Research ArticleStudy on the Seismic Active Earth Pressure by Variational LimitEquilibrium Method

Jiangong Chen12 Zejun Yang12 Richeng Hu12 and Haiquan Zhang12

1College of Civil Engineering Chongqing University Chongqing 400045 China2Key Laboratory of New Technology for Construction of Cities in Mountain Area Chongqing UniversityMinistry of Education Chongqing 400045 China

Correspondence should be addressed to Zejun Yang cquyzj1986163com

Received 8 January 2016 Revised 19 April 2016 Accepted 8 May 2016

Academic Editor Sergio De Rosa

Copyright copy 2016 Jiangong Chen et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the framework of limit equilibrium theory the isoperimetric model of functional extremum regarding the seismic active earthpressure is deduced according to the variational method On this basis Lagrange multipliers are introduced to convert the problemof seismic active earth pressure into the problem on the functional extremum of two undetermined function arguments Based onthe necessary conditions required for the existence of functional extremum the function of the slip surface and the normal stressdistribution on the slip surface is obtained and the functional extremum problem is further converted into a function optimizationproblem with two undetermined Lagrange multipliers The calculated results show that the slip surface is a plane and the seismicactive earth pressure is minimal when the action point is at the lower limit position As the action point moves upward the slipsurface becomes a logarithmic spiral and the corresponding value of seismic active earth pressure increases in a nonlinear mannerAnd the seismic active earth pressure is maximal at the upper limit position The interval estimation constructed by the minimumand maximum values of seismic active earth pressure can provide a reference for the aseismic design of gravity retaining walls

1 Introduction

The magnitude and distribution of active earth pressureon the retaining wall under the seismic loading are thetheoretical premises of the aseismic design for the retainingwall and play a vital role in evaluating the stability of theretaining wall in the seismic area The common calculationmethod of seismic active earth pressure is the Mononobe-Okabe (M-O) theory which is based on the Coulomb theoryand believes that the sliding soil wedge is a rigid body underthe seismic loadingThe pseudo-static approach is adopted tosimplify the seismic force into an inertia force acting on thesliding soil wedge and transcribe the dynamic problem as astatic problem [1] The M-O theory assumes that the backfillbehind the wall is cohesion-less soil and the slip surface is aplane and the theory cannot obtain the real action point ofresultant force without considering the equation of momentequilibrium thus restrictions are formed on this theory TheM-O theory has been improved by a number of scholars toexpand its application scope [2ndash6]

Both the M-O theory and the improvement methodbased thereon are on the basis of the assumption that the slipsurface is a plane which does not conform to the practicalsituation One rigorous method in math is the variationallimit equilibrium method by which the seismic active earthpressure on the retaining wall is attributed as the functionalextremum problem of two undetermined functions Oneundetermined function is expressed by the shape of the slipsurface while the other is the function of the normal stressdistribution on the slip surface They are numerically solvedby the variational method

The variational limit equilibrium method was firstproposed by Kopascy [7ndash9] Then it was introduced to thestability analysis of slope and foundation [10ndash22] And somescholars utilized the variational limit equilibrium methodto study the lateral earth pressure on the retaining wall[23ndash26] According to the variational method Shaojunstudied the shape of the slip surface of the sliding soilwedge behind the retaining wall and obtained the analytical

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 4158785 10 pageshttpdxdoiorg10115520164158785

2 Shock and Vibration

solution of the shape of the slip surface and the magnitudeof seismic active earth pressure [27] However only thecases where the retaining wall is vertical and the backfillsurface is horizontal without surcharge are considered in thecalculation models while the influences of wall-movementmodes of the retaining wall on the magnitude and the actionpoint position of seismic active earth pressure fail to betaken into account In fact the distribution of earth pressureon the retaining wall is nonlinear The magnitude and theaction point position of seismic active earth pressure dependon the coordinated deformation of soil-wall contact surfaceand vary with the change of wall-movement modes of theretaining wall [26] However in the design process it is oftenhard to accurately estimate the wall-movement modes of theretainingwall For the static and dynamic ultimate load actingon the retaining wall a reasonable approach is to contain theseismic active earth pressure under different wall-movementmodes in a certain range as possible for the engineeringdesigners to select and use In this paper the variationallimit equilibrium method is used to study the seismicactive earth pressure on the gravity retaining wall undergeneral conditions (the retaining wall is inclined and coarsethe backfill is cohesive soil the backfill surface is a curvedsurface with nonuniform surcharge) The interval of theseismic active earth pressure under different wall-movementmodes can be effectively estimated by the proposedapproach

2 Variational Analysis of SeismicActive Earth Pressure

21 Basic Assumption (1) The research problem is a planestrain problem (2) the soil behind the wall is Coulombmaterial which can be represented by the intensity parametercohesion 119888 and the internal friction angle 120593 (3) when thebackfill soil is in the critical active state a sliding soilwedge is formed and its slip surface passes through the wallheel (4) the retaining wall is rigid and its motion formsare unconstrained wherein the motion displacement canbe ignored compared with the wall height (5) the seismicaction is simplified to static load acting on the sliding wedgewith horizontal seismic coefficient 119896

119867and vertical seismic

coefficient 119896119881

22 Limit Equilibrium Equation of Sliding Soil Wedge Thecalculation model of active earth pressure under the seismicloading is shown in Figure 1 wherein the height of theretaining wall is119867 the retaining wall is inclined and coarsethe slope angle of the wall to vertical is 120572 the friction anglebetween soil and wall is 120575 the unit weight of soil is 120574 thecohesion is 119888 the internal fraction angle is 120593 the expressionof the backfill surface is 119892(119909) and the expression of theslip surface is 119904(119909) the expression of the vertical surchargedistribution on the backfill surface is 119902(119909) the expression ofthe tangential stress distribution on the slip surface is 120591(119909)and the expression of the normal stress distribution on theslip surface is 120590(119909) the seismic soil pressure on the wall is119864119886

kHq(x)

(1 minus kV)q(x)

kHWG

(1 minus kV)WG

Ea120585H

x2 x1

120590(x)

120591(x)

y = s(x)

y = g(x)

120572

120575

O X

B

A

Y

H

Figure 1 Calculation model of seismic active earth pressure

The equilibrium equation of the sliding soil wedge OABunder the limit equilibrium state can be written as follows

119864119886cos (120572 + 120575) + int

1199091

0

120591 minus 1205901199041015840minus 119896119867[120574 (119892 minus 119904) + 119902] 119889119909

minus int

0

1199092

119896119867[120574 (119892 minus 119896119909) + 119902] 119889119909 = 0

(1)

119864119886sin (120572 + 120575) + int

1199091

0

1205911199041015840+ 120590 minus (1 minus 119896

119881) [120574 (119892 minus 119904)

+ 119902] 119889119909 minus int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119889119909

= 0

(2)

119864119886cos 120575 120585119867

cos120572+ int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119909 119889119909

minus int

0

1199092

119896119867119902119892 119889119909 minus int

1199091

0

119896119867119902119892 119889119909 minus int

0

1199092

1

2

119896119867120574 (1198922

minus 11989621199092) 119889119909 minus int

1199091

0

1

2

119896119867120574 (1198922minus 1199042) 119889119909

minus int

1199091

0

(1199091199041015840minus 119904) 120591 + (119909 + 119904119904

1015840) 120590

minus (1 minus 119896119881) [119902119909 + 120574 (119892 minus 119904) 119909] 119889119909 = 0

(3)

wherein 1199041015840= 119889119904119889119909 119896 = minus cot120572 119909

1is 119883 coordinate of 119861

1199092is 119883 coordinate of 119860 and 119909

2= minus119867 tan120572 120585 is the position

coefficient of the action point of seismic active earth pressureand its value is the ratio of the vertical distance (from theaction point position to the wall heel) to the height of theretaining wall

Shock and Vibration 3

It is assumed that the tangential stress 120591 and the normalstress 120590 on the slip surface are subject to the Mohr-Coulombfailure criterion

120591 = 119888 + 1198991120590 (4)

wherein 1198991= tan120593

23 Variational Limit Equilibrium Model of Seismic ActiveEarth Pressure The problem to solve the seismic active earthpressure 119864ae can now be stated as follows The seismic activeearth pressure119864ae is themaximumvalue of119864

119886 corresponding

to the most dangerous (critical) slip surface Realizing themaximum value of 119864

119886is to find the function of the slip

surface 119904(119909) and the function of normal stress distribution120590(119909) on the slip surface subject to the three equations ofLE ((1)sim(3)) Among these three equations (3) is changed asobjective function and the other equations are the constraintconditions according to the variational method

The following objective function 119869 can be obtained after(4) is substituted into (3) Obviously to solve the maximumof 119864119886means the same as to solve the maximum of objective

function 119869

119869 = int

1199091

0

1198650119889119909 (5)

wherein

119869 =

119864119886120585119867 cos 120575cos120572

+ 1199110

1198650= (11989911199041015840119909 + 119909 + 119904119904

1015840minus 1198991119904) 120590 + 119888119909119904

1015840minus 119888119904 minus (1 minus 119896

119881)

sdot [(119892 minus 119904) 120574119909 + 119902119909] + 119896119867119902119892 +

1

2

119896119867120574 (1198922minus 1199042)

1199110= int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119909 minus 119896

119867119902119892

minus

1

2

119896119867120574 (1198922minus 11989621199092) 119889119909 = const

(6)

Two constraint conditions can be obtained after (4) and (5)are substituted into (1) and (2)

int

1199091

0

(11991111198650+ 1198651) 119889119909 = 119911

2

int

1199091

0

(11991131198650+ 1198652) 119889119909 = 119911

4

(7)

wherein

1198651= (1198991minus 1199041015840) 120590 + 119888 minus 119896

119867[120574 (119892 minus 119904) + 119902]

1198652= (11989911199041015840+ 1) 120590 + 119888119904

1015840minus (1 minus 119896

119881) [119902 + 120574 (119892 minus 119904)]

1199111=

cos120572 cos (120572 + 120575)

120585119867 cos 120575= const

1199112= 11991101199111+ int

0

1199092

119896119867[120574 (119892 minus 119896119909) + 119902] 119889119909 = const

1199113=

cos120572 cos (120572 + 120575)

120585119867 cos 120575= const

1199114= 11991101199113+ int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119889119909 = const

(8)

Equations (5) and (7) show the isoperimetric model of theconstrained variational extremum with an undeterminedboundary The starting point of the slip surface 119874 is a fixedpoint on the coordinates of (119909

0= 0 119910

0= 0) and the end point

119861 moves on the backfill surface 119892(119909) with the undeterminedcoordinates of (119909

1 119892(1199091)) The undetermined boundary is a

variational boundary The variational boundary refers to anunfixed boundary on which one or two ends respectivelymove on the given function

According to the variational method of the functionalwith constraints the following auxiliary functional 119869lowast isconstructed by Lagrangemultipliers to convert the functionalextremum problem under constraint conditions into a func-tional extremum problem without constraint conditions

119869lowast= int

1199091

0

119865119889119909

119865 = 1198650+ 12058211198651+ 12058221198652

(9)

wherein 1205821and 120582

2are Lagrange multipliers

Now the solution of the maximum of 119864119886is converted

into the solution of the maximum of auxiliary function 119869lowast

According to the necessary conditions for the existence ofthe extremum of auxiliary functional 119869lowast the function ofthe slip surface 119904(119909) and the function of the normal stressdistribution 120590(119909) on the slip surface must meet the followingrequirements

(1) The Euler differential equation of the auxiliary func-tion 119865 is

120597119865

120597120590

minus

119889

119889119909

(

120597119865

1205971205901015840) = 0 (10)

120597119865

120597119904

minus

119889

119889119909

(

120597119865

1205971199041015840) = 0 (11)

(2) The integral constraint equations are (7)(3) Boundary conditions are

Fixed boundary condition 119904 (0) = 0

Variational boundary condition 119904 (1199091) = 119892 (119909

1)

(12)

(4) Transversality condition at the variational boundaryis

(119865 minus 1199041015840 120597119865

1205971199041015840+ 1198921015840 120597119865

1205971199041015840)

10038161003816100381610038161003816100381610038161003816119909=1199091

= 0 (13)

4 Shock and Vibration

3 Variational Solution of SeismicActive Earth Pressure

31 Shape Function of the Slip Surface Thefollowing equationcan be obtained after (5) (7) and (9) are substituted into (10)

119889119904

119889119909

=

119909 minus 1198991119904 + 12058211198991+ 1205822

minus1198991119909 minus 119904 + 120582

1minus 12058221198991

(14)

