mononobe okabe

Green et al. (2003), Proceedings of The Sixth US Conference and Workshop on Lifeline EarthquakeEngineering (TCLEE2003), ASCE, August 10-13, 2003, Long Beach, CA

Paper Number 87:

Seismically Induced Lateral Earth Pressures on a CantileverRetaining Wall

Russell A. Green1, C. Guney Olgun2, Robert M. Ebeling3, and Wanda I. Cameron4

AbstractA series of non-linear dynamic response analyses of a cantilever retaining wall wereperformed to assess the appropriateness of the Mononobe-Okabe method fordetermining the seismically induced lateral earth pressures on the stem of the wall.For the wall analyzed, it was found that at very low levels of acceleration the inducedpressures were in general agreement with those predicted by the Mononobe-Okabemethod. However, as the accelerations increased to those expected in regions ofmoderate seismicity, the induced pressures are larger than those predicted by theMononobe-Okabe method. This deviation is attributed to the flexibility of theretaining wall system and to the observation that the driving soil wedge does notrespond monolithically, but rather as several wedges.

IntroductionEarth retaining structures constitute an integral part of the lifelines across the US,extensively being used for port facilities, cuts along highways, bridge abutments, etc.In current design standards and guidelines (e.g., ASCE 4-98), the Mononobe-Okabemethod (Mononobe and Matsuo, 1929; Okabe, 1924) is commonly specified fordetermining seismic earth pressures for which the retaining structure must resist.Inherent in the Mononobe-Okabe method are the assumptions that the earth retainingstructure and the driving soil wedge act as rigid bodies, which have been shown to bereasonable assumptions for large gravity type retaining structures (e.g., Seed andWhitman, 1970).

The focus of this paper is to assess the validity of the Mononobe-Okabe method fordetermining seismically induced lateral earth pressures on a more flexible cantileverretaining wall, particularly the stem portion of the wall. In this vein, a series of non-linear dynamic response analyses of a cantilever retaining wall were performed usingthe commercially available computer program FLAC (Itasca Consulting Group, Inc.).The analyses consisted of an incremental construction of the wall and placement ofthe backfill, followed by dynamic response analyses.

1 Associate Member, ASCE; Assistant Professor, Department of Civil and Environmental Engineering,University of Michigan, 2372 GG Brown Building, Ann Arbor, MI 48109-2125;[email protected] Doctoral Candidate, Department of Civil and Environmental Engineering, Virginia PolytechnicInstitute and State University, Blacksburg, VA.3 Senior Research Engineer, Information Technology Laboratory, US Army Engineer Research andDevelopment Center, Vicksburg, MS.4 Graduate Student Research Assistant, Department of Civil and Environmental Engineering,University of Michigan, Ann Arbor, MI.


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In the following, the earth pressures determined from the FLAC analyses arecompared with those predicted using the Mononobe-Okabe method. However, priorto the comparison, a brief review of the Mononobe-Okabe method is presented.

Mononobe-Okabe Earth PressuresThe Mononobe-Okabe method for determining seismically induced active andpassive lateral earth pressures is based on limit equilibrium and is an extension of theCoulomb theory for static stress conditions. The method entails three fundamentalassumptions (e.g., Seed and Whitman, 1970):

1. Wall movement is sufficient to ensure either active or passive conditions, asthe case may be.

2. The driving soil wedge inducing the lateral earth pressures is formed by aplanar failure surface starting at the heel of the wall and extending to the freesurface of the backfill. Along this failure plane the maximum shear strengthof the backfill is mobilized.

3. The driving soil wedge and the retaining structure act as rigid bodies andtherefore experience uniform accelerations throughout the respective bodies.

As demonstrated by Dr. Ignacio Arango (Seed and Whitman, 1970), the dynamicearth pressures may be determined from analogous static conditions. Accordingly, theMononobe-Okabe expressions for dynamic earth pressures can be derived from theCoulomb’s expressions for static earth pressures. The analogous static conditions areachieved by rotating the wall-backfill system by an angle ψ, such that the vector sumof the horizontal and vertical inertial coefficients (kh and kv, respectively) is orientedvertically, where tan(ψ) = kh / (1-kv). This procedure is illustrated in Figures 1 and 2for active and passive stress conditions, respectively. In regards to the mathematicalexpressions, the Mononobe-Okabe expressions can be derived from the Coulomb’sexpressions by replacing the static values for the total unit weight of the soil (γt),height of the wall (H), inclination of the backfill (β), and inclination of the wall facefrom the vertical (θ), with the corresponding dynamic values (i.e., γtd, Hd, βd, and θd).This substitution is demonstrated in the following set of equations.

