1

2

3

4

5


Z = 0

Z ne 0

Kh = 0

Kp

(a)

0

01

02

03

04


Z = 0

Z ne 0

Kh = 0

Ka

(b)

05

15

25

35

45


Z = 0

Z ne 0

Kh = 02

Kp

(c)

0

01

02

03

04

05


Z = 0

Z ne 0

Kh = 02

Ka

(d)




∘


ℎ= 0 and 119870







6 Conclusion





References




























References


























extended ‘‘mononobe-okabe’’ method for seismic design of

Documents