Polar coordinate transformations 119903 cos 120579 = 119909 + 1205822and

119903 sin 120579 = 119904 minus 1205821are brought in For the calculation model in

polar coordinates see Figure 2 Equation (14) is transcribedas a differential equation

119889119903

119889120579

= minus1198991119903 (15)

The general solution of the differential equation is

119903 = 1199115119890minus1198991120579 (16)

wherein 1199115is an arbitrary integration constant

The fixed boundary condition 119904(0) = 0 shows that thepole of the polar coordinate is (119903

0 1205790) After it is substituted

into (16) the expression of the logarithmic spiral of the slipsurface can be obtained

119903 = 11990301198901198991(1205790minus120579)

(17)

wherein

1199030= radic1205822

1+ 1205822

2

1205790= minus arctan(1205821

1205822

) 1205822gt 0

1205790= minus120587 minus arctan(1205821

1205822

) 1205822le 0

(18)

32 Normal Stress Distribution on the Slip Surface After(5) (7) and (9) are substituted into (11) the following equa-tion can be obtained

21198991120590 + (119899

1119909 + 119904 minus 120582

1+ 11989911205822) 1205901015840minus (1 minus 119896

119881) 120574119909

minus (1 minus 119896119881) 1205822120574 + 2119888 + 119896

119867120574119904 minus 119896

1198671205821120574 = 0

(19)

Introducing the Polar coordinate transformations 119903 cos 120579 =

119909 + 1205822and 119903 sin 120579 = 119904 minus 120582

1 (19) is transcribed as a differential

equation119889120590

119889120579

minus 21198991120590 = 2119888 minus (1 minus 119896

119881) 12057411990301198901198991(1205790minus120579) cos 120579

+ 11989611986712057411990301198901198991(1205790minus120579) sin 120579

(20)

The general solution of the differential equation is (when 1198991

=

0)

120590 = 119890int120579

1205791

21198991119889120579(1199116+ int

120579

1205791

(2119888

minus (1 minus 119896119881) 12057411990301198901198991(1205790minus120579) cos 120579 + 119896

11986712057411990301198901198991(1205790minus120579) sin 120579)

sdot 119890int120579

1205791

minus21198991119889120579119889120579)

(21)

kHq(x)

(1 minus kV)q(x)

kHWG

(1 minus kV)WG

Ea120585H

x2 x1

120590(x)

120591(x)

y = s(x)

y = g(x)

120572

120575

120579120579B

O998400

Y998400

X998400

r

OX

B

A

H

Y

r0

1205790

Figure 2 Calculationmodel of seismic active earth pressure in polarcoordinates

wherein 1199116is an integration constant 120579

1is an arbitrary angle

1205791= 0 is available

120590 = 119911611989021198991120579minus

(1 minus 119896119881) 12057411990301198901198991(1205790minus120579)

1 + 91198992

1

(sin 120579 minus 31198991cos 120579

+ 3119899111989031198991120579) minus

11989611986712057411990301198901198991(1205790minus120579)

1 + 91198992

1

(31198991sin 120579 + cos 120579

minus 11989031198991120579) minus

119888

1198991

(22)

According to the transversality condition at the variationalboundary the normal stress distribution at 119861 of the slipsurface can be obtained120590 (1199091)

=

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(23)

After it is substituted into (22) the following equation isobtained

1199116= 119890minus211989911205791

119888

1198991

+

11989611986712057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(31198991sin 1205791+ cos 120579

1minus 119890311989911205791)

+

(1 minus 119896119881) 12057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(sin 1205791minus 31198991cos 1205791+ 31198991119890311989911205791)

+

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(24)

33 Optimal Solution of Seismic Active Earth Pressure Theanalysis carried out so far indicates that the maximum value

Shock and Vibration 5

of119864119886can be obtained according to a pair of functions 119903(120579) and

120590(120579) determined by two undetermined Lagrange multipliers1205821and 120582

2 The problem is converted into the search of a pair

of constants 1205821and 120582

2through the two constraint equations

The two constraint equationsmdash(7)mdashmust be satisfied

1198911(1205821 1205822) = int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2= 0

1198912(1205821 1205822) = int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4= 0

(25)

The following equation is always correct

1198911(1205821 1205822) = 0

1198912(1205821 1205822) = 0

lArrrArr Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822) = 0 (26)

Therefore two integral equationsmdash(25)mdashare equivalent tothe following equation

Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822)

= [int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2]

2

+ [int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4]

2

= 0

(27)

ForΦ(1205821 1205822) ge 0 so

min [Φ (1205821 1205822)] = 0 (28)

The above analysis indicates that the solution of the twoconstraint equationsmdash(7)mdashcan be obtained through thesolution of the minimum value of the function Φ (theminimum value is 0)When theminimum value ofΦ is not 0it shows that the sliding soil wedge cannot keep balance andis improper

In this paper the fminsearch function provided by MAT-LAB is used to find the optimal solution For this functionit only requires establishing a function module of Φ andgiving an initial value to search the minimum value of thefunction Fminsearch applies the derivative-free method tofind the minimum value of the unconstrained multivariablefunction which is generally called unconstrained nonlinearoptimization Fminsearch finds the minimum of a scalarfunction of several variables starting at an initial estimateTherefore one major drawback of this optimization tool isthat the search would be very slow if the initial estimateis improper In order to overcome this drawback and inthe consideration of the condition that the center of thelogarithmic spiral is usually at the top left corner of theretaining wall the initial value is 120582

1= 1205822

= 119867 in theoptimization analysis of this paper and the search is quiterapid

4 Calculated Result and Parameter Analysis

41 Interval Estimation of Seismic Active Earth PressureUnder the general conditions that the retaining wall is

times105

120585d = 0449 120585u = 0612

00

05

10

15

20

25

min

Φ

02 04 06 08 1000120585

Figure 3 The minimum value of the function Φ

inclined and coarse the backfill is cohesive soil and thebackfill surface is inclined with surcharge the interval ofthe position coefficient 120585 of the action point under seismicloading is numerically solved Besides the influence of theaction point position on the shape of the slip surface and themagnitude of seismic active earth pressure is studied

The retaining wall has a height of 10m with the slopeangle of thewall to vertical120572 = 10

∘ the friction angle betweensoil and wall 120575 = 15

∘ the unit weight of soil 120574 = 18 kNm3cohesion 119888 = 10 kPa internal friction angle 120593 = 20

∘ slopeangle of the backfill soil 120573 = 5

∘ surcharge 119902 = 50 kNm andseismic coefficients 119896

119867= 005 and 119896

119881= 00

Figure 3 shows the curve of the minimum value of thefunction Φ varying with the position coefficient 120585 of theaction point As shown in the figure the position coefficient120585 of the action point of seismic active earth pressure is in aninterval (lower limit value 120585

119889= 0449 upper limit value 120585

119906=

0612) In the interval the minimum value of the functionΦ

is 0 In other words when the action point of seismic activeearth pressure falls within the range of 449sim612m from thewall bottom the counterforce provided by the retaining wallcan balance the soil mass behind the wall otherwise the soilmass will inevitably lose its balance no matter how big thecounterforce provided by the retaining wall is

There is a one-to-one corresponding relation between themagnitude of seismic active earth pressure and the shapeof slip surface (Figure 5) For the curves varying with theposition coefficient 120585 of the action point of seismic activeearth pressure see Figures 3 and 4 At the lower limit of120585 the slip surface is a plane with the minimal value ofseismic active earth pressure As the action point movesupward the slip surface becomes a logarithmic spiral surfaceand its curvature is gradually increased that is in thelogarithmic spiral equation 119903

0 is gradually decreased and

the corresponding value of seismic active earth pressure isincreased in a nonlinear manner At the upper limit of 120585the seismic active earth pressure is maximal The minimumand maximum values can be as an interval estimation of theseismic active earth pressure under different wall-movementmodes of the retaining wall

6 Shock and Vibration

Table 1 Comparison of calculated results of active earth pressure with that of M-O

120572∘ 120573∘ 119896119867

119896119881

The proposed method M-O method119864aekN 120585 119864

119886kN 120585

0 0 0 0 3910sim4121 0265sim0408 3910 130 5 005 005 4246sim4502 0286sim0449 4245 130 10 01 01 4708sim4950 0306sim0490 4707 1310 0 005 01 4605sim4752 0327sim0408 4601 1310 5 01 0 5043sim5253 0327sim0449 5043 1310 10 0 005 5599sim5728 0367sim0469 5596 1320 0 01 005 5504sim5628 0347sim0408 5503 1320 5 0 01 6049sim6161 0367sim0429 6047 1320 10 005 0 6827sim6963 0388sim0469 6825 13

120585d = 0449

120585u = 0612

0

2

4

6

8

10

12

Y (m

)

2 4 6 8 10 120X (m)

Figure 4 The shape of the slip surface

42 Comparison with the M-OTheory Solution To verify thecorrectness of the calculationmethod proposed in this papera comparison is made between the calculated result of theproposedmethod and the solution ofM-O theory Accordingto the assumption of M-O theory the values of cohesion 119888

and the surcharge 119902 are both 0 in the contrastive analysis Forother parameters and specific comparison results see Table 1

When the slip surface is a plane the seismic activeearth pressure is the lower limit value of the interval inthe table This value is equivalent to the solution of seismicactive earth pressure calculated by the M-O theory Thesolution obtained in the proposed method is degraded intoM-O theory solution thereby proving the correctness of theproposed method

When the slip surface is a logarithmic spiral surfacethe seismic active earth pressure calculated by the M-Otheory is relatively small and the assumed action pointposition is lower than the actual action point position undermost possible wall-movement modes The underestimationof the magnitude and the action point position of seismicactive earth pressure cause the potential safety hazard ofoverturning of the retaining wall designed by M-O theory

43 Parameter Analysis In this section discussions aremade regarding the influence of relevant parameters onthe magnitude and the action point position of seismicactive earth pressure as well as the overturning momenton the retaining wall Such parameters include the slopeangle 120572 of the retaining wall the slope angle 120573 of backfillsurface the surcharge 119902 on the backfill surface the horizontalseismic coefficient 119896

119867 and the vertical seismic coefficient

119896119881

431 Influence of Parameters on the Action Point PositionThe influence of relevant parameters on the position coef-ficient 120585 of action point of seismic active earth pressure isshown in Figures 6(a)ndash6(c) With the increase of the slopeangle120573 of backfill surface the action point positionmoves upwith the increase of120572 the action point position slightlymovesdown with the increase of the horizontal seismic coefficient119896119867and the vertical seismic coefficient 119896

119881 the action point

positionmoves up wherein the horizontal seismic coefficient119896119867has a significant influence on it with the increase of the

surcharge 119902 the action point positionmoves up in a nonlinearway

432 Influence of Parameters on the Seismic Active EarthPressure 119864

119886119890 Figures 7(a)ndash7(c) show the influence of rele-

vant parameters on the dimensionless seismic active earthpressure 119864ae(120574119867

2) The seismic active earth pressure 119864ae

is increased with the increase of the slope angle 120572 of theretaining wall and the slope angle 120573 of backfill surface119864ae is increased with the increase of the horizontal seismiccoefficient 119896

119867and decreased with the increase of the vertical

seismic coefficient 119896119881 119864ae is increased linearly with the

increase of the dimensionless surcharge 119902(120574119867) of slopesurface

433 Influence of Parameters on the OverturningMoment119872119886

Figures 8(a)ndash8(c) show the influence of relevant parameterson the dimensionless overturning moment 119872

1198861205741198673 The

overturning moment 119872119886is increased with the increase of

Shock and Vibration 7

860

870

880

890

900

910

Eae

(kN

mminus

1 )

046 048 050 052 054 056 058 060 062044120585

Figure 5 The magnitude of seismic active earth pressure

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

045

050

055120585

060

065

070

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

053

054

120585

055

056

057

058

059

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

050

055

060120585

065

070

010 020 030 040 050000q120574H

(c) 119902

Figure 6 Effects of relevant parameters on 120585

the slope angle 120572 of the retaining wall and the slope angle120573 of backfill surface 119872

119886is increased with the increase of

the horizontal seismic coefficient 119896119867and decreased with the

increase of the vertical seismic coefficient 119896119881119872119886is increased

linearly with the increase of the dimensionless surcharge load119902(120574119867) of slope surface

5 Conclusion

The seismic active earth pressure is studied in the paperbased on the variational limit equilibrium method and thefollowing conclusions are obtained

(1) In the framework of limit equilibrium theory theisoperimetric model of functional extremum regarding the