Active case:Static conditions (Coulomb's expression)

AtA KHP ⋅⋅⋅= 2

21 γ

2

2

2

)cos()cos()sin()sin(1)cos()(cos

)(cos

−⋅+−⋅+

+⋅+⋅

−=

θβθδβφδφδθθ

θφAK


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Dynamic conditions (Mononobe-Okabe expression)

( )

( ) AEvt

ddAdtdAE

KkH

KHP

⋅−⋅⋅⋅=

⋅⋅⋅=

121

,21

2

2

γ

θβγ

2

2

2

)cos()cos()sin()sin(1)cos()(cos)cos(

)(cos

−⋅++−−⋅+

+⋅++⋅⋅

−−=

θβψθδψβφδφψθδθψ

ψθφAEK

Passive case:Static conditions (Coulomb's expression)

PtP KHP ⋅⋅⋅= 2

21 γ

2

2

2

)cos()cos()sin()sin(1)cos()(cos

)(cos

−⋅−+⋅+

−⋅−⋅

+=

θβθδβφδφθδθ

θφPK

Dynamic conditions (Mononobe-Okabe expression)

( )

( ) PEvt

ddPdtdPE

KkH

KHP

⋅−⋅⋅⋅=

⋅⋅⋅=

121

,21

2

2

γ

θβγ

2

2

2

)cos()cos()sin()sin(1)cos()(cos)cos(

)(cos

−⋅+−−+⋅+

−⋅+−⋅⋅

−+=

θβψθδψβφδφψθδθψ

ψθφPEK

A plot of the Mononobe-Okabe active and passive earth pressure coefficients (KAEand KPE, respectively) as functions of the horizontal inertial coefficient (kh) areshown in Figure 3a for β = δ = θ = 0° and φ' = 35°. As may be observed from thisfigure, when kh = 0 (i.e., static conditions), the values of KAE and KPE are equivalentto Coulomb's active and passive coefficients (KA and KP, respectively). However, askh increases in value, KAE becomes greater than KA, and KPE becomes less than KP.This trend can be understood by referring back to Figures 1 and 2. In the active case


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(Figure 1), as kh increases, the analogous static condition is achieved by tilting thewall forward, thus increasing the inclination of the backfill and increasing thepressure induced on the wall. Correspondingly, in the passive case (Figure 2), as khincreases, the analogous static condition is achieved by tilting the wall backward, thusdecreasing the inclination of the backfill and decreasing the pressure induced on thewall.

The limiting pressures for both the active and passive cases occur when for theanalogous static conditions, the inclination of the backfill equals the angle of internalfriction (i.e., βd = φ'). At this point, the failure wedges become infinite in size, orsynonymously, the angle of the failure planes equal the static inclinations of thebackfill (i.e., αAE = β and αPE = β, where both αAE and αPE decrease as kh increasesfor the respective stress conditions). For the active case, no sized wall could restrainthe backfill from movement, while in the passive case, the pressure induced on a wallrestrained from movement becomes zero, as the backfill yields under its own inertialload.

FLAC Computed Earth Pressures on the Cantilever WallA series of non-linear dynamic response analyses were performed on the cantileverwall shown in Figure 4 using the finite difference program FLAC (Itasca ConsultingGroup, Inc.). The geometry and structural detailing of the wall analyzed weredetermined following the US Army Corps of Engineers static design procedures(Headquarters, US Army Corps of Engineers, 1989, 1992). Both the foundation andbackfill soils were modeled as being elasto-plastic, and interface elements were usedbetween the wall and the soil to allow relative movements and permanentdisplacements in the wall-soil system to occur. The wall and backfill werenumerically constructed in 2 ft lifts, allowing for equilibrium of the stresses to occurbetween lift placements. Additional details of the wall design and FLAC modelingare presented in Green and Ebeling (2002).

Upon the completion of the numerical construction of the wall and placement of thebackfill, the lateral pressures imposed on the stem of the wall were in good agreementwith the active earth pressures determined using the Coulomb expressions. A seriesof dynamic response analyses were then performed using the same acceleration timehistory scaled to different peak ground accelerations (pga). For all the analyses, thecomputed stresses on the stem of the wall were in very good agreement with thosepredicted by the Mononobe-Okabe expressions at the early part of the time historywhere the accelerations were very low. However, at the larger levels of shaking, theMononobe-Okabe expressions failed to predict the induced stresses on the stem of thewall. A comparison of the results is shown in Figure 3b. The data shown in this figureare the lateral earth pressure coefficients (KFLAC) back-calculated from the FLACresults using the following expression:

( )vt

FLACFLAC kH

PK

−⋅⋅⋅

=1

22γ


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where PFLAC is the resultant of the FLAC computed stresses imposed on the stem ofthe wall. KFLAC values were computed at times corresponding to the peaks in the timehistory of the horizontal inertial coefficient (kh) acting away from the backfill (i.e.,active-type conditions), wherein spurious high frequency spikes were filtered fromthe kh time history.