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

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International Journal of

2 Shock and Vibration

solution of the shape of the slip surface and the magnitudeof seismic active earth pressure [27] However only thecases where the retaining wall is vertical and the backfillsurface is horizontal without surcharge are considered in thecalculation models while the influences of wall-movementmodes of the retaining wall on the magnitude and the actionpoint position of seismic active earth pressure fail to betaken into account In fact the distribution of earth pressureon the retaining wall is nonlinear The magnitude and theaction point position of seismic active earth pressure dependon the coordinated deformation of soil-wall contact surfaceand vary with the change of wall-movement modes of theretaining wall [26] However in the design process it is oftenhard to accurately estimate the wall-movement modes of theretainingwall For the static and dynamic ultimate load actingon the retaining wall a reasonable approach is to contain theseismic active earth pressure under different wall-movementmodes in a certain range as possible for the engineeringdesigners to select and use In this paper the variationallimit equilibrium method is used to study the seismicactive earth pressure on the gravity retaining wall undergeneral conditions (the retaining wall is inclined and coarsethe backfill is cohesive soil the backfill surface is a curvedsurface with nonuniform surcharge) The interval of theseismic active earth pressure under different wall-movementmodes can be effectively estimated by the proposedapproach

2 Variational Analysis of SeismicActive Earth Pressure

21 Basic Assumption (1) The research problem is a planestrain problem (2) the soil behind the wall is Coulombmaterial which can be represented by the intensity parametercohesion 119888 and the internal friction angle 120593 (3) when thebackfill soil is in the critical active state a sliding soilwedge is formed and its slip surface passes through the wallheel (4) the retaining wall is rigid and its motion formsare unconstrained wherein the motion displacement canbe ignored compared with the wall height (5) the seismicaction is simplified to static load acting on the sliding wedgewith horizontal seismic coefficient 119896

119867and vertical seismic

coefficient 119896119881

22 Limit Equilibrium Equation of Sliding Soil Wedge Thecalculation model of active earth pressure under the seismicloading is shown in Figure 1 wherein the height of theretaining wall is119867 the retaining wall is inclined and coarsethe slope angle of the wall to vertical is 120572 the friction anglebetween soil and wall is 120575 the unit weight of soil is 120574 thecohesion is 119888 the internal fraction angle is 120593 the expressionof the backfill surface is 119892(119909) and the expression of theslip surface is 119904(119909) the expression of the vertical surchargedistribution on the backfill surface is 119902(119909) the expression ofthe tangential stress distribution on the slip surface is 120591(119909)and the expression of the normal stress distribution on theslip surface is 120590(119909) the seismic soil pressure on the wall is119864119886

kHq(x)

(1 minus kV)q(x)

kHWG

(1 minus kV)WG

Ea120585H

x2 x1

120590(x)

120591(x)

y = s(x)

y = g(x)

120572

120575

O X

B

A

Y

H

Figure 1 Calculation model of seismic active earth pressure

The equilibrium equation of the sliding soil wedge OABunder the limit equilibrium state can be written as follows

119864119886cos (120572 + 120575) + int

1199091

0

120591 minus 1205901199041015840minus 119896119867[120574 (119892 minus 119904) + 119902] 119889119909

minus int

0

1199092

119896119867[120574 (119892 minus 119896119909) + 119902] 119889119909 = 0

(1)

119864119886sin (120572 + 120575) + int

1199091

0

1205911199041015840+ 120590 minus (1 minus 119896

119881) [120574 (119892 minus 119904)

+ 119902] 119889119909 minus int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119889119909

= 0

(2)

119864119886cos 120575 120585119867

cos120572+ int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119909 119889119909

minus int

0

1199092

119896119867119902119892 119889119909 minus int

1199091

0

119896119867119902119892 119889119909 minus int

0

1199092

1

2

119896119867120574 (1198922

minus 11989621199092) 119889119909 minus int

1199091

0

1

2

119896119867120574 (1198922minus 1199042) 119889119909

minus int

1199091

0

(1199091199041015840minus 119904) 120591 + (119909 + 119904119904

1015840) 120590

minus (1 minus 119896119881) [119902119909 + 120574 (119892 minus 119904) 119909] 119889119909 = 0

(3)

wherein 1199041015840= 119889119904119889119909 119896 = minus cot120572 119909

1is 119883 coordinate of 119861

1199092is 119883 coordinate of 119860 and 119909

2= minus119867 tan120572 120585 is the position

coefficient of the action point of seismic active earth pressureand its value is the ratio of the vertical distance (from theaction point position to the wall heel) to the height of theretaining wall

Shock and Vibration 3

It is assumed that the tangential stress 120591 and the normalstress 120590 on the slip surface are subject to the Mohr-Coulombfailure criterion

120591 = 119888 + 1198991120590 (4)

wherein 1198991= tan120593

23 Variational Limit Equilibrium Model of Seismic ActiveEarth Pressure The problem to solve the seismic active earthpressure 119864ae can now be stated as follows The seismic activeearth pressure119864ae is themaximumvalue of119864

119886 corresponding

to the most dangerous (critical) slip surface Realizing themaximum value of 119864

119886is to find the function of the slip

surface 119904(119909) and the function of normal stress distribution120590(119909) on the slip surface subject to the three equations ofLE ((1)sim(3)) Among these three equations (3) is changed asobjective function and the other equations are the constraintconditions according to the variational method

The following objective function 119869 can be obtained after(4) is substituted into (3) Obviously to solve the maximumof 119864119886means the same as to solve the maximum of objective

function 119869

119869 = int

1199091

0

1198650119889119909 (5)

wherein

119869 =

119864119886120585119867 cos 120575cos120572

+ 1199110

1198650= (11989911199041015840119909 + 119909 + 119904119904

1015840minus 1198991119904) 120590 + 119888119909119904

1015840minus 119888119904 minus (1 minus 119896

119881)

sdot [(119892 minus 119904) 120574119909 + 119902119909] + 119896119867119902119892 +

1

2

119896119867120574 (1198922minus 1199042)

1199110= int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119909 minus 119896

119867119902119892

minus

1

2

119896119867120574 (1198922minus 11989621199092) 119889119909 = const

(6)

Two constraint conditions can be obtained after (4) and (5)are substituted into (1) and (2)

int

1199091

0

(11991111198650+ 1198651) 119889119909 = 119911

2

int

1199091

0

(11991131198650+ 1198652) 119889119909 = 119911

4

(7)

wherein

1198651= (1198991minus 1199041015840) 120590 + 119888 minus 119896

119867[120574 (119892 minus 119904) + 119902]

1198652= (11989911199041015840+ 1) 120590 + 119888119904

1015840minus (1 minus 119896

119881) [119902 + 120574 (119892 minus 119904)]

1199111=

cos120572 cos (120572 + 120575)

120585119867 cos 120575= const

1199112= 11991101199111+ int

0

1199092

119896119867[120574 (119892 minus 119896119909) + 119902] 119889119909 = const

1199113=

cos120572 cos (120572 + 120575)

120585119867 cos 120575= const

1199114= 11991101199113+ int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119889119909 = const

(8)

Equations (5) and (7) show the isoperimetric model of theconstrained variational extremum with an undeterminedboundary The starting point of the slip surface 119874 is a fixedpoint on the coordinates of (119909

0= 0 119910

0= 0) and the end point

119861 moves on the backfill surface 119892(119909) with the undeterminedcoordinates of (119909

1 119892(1199091)) The undetermined boundary is a

variational boundary The variational boundary refers to anunfixed boundary on which one or two ends respectivelymove on the given function

According to the variational method of the functionalwith constraints the following auxiliary functional 119869lowast isconstructed by Lagrangemultipliers to convert the functionalextremum problem under constraint conditions into a func-tional extremum problem without constraint conditions

119869lowast= int

1199091

0

119865119889119909

119865 = 1198650+ 12058211198651+ 12058221198652

(9)

wherein 1205821and 120582

2are Lagrange multipliers

Now the solution of the maximum of 119864119886is converted

into the solution of the maximum of auxiliary function 119869lowast

According to the necessary conditions for the existence ofthe extremum of auxiliary functional 119869lowast the function ofthe slip surface 119904(119909) and the function of the normal stressdistribution 120590(119909) on the slip surface must meet the followingrequirements

(1) The Euler differential equation of the auxiliary func-tion 119865 is

120597119865

120597120590

minus

119889

119889119909

(

120597119865

1205971205901015840) = 0 (10)

120597119865

120597119904

minus

119889

119889119909

(

120597119865

1205971199041015840) = 0 (11)

(2) The integral constraint equations are (7)(3) Boundary conditions are

Fixed boundary condition 119904 (0) = 0

Variational boundary condition 119904 (1199091) = 119892 (119909

1)

(12)

(4) Transversality condition at the variational boundaryis

(119865 minus 1199041015840 120597119865

1205971199041015840+ 1198921015840 120597119865

1205971199041015840)

10038161003816100381610038161003816100381610038161003816119909=1199091

= 0 (13)

4 Shock and Vibration

3 Variational Solution of SeismicActive Earth Pressure

31 Shape Function of the Slip Surface Thefollowing equationcan be obtained after (5) (7) and (9) are substituted into (10)

119889119904

119889119909

=

119909 minus 1198991119904 + 12058211198991+ 1205822

minus1198991119909 minus 119904 + 120582

1minus 12058221198991

(14)

Polar coordinate transformations 119903 cos 120579 = 119909 + 1205822and

119903 sin 120579 = 119904 minus 1205821are brought in For the calculation model in

polar coordinates see Figure 2 Equation (14) is transcribedas a differential equation

119889119903

119889120579

= minus1198991119903 (15)

The general solution of the differential equation is

119903 = 1199115119890minus1198991120579 (16)

wherein 1199115is an arbitrary integration constant

The fixed boundary condition 119904(0) = 0 shows that thepole of the polar coordinate is (119903

0 1205790) After it is substituted

into (16) the expression of the logarithmic spiral of the slipsurface can be obtained

119903 = 11990301198901198991(1205790minus120579)

(17)

wherein

1199030= radic1205822

1+ 1205822

2

1205790= minus arctan(1205821

1205822

) 1205822gt 0

1205790= minus120587 minus arctan(1205821

1205822

) 1205822le 0

(18)

32 Normal Stress Distribution on the Slip Surface After(5) (7) and (9) are substituted into (11) the following equa-tion can be obtained

21198991120590 + (119899

1119909 + 119904 minus 120582

1+ 11989911205822) 1205901015840minus (1 minus 119896

119881) 120574119909

minus (1 minus 119896119881) 1205822120574 + 2119888 + 119896

119867120574119904 minus 119896

1198671205821120574 = 0

(19)

Introducing the Polar coordinate transformations 119903 cos 120579 =

119909 + 1205822and 119903 sin 120579 = 119904 minus 120582

1 (19) is transcribed as a differential

equation119889120590

119889120579

minus 21198991120590 = 2119888 minus (1 minus 119896

119881) 12057411990301198901198991(1205790minus120579) cos 120579

+ 11989611986712057411990301198901198991(1205790minus120579) sin 120579

(20)

The general solution of the differential equation is (when 1198991

=

0)

120590 = 119890int120579

1205791

21198991119889120579(1199116+ int

120579

1205791

(2119888

minus (1 minus 119896119881) 12057411990301198901198991(1205790minus120579) cos 120579 + 119896

11986712057411990301198901198991(1205790minus120579) sin 120579)

sdot 119890int120579

1205791

minus21198991119889120579119889120579)

(21)

kHq(x)

(1 minus kV)q(x)

kHWG

(1 minus kV)WG

Ea120585H

x2 x1

120590(x)

120591(x)

y = s(x)

y = g(x)

120572

120575

120579120579B

O998400

Y998400

X998400

r

OX

B

A

H

Y

r0

1205790

Figure 2 Calculationmodel of seismic active earth pressure in polarcoordinates

wherein 1199116is an integration constant 120579

1is an arbitrary angle

1205791= 0 is available

120590 = 119911611989021198991120579minus

(1 minus 119896119881) 12057411990301198901198991(1205790minus120579)

1 + 91198992

1

(sin 120579 minus 31198991cos 120579

+ 3119899111989031198991120579) minus

11989611986712057411990301198901198991(1205790minus120579)

1 + 91198992

1

(31198991sin 120579 + cos 120579

minus 11989031198991120579) minus

119888

1198991

(22)

According to the transversality condition at the variationalboundary the normal stress distribution at 119861 of the slipsurface can be obtained120590 (1199091)

=

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(23)

After it is substituted into (22) the following equation isobtained

1199116= 119890minus211989911205791

119888

1198991

+

11989611986712057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(31198991sin 1205791+ cos 120579

1minus 119890311989911205791)

+

(1 minus 119896119881) 12057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(sin 1205791minus 31198991cos 1205791+ 31198991119890311989911205791)

+

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(24)

33 Optimal Solution of Seismic Active Earth Pressure Theanalysis carried out so far indicates that the maximum value

Shock and Vibration 5

of119864119886can be obtained according to a pair of functions 119903(120579) and

120590(120579) determined by two undetermined Lagrange multipliers1205821and 120582

2 The problem is converted into the search of a pair

of constants 1205821and 120582

2through the two constraint equations

The two constraint equationsmdash(7)mdashmust be satisfied

1198911(1205821 1205822) = int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2= 0

1198912(1205821 1205822) = int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4= 0

(25)

The following equation is always correct

1198911(1205821 1205822) = 0

1198912(1205821 1205822) = 0

lArrrArr Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822) = 0 (26)

Therefore two integral equationsmdash(25)mdashare equivalent tothe following equation

Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822)

= [int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2]

2

+ [int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4]

2

= 0

(27)

ForΦ(1205821 1205822) ge 0 so

min [Φ (1205821 1205822)] = 0 (28)

The above analysis indicates that the solution of the twoconstraint equationsmdash(7)mdashcan be obtained through thesolution of the minimum value of the function Φ (theminimum value is 0)When theminimum value ofΦ is not 0it shows that the sliding soil wedge cannot keep balance andis improper

In this paper the fminsearch function provided by MAT-LAB is used to find the optimal solution For this functionit only requires establishing a function module of Φ andgiving an initial value to search the minimum value of thefunction Fminsearch applies the derivative-free method tofind the minimum value of the unconstrained multivariablefunction which is generally called unconstrained nonlinearoptimization Fminsearch finds the minimum of a scalarfunction of several variables starting at an initial estimateTherefore one major drawback of this optimization tool isthat the search would be very slow if the initial estimateis improper In order to overcome this drawback and inthe consideration of the condition that the center of thelogarithmic spiral is usually at the top left corner of theretaining wall the initial value is 120582

1= 1205822

= 119867 in theoptimization analysis of this paper and the search is quiterapid

4 Calculated Result and Parameter Analysis

41 Interval Estimation of Seismic Active Earth PressureUnder the general conditions that the retaining wall is

times105

120585d = 0449 120585u = 0612

00

05

10

15

20

25

min

Φ

02 04 06 08 1000120585

Figure 3 The minimum value of the function Φ

inclined and coarse the backfill is cohesive soil and thebackfill surface is inclined with surcharge the interval ofthe position coefficient 120585 of the action point under seismicloading is numerically solved Besides the influence of theaction point position on the shape of the slip surface and themagnitude of seismic active earth pressure is studied

The retaining wall has a height of 10m with the slopeangle of thewall to vertical120572 = 10

∘ the friction angle betweensoil and wall 120575 = 15

∘ the unit weight of soil 120574 = 18 kNm3cohesion 119888 = 10 kPa internal friction angle 120593 = 20

∘ slopeangle of the backfill soil 120573 = 5

∘ surcharge 119902 = 50 kNm andseismic coefficients 119896

119867= 005 and 119896

119881= 00

Figure 3 shows the curve of the minimum value of thefunction Φ varying with the position coefficient 120585 of theaction point As shown in the figure the position coefficient120585 of the action point of seismic active earth pressure is in aninterval (lower limit value 120585

119889= 0449 upper limit value 120585

119906=

0612) In the interval the minimum value of the functionΦ

is 0 In other words when the action point of seismic activeearth pressure falls within the range of 449sim612m from thewall bottom the counterforce provided by the retaining wallcan balance the soil mass behind the wall otherwise the soilmass will inevitably lose its balance no matter how big thecounterforce provided by the retaining wall is

There is a one-to-one corresponding relation between themagnitude of seismic active earth pressure and the shapeof slip surface (Figure 5) For the curves varying with theposition coefficient 120585 of the action point of seismic activeearth pressure see Figures 3 and 4 At the lower limit of120585 the slip surface is a plane with the minimal value ofseismic active earth pressure As the action point movesupward the slip surface becomes a logarithmic spiral surfaceand its curvature is gradually increased that is in thelogarithmic spiral equation 119903

0 is gradually decreased and

the corresponding value of seismic active earth pressure isincreased in a nonlinear manner At the upper limit of 120585the seismic active earth pressure is maximal The minimumand maximum values can be as an interval estimation of theseismic active earth pressure under different wall-movementmodes of the retaining wall

6 Shock and Vibration

Table 1 Comparison of calculated results of active earth pressure with that of M-O

120572∘ 120573∘ 119896119867

119896119881

The proposed method M-O method119864aekN 120585 119864

119886kN 120585

0 0 0 0 3910sim4121 0265sim0408 3910 130 5 005 005 4246sim4502 0286sim0449 4245 130 10 01 01 4708sim4950 0306sim0490 4707 1310 0 005 01 4605sim4752 0327sim0408 4601 1310 5 01 0 5043sim5253 0327sim0449 5043 1310 10 0 005 5599sim5728 0367sim0469 5596 1320 0 01 005 5504sim5628 0347sim0408 5503 1320 5 0 01 6049sim6161 0367sim0429 6047 1320 10 005 0 6827sim6963 0388sim0469 6825 13

120585d = 0449

120585u = 0612

0

2

4

6

8

10

12

Y (m

)

2 4 6 8 10 120X (m)

Figure 4 The shape of the slip surface

42 Comparison with the M-OTheory Solution To verify thecorrectness of the calculationmethod proposed in this papera comparison is made between the calculated result of theproposedmethod and the solution ofM-O theory Accordingto the assumption of M-O theory the values of cohesion 119888

and the surcharge 119902 are both 0 in the contrastive analysis Forother parameters and specific comparison results see Table 1

When the slip surface is a plane the seismic activeearth pressure is the lower limit value of the interval inthe table This value is equivalent to the solution of seismicactive earth pressure calculated by the M-O theory Thesolution obtained in the proposed method is degraded intoM-O theory solution thereby proving the correctness of theproposed method

When the slip surface is a logarithmic spiral surfacethe seismic active earth pressure calculated by the M-Otheory is relatively small and the assumed action pointposition is lower than the actual action point position undermost possible wall-movement modes The underestimationof the magnitude and the action point position of seismicactive earth pressure cause the potential safety hazard ofoverturning of the retaining wall designed by M-O theory

43 Parameter Analysis In this section discussions aremade regarding the influence of relevant parameters onthe magnitude and the action point position of seismicactive earth pressure as well as the overturning momenton the retaining wall Such parameters include the slopeangle 120572 of the retaining wall the slope angle 120573 of backfillsurface the surcharge 119902 on the backfill surface the horizontalseismic coefficient 119896

119867 and the vertical seismic coefficient

119896119881

431 Influence of Parameters on the Action Point PositionThe influence of relevant parameters on the position coef-ficient 120585 of action point of seismic active earth pressure isshown in Figures 6(a)ndash6(c) With the increase of the slopeangle120573 of backfill surface the action point positionmoves upwith the increase of120572 the action point position slightlymovesdown with the increase of the horizontal seismic coefficient119896119867and the vertical seismic coefficient 119896

119881 the action point

positionmoves up wherein the horizontal seismic coefficient119896119867has a significant influence on it with the increase of the

surcharge 119902 the action point positionmoves up in a nonlinearway

432 Influence of Parameters on the Seismic Active EarthPressure 119864

119886119890 Figures 7(a)ndash7(c) show the influence of rele-

vant parameters on the dimensionless seismic active earthpressure 119864ae(120574119867

2) The seismic active earth pressure 119864ae

is increased with the increase of the slope angle 120572 of theretaining wall and the slope angle 120573 of backfill surface119864ae is increased with the increase of the horizontal seismiccoefficient 119896

119867and decreased with the increase of the vertical

seismic coefficient 119896119881 119864ae is increased linearly with the

increase of the dimensionless surcharge 119902(120574119867) of slopesurface

433 Influence of Parameters on the OverturningMoment119872119886

Figures 8(a)ndash8(c) show the influence of relevant parameterson the dimensionless overturning moment 119872

1198861205741198673 The

overturning moment 119872119886is increased with the increase of

Shock and Vibration 7

860

870

880

890

900

910

Eae

(kN

mminus

1 )

046 048 050 052 054 056 058 060 062044120585

Figure 5 The magnitude of seismic active earth pressure

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

045

050

055120585

060

065

070

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

053

054

120585

055

056

057

058

059

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

050

055

060120585

065

070

010 020 030 040 050000q120574H

(c) 119902

Figure 6 Effects of relevant parameters on 120585

the slope angle 120572 of the retaining wall and the slope angle120573 of backfill surface 119872

119886is increased with the increase of

the horizontal seismic coefficient 119896119867and decreased with the

increase of the vertical seismic coefficient 119896119881119872119886is increased

linearly with the increase of the dimensionless surcharge load119902(120574119867) of slope surface

5 Conclusion

The seismic active earth pressure is studied in the paperbased on the variational limit equilibrium method and thefollowing conclusions are obtained

(1) In the framework of limit equilibrium theory theisoperimetric model of functional extremum regarding the

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

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Shock and Vibration 3

It is assumed that the tangential stress 120591 and the normalstress 120590 on the slip surface are subject to the Mohr-Coulombfailure criterion

120591 = 119888 + 1198991120590 (4)

wherein 1198991= tan120593

23 Variational Limit Equilibrium Model of Seismic ActiveEarth Pressure The problem to solve the seismic active earthpressure 119864ae can now be stated as follows The seismic activeearth pressure119864ae is themaximumvalue of119864

119886 corresponding

to the most dangerous (critical) slip surface Realizing themaximum value of 119864

119886is to find the function of the slip

surface 119904(119909) and the function of normal stress distribution120590(119909) on the slip surface subject to the three equations ofLE ((1)sim(3)) Among these three equations (3) is changed asobjective function and the other equations are the constraintconditions according to the variational method

The following objective function 119869 can be obtained after(4) is substituted into (3) Obviously to solve the maximumof 119864119886means the same as to solve the maximum of objective

function 119869

119869 = int

1199091

0

1198650119889119909 (5)

wherein

119869 =

119864119886120585119867 cos 120575cos120572

+ 1199110

1198650= (11989911199041015840119909 + 119909 + 119904119904

1015840minus 1198991119904) 120590 + 119888119909119904

1015840minus 119888119904 minus (1 minus 119896

119881)

sdot [(119892 minus 119904) 120574119909 + 119902119909] + 119896119867119902119892 +

1

2

119896119867120574 (1198922minus 1199042)

1199110= int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119909 minus 119896

119867119902119892

minus

1

2

119896119867120574 (1198922minus 11989621199092) 119889119909 = const

(6)

Two constraint conditions can be obtained after (4) and (5)are substituted into (1) and (2)

int

1199091

0

(11991111198650+ 1198651) 119889119909 = 119911

2

int

1199091

0

(11991131198650+ 1198652) 119889119909 = 119911

4

(7)

wherein

1198651= (1198991minus 1199041015840) 120590 + 119888 minus 119896

119867[120574 (119892 minus 119904) + 119902]

1198652= (11989911199041015840+ 1) 120590 + 119888119904

1015840minus (1 minus 119896

119881) [119902 + 120574 (119892 minus 119904)]

1199111=

cos120572 cos (120572 + 120575)

120585119867 cos 120575= const

1199112= 11991101199111+ int

0

1199092

119896119867[120574 (119892 minus 119896119909) + 119902] 119889119909 = const

1199113=

cos120572 cos (120572 + 120575)

120585119867 cos 120575= const

1199114= 11991101199113+ int

0

1199092

(1 minus 119896119881) [120574 (119892 minus 119896119909) + 119902] 119889119909 = const

(8)

Equations (5) and (7) show the isoperimetric model of theconstrained variational extremum with an undeterminedboundary The starting point of the slip surface 119874 is a fixedpoint on the coordinates of (119909

0= 0 119910

0= 0) and the end point

119861 moves on the backfill surface 119892(119909) with the undeterminedcoordinates of (119909

1 119892(1199091)) The undetermined boundary is a

variational boundary The variational boundary refers to anunfixed boundary on which one or two ends respectivelymove on the given function

According to the variational method of the functionalwith constraints the following auxiliary functional 119869lowast isconstructed by Lagrangemultipliers to convert the functionalextremum problem under constraint conditions into a func-tional extremum problem without constraint conditions

119869lowast= int

1199091

0

119865119889119909

119865 = 1198650+ 12058211198651+ 12058221198652

(9)

wherein 1205821and 120582

2are Lagrange multipliers

Now the solution of the maximum of 119864119886is converted

into the solution of the maximum of auxiliary function 119869lowast

According to the necessary conditions for the existence ofthe extremum of auxiliary functional 119869lowast the function ofthe slip surface 119904(119909) and the function of the normal stressdistribution 120590(119909) on the slip surface must meet the followingrequirements

(1) The Euler differential equation of the auxiliary func-tion 119865 is

120597119865

120597120590

minus

119889

119889119909

(

120597119865

1205971205901015840) = 0 (10)

120597119865

120597119904

minus

119889

119889119909

(

120597119865

1205971199041015840) = 0 (11)

(2) The integral constraint equations are (7)(3) Boundary conditions are

Fixed boundary condition 119904 (0) = 0

Variational boundary condition 119904 (1199091) = 119892 (119909

1)

(12)

(4) Transversality condition at the variational boundaryis

(119865 minus 1199041015840 120597119865

1205971199041015840+ 1198921015840 120597119865

1205971199041015840)

10038161003816100381610038161003816100381610038161003816119909=1199091

= 0 (13)

4 Shock and Vibration

3 Variational Solution of SeismicActive Earth Pressure

31 Shape Function of the Slip Surface Thefollowing equationcan be obtained after (5) (7) and (9) are substituted into (10)

119889119904

119889119909

=

119909 minus 1198991119904 + 12058211198991+ 1205822

minus1198991119909 minus 119904 + 120582

1minus 12058221198991

(14)

Polar coordinate transformations 119903 cos 120579 = 119909 + 1205822and

119903 sin 120579 = 119904 minus 1205821are brought in For the calculation model in

polar coordinates see Figure 2 Equation (14) is transcribedas a differential equation

119889119903

119889120579

= minus1198991119903 (15)

The general solution of the differential equation is

119903 = 1199115119890minus1198991120579 (16)

wherein 1199115is an arbitrary integration constant

The fixed boundary condition 119904(0) = 0 shows that thepole of the polar coordinate is (119903

0 1205790) After it is substituted

into (16) the expression of the logarithmic spiral of the slipsurface can be obtained

119903 = 11990301198901198991(1205790minus120579)

(17)

wherein

1199030= radic1205822

1+ 1205822

2

1205790= minus arctan(1205821

1205822

) 1205822gt 0

1205790= minus120587 minus arctan(1205821

1205822

) 1205822le 0

(18)

32 Normal Stress Distribution on the Slip Surface After(5) (7) and (9) are substituted into (11) the following equa-tion can be obtained

21198991120590 + (119899

1119909 + 119904 minus 120582

1+ 11989911205822) 1205901015840minus (1 minus 119896

119881) 120574119909

minus (1 minus 119896119881) 1205822120574 + 2119888 + 119896

119867120574119904 minus 119896

1198671205821120574 = 0

(19)

Introducing the Polar coordinate transformations 119903 cos 120579 =

119909 + 1205822and 119903 sin 120579 = 119904 minus 120582

1 (19) is transcribed as a differential

equation119889120590

119889120579

minus 21198991120590 = 2119888 minus (1 minus 119896

119881) 12057411990301198901198991(1205790minus120579) cos 120579

+ 11989611986712057411990301198901198991(1205790minus120579) sin 120579

(20)

The general solution of the differential equation is (when 1198991

=

0)

120590 = 119890int120579

1205791

21198991119889120579(1199116+ int

120579

1205791

(2119888

minus (1 minus 119896119881) 12057411990301198901198991(1205790minus120579) cos 120579 + 119896

11986712057411990301198901198991(1205790minus120579) sin 120579)

sdot 119890int120579

1205791

minus21198991119889120579119889120579)

(21)

kHq(x)

(1 minus kV)q(x)

kHWG

(1 minus kV)WG

Ea120585H

x2 x1

120590(x)

120591(x)

y = s(x)

y = g(x)

120572

120575

120579120579B

O998400

Y998400

X998400

r

OX

B

A

H

Y

r0

1205790

Figure 2 Calculationmodel of seismic active earth pressure in polarcoordinates

wherein 1199116is an integration constant 120579

1is an arbitrary angle

1205791= 0 is available

120590 = 119911611989021198991120579minus

(1 minus 119896119881) 12057411990301198901198991(1205790minus120579)

1 + 91198992

1

(sin 120579 minus 31198991cos 120579

+ 3119899111989031198991120579) minus

11989611986712057411990301198901198991(1205790minus120579)

1 + 91198992

1

(31198991sin 120579 + cos 120579

minus 11989031198991120579) minus

119888

1198991

(22)

According to the transversality condition at the variationalboundary the normal stress distribution at 119861 of the slipsurface can be obtained120590 (1199091)

=

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(23)

After it is substituted into (22) the following equation isobtained

1199116= 119890minus211989911205791

119888

1198991

+

11989611986712057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(31198991sin 1205791+ cos 120579

1minus 119890311989911205791)

+

(1 minus 119896119881) 12057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(sin 1205791minus 31198991cos 1205791+ 31198991119890311989911205791)

+

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(24)

33 Optimal Solution of Seismic Active Earth Pressure Theanalysis carried out so far indicates that the maximum value

Shock and Vibration 5

of119864119886can be obtained according to a pair of functions 119903(120579) and

120590(120579) determined by two undetermined Lagrange multipliers1205821and 120582

2 The problem is converted into the search of a pair

of constants 1205821and 120582

2through the two constraint equations

The two constraint equationsmdash(7)mdashmust be satisfied

1198911(1205821 1205822) = int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2= 0

1198912(1205821 1205822) = int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4= 0

(25)

The following equation is always correct

1198911(1205821 1205822) = 0

1198912(1205821 1205822) = 0

lArrrArr Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822) = 0 (26)

Therefore two integral equationsmdash(25)mdashare equivalent tothe following equation

Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822)

= [int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2]

2

+ [int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4]

2

= 0

(27)

ForΦ(1205821 1205822) ge 0 so

min [Φ (1205821 1205822)] = 0 (28)

The above analysis indicates that the solution of the twoconstraint equationsmdash(7)mdashcan be obtained through thesolution of the minimum value of the function Φ (theminimum value is 0)When theminimum value ofΦ is not 0it shows that the sliding soil wedge cannot keep balance andis improper

In this paper the fminsearch function provided by MAT-LAB is used to find the optimal solution For this functionit only requires establishing a function module of Φ andgiving an initial value to search the minimum value of thefunction Fminsearch applies the derivative-free method tofind the minimum value of the unconstrained multivariablefunction which is generally called unconstrained nonlinearoptimization Fminsearch finds the minimum of a scalarfunction of several variables starting at an initial estimateTherefore one major drawback of this optimization tool isthat the search would be very slow if the initial estimateis improper In order to overcome this drawback and inthe consideration of the condition that the center of thelogarithmic spiral is usually at the top left corner of theretaining wall the initial value is 120582

1= 1205822

= 119867 in theoptimization analysis of this paper and the search is quiterapid

4 Calculated Result and Parameter Analysis

41 Interval Estimation of Seismic Active Earth PressureUnder the general conditions that the retaining wall is

times105

120585d = 0449 120585u = 0612

00

05

10

15

20

25

min

Φ

02 04 06 08 1000120585

Figure 3 The minimum value of the function Φ

inclined and coarse the backfill is cohesive soil and thebackfill surface is inclined with surcharge the interval ofthe position coefficient 120585 of the action point under seismicloading is numerically solved Besides the influence of theaction point position on the shape of the slip surface and themagnitude of seismic active earth pressure is studied

The retaining wall has a height of 10m with the slopeangle of thewall to vertical120572 = 10

∘ the friction angle betweensoil and wall 120575 = 15

∘ the unit weight of soil 120574 = 18 kNm3cohesion 119888 = 10 kPa internal friction angle 120593 = 20

∘ slopeangle of the backfill soil 120573 = 5

∘ surcharge 119902 = 50 kNm andseismic coefficients 119896

119867= 005 and 119896

119881= 00

Figure 3 shows the curve of the minimum value of thefunction Φ varying with the position coefficient 120585 of theaction point As shown in the figure the position coefficient120585 of the action point of seismic active earth pressure is in aninterval (lower limit value 120585

119889= 0449 upper limit value 120585

119906=

0612) In the interval the minimum value of the functionΦ

is 0 In other words when the action point of seismic activeearth pressure falls within the range of 449sim612m from thewall bottom the counterforce provided by the retaining wallcan balance the soil mass behind the wall otherwise the soilmass will inevitably lose its balance no matter how big thecounterforce provided by the retaining wall is

There is a one-to-one corresponding relation between themagnitude of seismic active earth pressure and the shapeof slip surface (Figure 5) For the curves varying with theposition coefficient 120585 of the action point of seismic activeearth pressure see Figures 3 and 4 At the lower limit of120585 the slip surface is a plane with the minimal value ofseismic active earth pressure As the action point movesupward the slip surface becomes a logarithmic spiral surfaceand its curvature is gradually increased that is in thelogarithmic spiral equation 119903

0 is gradually decreased and

the corresponding value of seismic active earth pressure isincreased in a nonlinear manner At the upper limit of 120585the seismic active earth pressure is maximal The minimumand maximum values can be as an interval estimation of theseismic active earth pressure under different wall-movementmodes of the retaining wall

6 Shock and Vibration

Table 1 Comparison of calculated results of active earth pressure with that of M-O

120572∘ 120573∘ 119896119867

119896119881

The proposed method M-O method119864aekN 120585 119864

119886kN 120585

0 0 0 0 3910sim4121 0265sim0408 3910 130 5 005 005 4246sim4502 0286sim0449 4245 130 10 01 01 4708sim4950 0306sim0490 4707 1310 0 005 01 4605sim4752 0327sim0408 4601 1310 5 01 0 5043sim5253 0327sim0449 5043 1310 10 0 005 5599sim5728 0367sim0469 5596 1320 0 01 005 5504sim5628 0347sim0408 5503 1320 5 0 01 6049sim6161 0367sim0429 6047 1320 10 005 0 6827sim6963 0388sim0469 6825 13

120585d = 0449

120585u = 0612

0

2

4

6

8

10

12

Y (m

)

2 4 6 8 10 120X (m)

Figure 4 The shape of the slip surface

42 Comparison with the M-OTheory Solution To verify thecorrectness of the calculationmethod proposed in this papera comparison is made between the calculated result of theproposedmethod and the solution ofM-O theory Accordingto the assumption of M-O theory the values of cohesion 119888

and the surcharge 119902 are both 0 in the contrastive analysis Forother parameters and specific comparison results see Table 1

When the slip surface is a plane the seismic activeearth pressure is the lower limit value of the interval inthe table This value is equivalent to the solution of seismicactive earth pressure calculated by the M-O theory Thesolution obtained in the proposed method is degraded intoM-O theory solution thereby proving the correctness of theproposed method

When the slip surface is a logarithmic spiral surfacethe seismic active earth pressure calculated by the M-Otheory is relatively small and the assumed action pointposition is lower than the actual action point position undermost possible wall-movement modes The underestimationof the magnitude and the action point position of seismicactive earth pressure cause the potential safety hazard ofoverturning of the retaining wall designed by M-O theory

43 Parameter Analysis In this section discussions aremade regarding the influence of relevant parameters onthe magnitude and the action point position of seismicactive earth pressure as well as the overturning momenton the retaining wall Such parameters include the slopeangle 120572 of the retaining wall the slope angle 120573 of backfillsurface the surcharge 119902 on the backfill surface the horizontalseismic coefficient 119896

119867 and the vertical seismic coefficient

119896119881

431 Influence of Parameters on the Action Point PositionThe influence of relevant parameters on the position coef-ficient 120585 of action point of seismic active earth pressure isshown in Figures 6(a)ndash6(c) With the increase of the slopeangle120573 of backfill surface the action point positionmoves upwith the increase of120572 the action point position slightlymovesdown with the increase of the horizontal seismic coefficient119896119867and the vertical seismic coefficient 119896

119881 the action point

positionmoves up wherein the horizontal seismic coefficient119896119867has a significant influence on it with the increase of the

surcharge 119902 the action point positionmoves up in a nonlinearway

432 Influence of Parameters on the Seismic Active EarthPressure 119864

119886119890 Figures 7(a)ndash7(c) show the influence of rele-

vant parameters on the dimensionless seismic active earthpressure 119864ae(120574119867

2) The seismic active earth pressure 119864ae

is increased with the increase of the slope angle 120572 of theretaining wall and the slope angle 120573 of backfill surface119864ae is increased with the increase of the horizontal seismiccoefficient 119896

119867and decreased with the increase of the vertical

seismic coefficient 119896119881 119864ae is increased linearly with the

increase of the dimensionless surcharge 119902(120574119867) of slopesurface

433 Influence of Parameters on the OverturningMoment119872119886

Figures 8(a)ndash8(c) show the influence of relevant parameterson the dimensionless overturning moment 119872

1198861205741198673 The

overturning moment 119872119886is increased with the increase of

Shock and Vibration 7

860

870

880

890

900

910

Eae

(kN

mminus

1 )

046 048 050 052 054 056 058 060 062044120585

Figure 5 The magnitude of seismic active earth pressure

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

045

050

055120585

060

065

070

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

053

054

120585

055

056

057

058

059

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

050

055

060120585

065

070

010 020 030 040 050000q120574H

(c) 119902

Figure 6 Effects of relevant parameters on 120585

the slope angle 120572 of the retaining wall and the slope angle120573 of backfill surface 119872

119886is increased with the increase of

the horizontal seismic coefficient 119896119867and decreased with the

increase of the vertical seismic coefficient 119896119881119872119886is increased

linearly with the increase of the dimensionless surcharge load119902(120574119867) of slope surface

5 Conclusion

The seismic active earth pressure is studied in the paperbased on the variational limit equilibrium method and thefollowing conclusions are obtained

(1) In the framework of limit equilibrium theory theisoperimetric model of functional extremum regarding the

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

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4 Shock and Vibration

3 Variational Solution of SeismicActive Earth Pressure

31 Shape Function of the Slip Surface Thefollowing equationcan be obtained after (5) (7) and (9) are substituted into (10)

119889119904

119889119909

=

119909 minus 1198991119904 + 12058211198991+ 1205822

minus1198991119909 minus 119904 + 120582

1minus 12058221198991

(14)

Polar coordinate transformations 119903 cos 120579 = 119909 + 1205822and

119903 sin 120579 = 119904 minus 1205821are brought in For the calculation model in

polar coordinates see Figure 2 Equation (14) is transcribedas a differential equation

119889119903

119889120579

= minus1198991119903 (15)

The general solution of the differential equation is

119903 = 1199115119890minus1198991120579 (16)

wherein 1199115is an arbitrary integration constant

The fixed boundary condition 119904(0) = 0 shows that thepole of the polar coordinate is (119903

0 1205790) After it is substituted

into (16) the expression of the logarithmic spiral of the slipsurface can be obtained

119903 = 11990301198901198991(1205790minus120579)

(17)

wherein

1199030= radic1205822

1+ 1205822

2

1205790= minus arctan(1205821

1205822

) 1205822gt 0

1205790= minus120587 minus arctan(1205821

1205822

) 1205822le 0

(18)

32 Normal Stress Distribution on the Slip Surface After(5) (7) and (9) are substituted into (11) the following equa-tion can be obtained

21198991120590 + (119899

1119909 + 119904 minus 120582

1+ 11989911205822) 1205901015840minus (1 minus 119896

119881) 120574119909

minus (1 minus 119896119881) 1205822120574 + 2119888 + 119896

119867120574119904 minus 119896

1198671205821120574 = 0

(19)

Introducing the Polar coordinate transformations 119903 cos 120579 =

119909 + 1205822and 119903 sin 120579 = 119904 minus 120582

1 (19) is transcribed as a differential

equation119889120590

119889120579

minus 21198991120590 = 2119888 minus (1 minus 119896

119881) 12057411990301198901198991(1205790minus120579) cos 120579

+ 11989611986712057411990301198901198991(1205790minus120579) sin 120579

(20)

The general solution of the differential equation is (when 1198991

=

0)

120590 = 119890int120579

1205791

21198991119889120579(1199116+ int

120579

1205791

(2119888

minus (1 minus 119896119881) 12057411990301198901198991(1205790minus120579) cos 120579 + 119896

11986712057411990301198901198991(1205790minus120579) sin 120579)

sdot 119890int120579

1205791

minus21198991119889120579119889120579)

(21)

kHq(x)

(1 minus kV)q(x)

kHWG

(1 minus kV)WG

Ea120585H

x2 x1

120590(x)

120591(x)

y = s(x)

y = g(x)

120572

120575

120579120579B

O998400

Y998400

X998400

r

OX

B

A

H

Y

r0

1205790

Figure 2 Calculationmodel of seismic active earth pressure in polarcoordinates

wherein 1199116is an integration constant 120579

1is an arbitrary angle

1205791= 0 is available

120590 = 119911611989021198991120579minus

(1 minus 119896119881) 12057411990301198901198991(1205790minus120579)

1 + 91198992

1

(sin 120579 minus 31198991cos 120579

+ 3119899111989031198991120579) minus

11989611986712057411990301198901198991(1205790minus120579)

1 + 91198992

1

(31198991sin 120579 + cos 120579

minus 11989031198991120579) minus

119888

1198991

(22)

According to the transversality condition at the variationalboundary the normal stress distribution at 119861 of the slipsurface can be obtained120590 (1199091)

=

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(23)

After it is substituted into (22) the following equation isobtained

1199116= 119890minus211989911205791

119888

1198991

+

11989611986712057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(31198991sin 1205791+ cos 120579

1minus 119890311989911205791)

+

(1 minus 119896119881) 12057411990301198901198991(1205790minus1205791)

1 + 91198992

1

(sin 1205791minus 31198991cos 1205791+ 31198991119890311989911205791)

+

sin 1205791(119888 minus 119896

119867119902 (1199091)) + cos 120579

1[(1 minus 119896

119881) 119902 (1199091) minus 119888119892

1015840(1199091)]

sin 1205791(1198921015840(1199091) minus 1198991) + cos 120579

1(11989911198921015840(1199091) + 1)

(24)

33 Optimal Solution of Seismic Active Earth Pressure Theanalysis carried out so far indicates that the maximum value

Shock and Vibration 5

of119864119886can be obtained according to a pair of functions 119903(120579) and

120590(120579) determined by two undetermined Lagrange multipliers1205821and 120582

2 The problem is converted into the search of a pair

of constants 1205821and 120582

2through the two constraint equations

The two constraint equationsmdash(7)mdashmust be satisfied

1198911(1205821 1205822) = int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2= 0

1198912(1205821 1205822) = int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4= 0

(25)

The following equation is always correct

1198911(1205821 1205822) = 0

1198912(1205821 1205822) = 0

lArrrArr Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822) = 0 (26)

Therefore two integral equationsmdash(25)mdashare equivalent tothe following equation

Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822)

= [int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2]

2

+ [int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4]

2

= 0

(27)

ForΦ(1205821 1205822) ge 0 so

min [Φ (1205821 1205822)] = 0 (28)

The above analysis indicates that the solution of the twoconstraint equationsmdash(7)mdashcan be obtained through thesolution of the minimum value of the function Φ (theminimum value is 0)When theminimum value ofΦ is not 0it shows that the sliding soil wedge cannot keep balance andis improper

In this paper the fminsearch function provided by MAT-LAB is used to find the optimal solution For this functionit only requires establishing a function module of Φ andgiving an initial value to search the minimum value of thefunction Fminsearch applies the derivative-free method tofind the minimum value of the unconstrained multivariablefunction which is generally called unconstrained nonlinearoptimization Fminsearch finds the minimum of a scalarfunction of several variables starting at an initial estimateTherefore one major drawback of this optimization tool isthat the search would be very slow if the initial estimateis improper In order to overcome this drawback and inthe consideration of the condition that the center of thelogarithmic spiral is usually at the top left corner of theretaining wall the initial value is 120582

1= 1205822

= 119867 in theoptimization analysis of this paper and the search is quiterapid

4 Calculated Result and Parameter Analysis

41 Interval Estimation of Seismic Active Earth PressureUnder the general conditions that the retaining wall is

times105

120585d = 0449 120585u = 0612

00

05

10

15

20

25

min

Φ

02 04 06 08 1000120585

Figure 3 The minimum value of the function Φ

inclined and coarse the backfill is cohesive soil and thebackfill surface is inclined with surcharge the interval ofthe position coefficient 120585 of the action point under seismicloading is numerically solved Besides the influence of theaction point position on the shape of the slip surface and themagnitude of seismic active earth pressure is studied

The retaining wall has a height of 10m with the slopeangle of thewall to vertical120572 = 10

∘ the friction angle betweensoil and wall 120575 = 15

∘ the unit weight of soil 120574 = 18 kNm3cohesion 119888 = 10 kPa internal friction angle 120593 = 20

∘ slopeangle of the backfill soil 120573 = 5

∘ surcharge 119902 = 50 kNm andseismic coefficients 119896

119867= 005 and 119896

119881= 00

Figure 3 shows the curve of the minimum value of thefunction Φ varying with the position coefficient 120585 of theaction point As shown in the figure the position coefficient120585 of the action point of seismic active earth pressure is in aninterval (lower limit value 120585

119889= 0449 upper limit value 120585

119906=

0612) In the interval the minimum value of the functionΦ

is 0 In other words when the action point of seismic activeearth pressure falls within the range of 449sim612m from thewall bottom the counterforce provided by the retaining wallcan balance the soil mass behind the wall otherwise the soilmass will inevitably lose its balance no matter how big thecounterforce provided by the retaining wall is

There is a one-to-one corresponding relation between themagnitude of seismic active earth pressure and the shapeof slip surface (Figure 5) For the curves varying with theposition coefficient 120585 of the action point of seismic activeearth pressure see Figures 3 and 4 At the lower limit of120585 the slip surface is a plane with the minimal value ofseismic active earth pressure As the action point movesupward the slip surface becomes a logarithmic spiral surfaceand its curvature is gradually increased that is in thelogarithmic spiral equation 119903

0 is gradually decreased and

the corresponding value of seismic active earth pressure isincreased in a nonlinear manner At the upper limit of 120585the seismic active earth pressure is maximal The minimumand maximum values can be as an interval estimation of theseismic active earth pressure under different wall-movementmodes of the retaining wall

6 Shock and Vibration

Table 1 Comparison of calculated results of active earth pressure with that of M-O

120572∘ 120573∘ 119896119867

119896119881

The proposed method M-O method119864aekN 120585 119864

119886kN 120585

0 0 0 0 3910sim4121 0265sim0408 3910 130 5 005 005 4246sim4502 0286sim0449 4245 130 10 01 01 4708sim4950 0306sim0490 4707 1310 0 005 01 4605sim4752 0327sim0408 4601 1310 5 01 0 5043sim5253 0327sim0449 5043 1310 10 0 005 5599sim5728 0367sim0469 5596 1320 0 01 005 5504sim5628 0347sim0408 5503 1320 5 0 01 6049sim6161 0367sim0429 6047 1320 10 005 0 6827sim6963 0388sim0469 6825 13

120585d = 0449

120585u = 0612

0

2

4

6

8

10

12

Y (m

)

2 4 6 8 10 120X (m)

Figure 4 The shape of the slip surface

42 Comparison with the M-OTheory Solution To verify thecorrectness of the calculationmethod proposed in this papera comparison is made between the calculated result of theproposedmethod and the solution ofM-O theory Accordingto the assumption of M-O theory the values of cohesion 119888

and the surcharge 119902 are both 0 in the contrastive analysis Forother parameters and specific comparison results see Table 1

When the slip surface is a plane the seismic activeearth pressure is the lower limit value of the interval inthe table This value is equivalent to the solution of seismicactive earth pressure calculated by the M-O theory Thesolution obtained in the proposed method is degraded intoM-O theory solution thereby proving the correctness of theproposed method

When the slip surface is a logarithmic spiral surfacethe seismic active earth pressure calculated by the M-Otheory is relatively small and the assumed action pointposition is lower than the actual action point position undermost possible wall-movement modes The underestimationof the magnitude and the action point position of seismicactive earth pressure cause the potential safety hazard ofoverturning of the retaining wall designed by M-O theory

43 Parameter Analysis In this section discussions aremade regarding the influence of relevant parameters onthe magnitude and the action point position of seismicactive earth pressure as well as the overturning momenton the retaining wall Such parameters include the slopeangle 120572 of the retaining wall the slope angle 120573 of backfillsurface the surcharge 119902 on the backfill surface the horizontalseismic coefficient 119896

119867 and the vertical seismic coefficient

119896119881

431 Influence of Parameters on the Action Point PositionThe influence of relevant parameters on the position coef-ficient 120585 of action point of seismic active earth pressure isshown in Figures 6(a)ndash6(c) With the increase of the slopeangle120573 of backfill surface the action point positionmoves upwith the increase of120572 the action point position slightlymovesdown with the increase of the horizontal seismic coefficient119896119867and the vertical seismic coefficient 119896

119881 the action point

positionmoves up wherein the horizontal seismic coefficient119896119867has a significant influence on it with the increase of the

surcharge 119902 the action point positionmoves up in a nonlinearway

432 Influence of Parameters on the Seismic Active EarthPressure 119864

119886119890 Figures 7(a)ndash7(c) show the influence of rele-

vant parameters on the dimensionless seismic active earthpressure 119864ae(120574119867

2) The seismic active earth pressure 119864ae

is increased with the increase of the slope angle 120572 of theretaining wall and the slope angle 120573 of backfill surface119864ae is increased with the increase of the horizontal seismiccoefficient 119896

119867and decreased with the increase of the vertical

seismic coefficient 119896119881 119864ae is increased linearly with the

increase of the dimensionless surcharge 119902(120574119867) of slopesurface

433 Influence of Parameters on the OverturningMoment119872119886

Figures 8(a)ndash8(c) show the influence of relevant parameterson the dimensionless overturning moment 119872

1198861205741198673 The

overturning moment 119872119886is increased with the increase of

Shock and Vibration 7

860

870

880

890

900

910

Eae

(kN

mminus

1 )

046 048 050 052 054 056 058 060 062044120585

Figure 5 The magnitude of seismic active earth pressure

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

045

050

055120585

060

065

070

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

053

054

120585

055

056

057

058

059

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

050

055

060120585

065

070

010 020 030 040 050000q120574H

(c) 119902

Figure 6 Effects of relevant parameters on 120585

the slope angle 120572 of the retaining wall and the slope angle120573 of backfill surface 119872

119886is increased with the increase of

the horizontal seismic coefficient 119896119867and decreased with the

increase of the vertical seismic coefficient 119896119881119872119886is increased

linearly with the increase of the dimensionless surcharge load119902(120574119867) of slope surface

5 Conclusion

The seismic active earth pressure is studied in the paperbased on the variational limit equilibrium method and thefollowing conclusions are obtained

(1) In the framework of limit equilibrium theory theisoperimetric model of functional extremum regarding the

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

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International Journal of

Shock and Vibration 5

of119864119886can be obtained according to a pair of functions 119903(120579) and

120590(120579) determined by two undetermined Lagrange multipliers1205821and 120582

2 The problem is converted into the search of a pair

of constants 1205821and 120582

2through the two constraint equations

The two constraint equationsmdash(7)mdashmust be satisfied

1198911(1205821 1205822) = int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2= 0

1198912(1205821 1205822) = int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4= 0

(25)

The following equation is always correct

1198911(1205821 1205822) = 0

1198912(1205821 1205822) = 0

lArrrArr Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822) = 0 (26)

Therefore two integral equationsmdash(25)mdashare equivalent tothe following equation

Φ(1205821 1205822) = 1198912

1(1205821 1205822) + 1198912

2(1205821 1205822)

= [int

1199091

0

(11991111198650+ 1198651) 119889119909 minus 119911

2]

2

+ [int

1199091

0

(11991131198650+ 1198652) 119889119909 minus 119911

4]

2

= 0

(27)

ForΦ(1205821 1205822) ge 0 so

min [Φ (1205821 1205822)] = 0 (28)

The above analysis indicates that the solution of the twoconstraint equationsmdash(7)mdashcan be obtained through thesolution of the minimum value of the function Φ (theminimum value is 0)When theminimum value ofΦ is not 0it shows that the sliding soil wedge cannot keep balance andis improper

In this paper the fminsearch function provided by MAT-LAB is used to find the optimal solution For this functionit only requires establishing a function module of Φ andgiving an initial value to search the minimum value of thefunction Fminsearch applies the derivative-free method tofind the minimum value of the unconstrained multivariablefunction which is generally called unconstrained nonlinearoptimization Fminsearch finds the minimum of a scalarfunction of several variables starting at an initial estimateTherefore one major drawback of this optimization tool isthat the search would be very slow if the initial estimateis improper In order to overcome this drawback and inthe consideration of the condition that the center of thelogarithmic spiral is usually at the top left corner of theretaining wall the initial value is 120582

1= 1205822

= 119867 in theoptimization analysis of this paper and the search is quiterapid

4 Calculated Result and Parameter Analysis

41 Interval Estimation of Seismic Active Earth PressureUnder the general conditions that the retaining wall is

times105

120585d = 0449 120585u = 0612

00

05

10

15

20

25

min

Φ

02 04 06 08 1000120585

Figure 3 The minimum value of the function Φ

inclined and coarse the backfill is cohesive soil and thebackfill surface is inclined with surcharge the interval ofthe position coefficient 120585 of the action point under seismicloading is numerically solved Besides the influence of theaction point position on the shape of the slip surface and themagnitude of seismic active earth pressure is studied

The retaining wall has a height of 10m with the slopeangle of thewall to vertical120572 = 10

∘ the friction angle betweensoil and wall 120575 = 15

∘ the unit weight of soil 120574 = 18 kNm3cohesion 119888 = 10 kPa internal friction angle 120593 = 20

∘ slopeangle of the backfill soil 120573 = 5

∘ surcharge 119902 = 50 kNm andseismic coefficients 119896

119867= 005 and 119896

119881= 00

Figure 3 shows the curve of the minimum value of thefunction Φ varying with the position coefficient 120585 of theaction point As shown in the figure the position coefficient120585 of the action point of seismic active earth pressure is in aninterval (lower limit value 120585

119889= 0449 upper limit value 120585

119906=

0612) In the interval the minimum value of the functionΦ

is 0 In other words when the action point of seismic activeearth pressure falls within the range of 449sim612m from thewall bottom the counterforce provided by the retaining wallcan balance the soil mass behind the wall otherwise the soilmass will inevitably lose its balance no matter how big thecounterforce provided by the retaining wall is

There is a one-to-one corresponding relation between themagnitude of seismic active earth pressure and the shapeof slip surface (Figure 5) For the curves varying with theposition coefficient 120585 of the action point of seismic activeearth pressure see Figures 3 and 4 At the lower limit of120585 the slip surface is a plane with the minimal value ofseismic active earth pressure As the action point movesupward the slip surface becomes a logarithmic spiral surfaceand its curvature is gradually increased that is in thelogarithmic spiral equation 119903

0 is gradually decreased and

the corresponding value of seismic active earth pressure isincreased in a nonlinear manner At the upper limit of 120585the seismic active earth pressure is maximal The minimumand maximum values can be as an interval estimation of theseismic active earth pressure under different wall-movementmodes of the retaining wall

6 Shock and Vibration

Table 1 Comparison of calculated results of active earth pressure with that of M-O

120572∘ 120573∘ 119896119867

119896119881

The proposed method M-O method119864aekN 120585 119864

119886kN 120585

0 0 0 0 3910sim4121 0265sim0408 3910 130 5 005 005 4246sim4502 0286sim0449 4245 130 10 01 01 4708sim4950 0306sim0490 4707 1310 0 005 01 4605sim4752 0327sim0408 4601 1310 5 01 0 5043sim5253 0327sim0449 5043 1310 10 0 005 5599sim5728 0367sim0469 5596 1320 0 01 005 5504sim5628 0347sim0408 5503 1320 5 0 01 6049sim6161 0367sim0429 6047 1320 10 005 0 6827sim6963 0388sim0469 6825 13

120585d = 0449

120585u = 0612

0

2

4

6

8

10

12

Y (m

)

2 4 6 8 10 120X (m)

Figure 4 The shape of the slip surface

42 Comparison with the M-OTheory Solution To verify thecorrectness of the calculationmethod proposed in this papera comparison is made between the calculated result of theproposedmethod and the solution ofM-O theory Accordingto the assumption of M-O theory the values of cohesion 119888

and the surcharge 119902 are both 0 in the contrastive analysis Forother parameters and specific comparison results see Table 1

When the slip surface is a plane the seismic activeearth pressure is the lower limit value of the interval inthe table This value is equivalent to the solution of seismicactive earth pressure calculated by the M-O theory Thesolution obtained in the proposed method is degraded intoM-O theory solution thereby proving the correctness of theproposed method

When the slip surface is a logarithmic spiral surfacethe seismic active earth pressure calculated by the M-Otheory is relatively small and the assumed action pointposition is lower than the actual action point position undermost possible wall-movement modes The underestimationof the magnitude and the action point position of seismicactive earth pressure cause the potential safety hazard ofoverturning of the retaining wall designed by M-O theory

43 Parameter Analysis In this section discussions aremade regarding the influence of relevant parameters onthe magnitude and the action point position of seismicactive earth pressure as well as the overturning momenton the retaining wall Such parameters include the slopeangle 120572 of the retaining wall the slope angle 120573 of backfillsurface the surcharge 119902 on the backfill surface the horizontalseismic coefficient 119896

119867 and the vertical seismic coefficient

119896119881

431 Influence of Parameters on the Action Point PositionThe influence of relevant parameters on the position coef-ficient 120585 of action point of seismic active earth pressure isshown in Figures 6(a)ndash6(c) With the increase of the slopeangle120573 of backfill surface the action point positionmoves upwith the increase of120572 the action point position slightlymovesdown with the increase of the horizontal seismic coefficient119896119867and the vertical seismic coefficient 119896

119881 the action point

positionmoves up wherein the horizontal seismic coefficient119896119867has a significant influence on it with the increase of the

surcharge 119902 the action point positionmoves up in a nonlinearway

432 Influence of Parameters on the Seismic Active EarthPressure 119864

119886119890 Figures 7(a)ndash7(c) show the influence of rele-

vant parameters on the dimensionless seismic active earthpressure 119864ae(120574119867

2) The seismic active earth pressure 119864ae

is increased with the increase of the slope angle 120572 of theretaining wall and the slope angle 120573 of backfill surface119864ae is increased with the increase of the horizontal seismiccoefficient 119896

119867and decreased with the increase of the vertical

seismic coefficient 119896119881 119864ae is increased linearly with the

increase of the dimensionless surcharge 119902(120574119867) of slopesurface

433 Influence of Parameters on the OverturningMoment119872119886

Figures 8(a)ndash8(c) show the influence of relevant parameterson the dimensionless overturning moment 119872

1198861205741198673 The

overturning moment 119872119886is increased with the increase of

Shock and Vibration 7

860

870

880

890

900

910

Eae

(kN

mminus

1 )

046 048 050 052 054 056 058 060 062044120585

Figure 5 The magnitude of seismic active earth pressure

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

045

050

055120585

060

065

070

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

053

054

120585

055

056

057

058

059

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

050

055

060120585

065

070

010 020 030 040 050000q120574H

(c) 119902

Figure 6 Effects of relevant parameters on 120585

the slope angle 120572 of the retaining wall and the slope angle120573 of backfill surface 119872

119886is increased with the increase of

the horizontal seismic coefficient 119896119867and decreased with the

increase of the vertical seismic coefficient 119896119881119872119886is increased

linearly with the increase of the dimensionless surcharge load119902(120574119867) of slope surface

5 Conclusion

The seismic active earth pressure is studied in the paperbased on the variational limit equilibrium method and thefollowing conclusions are obtained

(1) In the framework of limit equilibrium theory theisoperimetric model of functional extremum regarding the

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

Table 1 Comparison of calculated results of active earth pressure with that of M-O

120572∘ 120573∘ 119896119867

119896119881

The proposed method M-O method119864aekN 120585 119864

119886kN 120585

0 0 0 0 3910sim4121 0265sim0408 3910 130 5 005 005 4246sim4502 0286sim0449 4245 130 10 01 01 4708sim4950 0306sim0490 4707 1310 0 005 01 4605sim4752 0327sim0408 4601 1310 5 01 0 5043sim5253 0327sim0449 5043 1310 10 0 005 5599sim5728 0367sim0469 5596 1320 0 01 005 5504sim5628 0347sim0408 5503 1320 5 0 01 6049sim6161 0367sim0429 6047 1320 10 005 0 6827sim6963 0388sim0469 6825 13

120585d = 0449

120585u = 0612

0

2

4

6

8

10

12

Y (m

)

2 4 6 8 10 120X (m)

Figure 4 The shape of the slip surface

42 Comparison with the M-OTheory Solution To verify thecorrectness of the calculationmethod proposed in this papera comparison is made between the calculated result of theproposedmethod and the solution ofM-O theory Accordingto the assumption of M-O theory the values of cohesion 119888

and the surcharge 119902 are both 0 in the contrastive analysis Forother parameters and specific comparison results see Table 1

When the slip surface is a plane the seismic activeearth pressure is the lower limit value of the interval inthe table This value is equivalent to the solution of seismicactive earth pressure calculated by the M-O theory Thesolution obtained in the proposed method is degraded intoM-O theory solution thereby proving the correctness of theproposed method

When the slip surface is a logarithmic spiral surfacethe seismic active earth pressure calculated by the M-Otheory is relatively small and the assumed action pointposition is lower than the actual action point position undermost possible wall-movement modes The underestimationof the magnitude and the action point position of seismicactive earth pressure cause the potential safety hazard ofoverturning of the retaining wall designed by M-O theory

43 Parameter Analysis In this section discussions aremade regarding the influence of relevant parameters onthe magnitude and the action point position of seismicactive earth pressure as well as the overturning momenton the retaining wall Such parameters include the slopeangle 120572 of the retaining wall the slope angle 120573 of backfillsurface the surcharge 119902 on the backfill surface the horizontalseismic coefficient 119896

119867 and the vertical seismic coefficient

119896119881

431 Influence of Parameters on the Action Point PositionThe influence of relevant parameters on the position coef-ficient 120585 of action point of seismic active earth pressure isshown in Figures 6(a)ndash6(c) With the increase of the slopeangle120573 of backfill surface the action point positionmoves upwith the increase of120572 the action point position slightlymovesdown with the increase of the horizontal seismic coefficient119896119867and the vertical seismic coefficient 119896

119881 the action point

positionmoves up wherein the horizontal seismic coefficient119896119867has a significant influence on it with the increase of the

surcharge 119902 the action point positionmoves up in a nonlinearway

432 Influence of Parameters on the Seismic Active EarthPressure 119864

119886119890 Figures 7(a)ndash7(c) show the influence of rele-

vant parameters on the dimensionless seismic active earthpressure 119864ae(120574119867

2) The seismic active earth pressure 119864ae

is increased with the increase of the slope angle 120572 of theretaining wall and the slope angle 120573 of backfill surface119864ae is increased with the increase of the horizontal seismiccoefficient 119896

119867and decreased with the increase of the vertical

seismic coefficient 119896119881 119864ae is increased linearly with the

increase of the dimensionless surcharge 119902(120574119867) of slopesurface

433 Influence of Parameters on the OverturningMoment119872119886

Figures 8(a)ndash8(c) show the influence of relevant parameterson the dimensionless overturning moment 119872

1198861205741198673 The

overturning moment 119872119886is increased with the increase of

Shock and Vibration 7

860

870

880

890

900

910

Eae

(kN

mminus

1 )

046 048 050 052 054 056 058 060 062044120585

Figure 5 The magnitude of seismic active earth pressure

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

045

050

055120585

060

065

070

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

053

054

120585

055

056

057

058

059

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

050

055

060120585

065

070

010 020 030 040 050000q120574H

(c) 119902

Figure 6 Effects of relevant parameters on 120585

the slope angle 120572 of the retaining wall and the slope angle120573 of backfill surface 119872

119886is increased with the increase of

the horizontal seismic coefficient 119896119867and decreased with the

increase of the vertical seismic coefficient 119896119881119872119886is increased

linearly with the increase of the dimensionless surcharge load119902(120574119867) of slope surface

5 Conclusion

The seismic active earth pressure is studied in the paperbased on the variational limit equilibrium method and thefollowing conclusions are obtained

(1) In the framework of limit equilibrium theory theisoperimetric model of functional extremum regarding the

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

860

870

880

890

900

910

Eae

(kN

mminus

1 )

046 048 050 052 054 056 058 060 062044120585

Figure 5 The magnitude of seismic active earth pressure

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

045

050

055120585

060

065

070

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

053

054

120585

055

056

057

058

059

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

050

055

060120585

065

070

010 020 030 040 050000q120574H

(c) 119902

Figure 6 Effects of relevant parameters on 120585

the slope angle 120572 of the retaining wall and the slope angle120573 of backfill surface 119872

119886is increased with the increase of

the horizontal seismic coefficient 119896119867and decreased with the

increase of the vertical seismic coefficient 119896119881119872119886is increased

linearly with the increase of the dimensionless surcharge load119902(120574119867) of slope surface

5 Conclusion

The seismic active earth pressure is studied in the paperbased on the variational limit equilibrium method and thefollowing conclusions are obtained

(1) In the framework of limit equilibrium theory theisoperimetric model of functional extremum regarding the

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

120573 = 0∘

120573 = 10∘

120573 = 20∘

5 10 15 200120572 (∘)

015

020

025

030

035

040

045E

ae120574H

2

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

002 004 006 008 010000kH (g)

020

022

024

026

028

030

032

Eae120574H

2

(b) 119896119867 and 119896119881

020

030

040

050

060

010 020 030 040 050000q120574H

Eae120574H

2

(c) 119902

Figure 7 Effects of relevant parameters on 119864ae

seismic active earth pressure under general conditions (theretaining wall is inclined and coarse the backfill is cohesivesoil the backfill surface is a curved surface with nonuniformsurcharge) is deduced

(2) With the introduction of Lagrange multipliers andin combination with the necessary conditions required forthe existence of functional extremum the solution of seis-mic active earth pressure is transcribed as an optimizationproblem of two undetermined Lagrange multipliers with thefminsearch function provided by MATLAB an optimizationsolution of two Lagrange multipliers is founded to obtain themagnitude and the action point position of seismic activeearth pressure as well as the shape of slip surface

(3) With the continuous changes of wall-movementmodes of the retaining wall the position coefficient 120585 of theaction point of seismic active earth pressure is an intervalwithlower and upper limit values When 120585 is the lower limit valuethe slip surface is a plane and the seismic active earth pressureisminimal As the action pointmoves upward the slip surfacebecomes a logarithmic spiral surface and its curvature isgradually increased with the corresponding value of seismicactive earth pressure increased in a nonlinear manner At the

upper limit of 120585 the seismic active earth pressure is maximalthe minimum and maximum values can be estimated as aninterval of the seismic active earth pressure under differentwall-movement modes of the retaining wall

(4) The contrastive analysis shows that when the slipsurface is a plane the solution of seismic active earth pressureobtained by the proposed method is in line with the M-Otheory solution verifying the correctness of the proposedmethod when the slip surface is a logarithmic spiral surfacethe underestimation of the magnitude and the action pointposition of seismic active earth pressure cause the potentialsafety hazard of overturning of the retaining wall designed byM-O theory

(5) The parameter analysis shows that the action pointposition moves up with the increase of the slope angle 120573 ofbackfill surface the horizontal seismic coefficient 119896

119867 and

the surcharge 119902 on the backfill surface However the verticalseismic coefficient 119896

119881and the slope angle 120572 of the retaining

wall have little influence on it so they can be neglectedThe seismic active earth pressure 119864ae and the overturningmoment119872

119886are increased with the increase of the slope angle

120572 of the retainingwall the slope angle120573 of backfill surface the

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

120573 = 0∘

120573 = 10∘

120573 = 20∘

005

010

015

020

025

030M

a120574H

3

5 10 15 200120572 (∘)

(a) 120572 and 120573

kV = 00

kV = 005

kV = 01

010

011

012

013

014

015

016

017

018

Ma120574H

3

002 004 006 008 010000kH (g)

(b) 119896119867 and 119896119881

010 020 030 040 050000q120574H

010

020

030

040

Ma120574H

3

(c) 119902

Figure 8 Effects of relevant parameters on119872119886

horizontal seismic coefficient 119896119867 and the surcharge 119902 on the

backfill surface and decreasedwith the increase of the verticalseismic coefficient 119896

119881

Competing Interests

The authors declare that they have no competing interests

References

[1] S Caltabiano E Cascone and M Maugeri ldquoStatic and seismiclimit equilibrium analysis of sliding retaining walls underdifferent surcharge conditionsrdquo Soil Dynamics and EarthquakeEngineering vol 37 pp 38ndash55 2012

[2] S Saran and R P Gupta ldquoSeismic earth pressures behindretaining wallsrdquo Indian Geotechnical Journal vol 33 no 3 pp195ndash213 2003

[3] G Mylonakis P Kloukinas and C Papantonopoulos ldquoAnalternative to theMononobendashOkabe equations for seismic earthpressuresrdquo Soil Dynamics and Earthquake Engineering vol 27no 10 pp 957ndash969 2007

[4] 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

[5] M Yazdani A Azad A H Farshi and S Talatahari ldquoExtendedlsquomononobe-okabersquo method for seismic design of retainingwallsrdquo Journal of Applied Mathematics vol 2013 Article ID136132 10 pages 2013

[6] Y-L Lin W-M Leng G-L Yang L-H Zhao L Li and J-S Yang ldquoSeismic active earth pressure of cohesive-frictionalsoil on retaining wall based on a slice analysis methodrdquo SoilDynamics and Earthquake Engineering vol 70 pp 133ndash1472015

[7] J Kopascy ldquoUber die Bruchflachen und Bruchspannungen inden Erdbautenrdquo in Gedenkbuch Fur Prof Dr J Jaky K SzechyEd pp 81ndash99 Akademiaikiado Budapest Hungary 1955

[8] J Kopascy ldquoThree-dimensional stress distribution and slipsurfaces in earth works at rupturerdquo in Proceedings of the 4thInternational Conference on Soil Mechanics and FoundationsEngineering vol 1 pp 339ndash342 London UK 1957

[9] J Kopascy ldquoDistribution des contraintesala rupture forme dela surface de glissement et hauteur theorique des talusrdquo inProceedings of the 5th International Conference on SoilMechanics

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

and Foundations Engineering vol 2 pp 641ndash650 Paris France1961

[10] W F Chen and N Snitbhan ldquoOn slip surface and slope stabilityanalysisrdquo Soils and Foundations vol 15 no 3 pp 41ndash49 1975

[11] M Garber and R Baker ldquoBearing capacity by variationalmethodrdquo Journal of Geotechnical Engineering Division vol 103no 11 pp 1209ndash1225 1977

[12] M Garber and R Baker ldquoExtreme-value problems of limitingequilibriumrdquo Journal of the Geotechnical Engineering Divisionvol 105 no 10 pp 1155ndash1171 1979

[13] A J Spencer and T C OrsquoMahony ldquoAn application of thecalculus of variations to rectilinear flow of granular materialsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 9 no 3 pp 225ndash235 1985

[14] D Leshchinsky ldquoSlope stability analysis generalized approachrdquoJournal of Geotechnical Engineering vol 116 no 5 pp 851ndash8671990

[15] D Leshchinsky and C-C Huang ldquoGeneralized three-dimensional slope-stability analysisrdquo Journal of GeotechnicalEngineering vol 118 no 11 pp 1748ndash1764 1992

[16] R Baker ldquoSufficient conditions for existence of physicallysignificant solutions in limiting equilibrium slope stabilityanalysisrdquo International Journal of Solids and Structures vol 40no 13-14 pp 3717ndash3735 2003

[17] R Baker ldquoStability chart for zero tensile strength Hoek-Brownmaterialsmdashthe variational solution and its engineering implica-tionsrdquo Soils and Foundations vol 44 no 3 pp 125ndash132 2004

[18] R Baker ldquoVariational slope stability analysis of materials withnonlinear failure criterionrdquo Electronic Journal of GeotechnicalEngineering vol 10 pp 1ndash22 2005

[19] R Baker ldquoA relation between safety factors with respect tostrength and height of slopesrdquo Computers and Geotechnics vol33 no 4-5 pp 275ndash277 2006

[20] L-YWuandY-F Tsai ldquoVariational stability analysis of cohesiveslope by applying boundary integral equation methodrdquo Journalof Mechanics vol 21 no 3 pp 187ndash198 2005

[21] Y M Cheng D Z Li N Li Y Y Lee and S K Au ldquoSolutionof some engineering partial differential equations governed bythe minimal of a functional by global optimization methodrdquoJournal of Mechanics vol 29 no 3 pp 507ndash516 2012

[22] X Li ldquoBearing capacity factors for eccentrically loaded stripfootings using variational analysisrdquo Mathematical Problems inEngineering vol 2013 Article ID 640273 17 pages 2013

[23] A H Soubra and R Kastner ldquoInfluence of seepage flow onthe passive earth pressuresrdquo in Proceeding of the InternationalConference on Retaining Structures pp 67ndash76 ICE AmsterdamThe Netherlands 1992

[24] A-H Soubra R Kastner and A Benmansour ldquoPassive earthpressures in the presence of hydraulic gradientsrdquo Geotechniquevol 49 no 3 pp 319ndash330 1999

[25] O Puła W Puła and A Wolny ldquoOn the variational solutionof a limiting equilibrium problem involving an anchored wallrdquoComputers and Geotechnics vol 32 no 2 pp 107ndash121 2005

[26] L Xinggao and L Weining ldquoStudy on the action of theactive earth pressure by variational limit equilibrium methodrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 34 no 10 pp 991ndash1008 2010

[27] M Shaojun Study on Calculation of Earth Pressure behindRetaining Wall under Static and Dynamic Loads and Its RelatedIssues Zhejiang University Zhejiang China 2012 (Chinese)

International Journal of

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of