The reason for the deviation of the FLAC computed stresses and those computed bythe Mononobe-Okabe expressions can be understood from examining Figure 5.Shown in this figure is the deformed mesh from one of the FLAC analyses, whereinthe deformations are magnified by a factor of 3. At large values of kh directed awayfrom the backfill, the induced inertial forces on the structural wedge cause it tosimultaneously bend, rotate, and potentially slide away from the backfill, at whichtime a small wedge of soil or graben moves vertically downward. (The structuralwedge consists of the cantilever wall and the backfill contained within; see Figure 4.)As the direction of kh reverses (i.e., changes direction from away to towards thebackfill), the graben prevents the structural wedge from returning to its undeformedshape, in effect locking in the elastic stresses resulting from the bending and rotationof the structural wedge.

This process is illustrated by the dashed arrows and corresponding data points inFigure 3a, wherein the initial stresses imposed on the stem of the wall correspond toactive conditions. As kh increases in the direction away from the backfill, the stresseson the stem increase according to the Mononobe-Okabe expressions for activeconditions. However, upon reversal of the direction of kh, the stresses imposed on thestem do not decrease as predicted by Mononobe-Okabe expression, but rather remainrelatively constant. This stepwise increase in the locked-in stresses continues until theresidual stresses imposed on the stem correspond to at-rest (or Ko) conditions, whilethe dynamically induced inertial stresses are superimposed on the locked-in residualstresses. The increase in residual stresses is clearly shown in Figure 6, wherein plotsare shown of both the time history of kh and of the resultant of the lateral stresses(PFLAC) imposed on the stem of the retaining wall.

The locked-in residual stresses on the wall are not released by the slippage of the wallaway from the backfill. This is because the "driving soil wedge" is not monolithic,but rather, in this case, consists of a graben and five driving soil wedges, with thelater tending to move downward and away from the backfill as the wall slidesoutward. As a result, the graben "rides along" with the driving soil wedgesmaintaining its role of locking in the residual stresses.

Summary and ConclusionsFor the cantilever retaining wall numerically modeled and analyzed, the stressesinduced on the stem of the wall did not correspond with those predicted by theMononobe-Okabe method. The reason for this deviation is attributed to the relativeflexibility of the structural wedge and to the non-monolithic motion of the drivingsoil wedge, both of which violate assumptions inherent in the Mononobe-Okabemethod. The dynamic response of the wall-backfill system was such that there was an


Paper Number 87:

incremental increase from active to at-rest stress conditions in the residual stressesimposed on the stem of the retaining wall. The conclusions drawn from this studymay not apply to retaining walls systems of differing geometry and/or materialproperties. Further research is required in order to draw more general conclusionsregarding the appropriateness of the Mononobe-Okabe method to evaluate thedynamic pressures induced on cantilever retaining walls.

AcknowledgementsA portion of this study was funded by the Headquarters, US Army Corps ofEngineers (HQUSACE) Civil Works Earthquake Engineering Research Program(EQEN). Permission was granted by the Chief of the US Army Corps of Engineers topublish this information. The first author benefited from several enlighteningdiscussions with Professor Radoslaw Michalowski, University of Michigan,regarding the derivation of the Mononobe-Okabe expressions.

ReferencesASCE 4-86: Seismic Analysis of Safety-Related Nuclear Structures and Commentaryon Standard for Seismic Analysis of Safety Related Nuclear Structures, AmericanSociety of Civil Engineers.

Green, R.A. and R.M. Ebeling (2002). Seismic Analysis of Cantilever RetainingWalls, Phase I, ERDC/ITL TR-02-3, Information Technology Laboratory, US ArmyCorps of Engineers, Engineer Research and Development Center, Vicksburg, MS.http://libweb.wes.army.mil/uhtbin/hyperion/ITL-TR-02-3.pdf

Headquarters, US Army Corps of Engineers (1989). Retaining and Flood Walls, EM1110-2-2502, Washington, DC.

Headquarters, US Army Corps of Engineers (1992). Strength Design for Reinforced-Concrete Hydraulic Structures, EM 1110-2-2104, Washington, DC.

Itasca Consulting Group, Inc., FLAC (Fast Lagrangian Analysis of Continua),Minneapolis, MN.

Mononobe, N. and H. Matsuo (1929). On the Determination of Earth Pressure DuringEarthquake, Proceedings: World Engineering Congress, Tokyo, Vol IX, Part 1, 177-185.

Okabe, S. (1924). General Theory on Earth Pressure and Seismic Stability ofRetaining Wall and Dam, Journal Japan Society of Civil Engineering, 10(6), 1277-1323, plus figures.

Seed, H.B. and R.V. Whitman (1970). Design of Earth Retaining Structures forDynamic Loads, Lateral Stresses in the Ground and Design of Earth-RetainingStructures, ASCE, 103-147.


Paper Number 87:

a)

b)

Figure 1. Active earth pressures for a) static conditions and b) analogous staticconditions for the dynamic case.

β

θ

αA

H δPA

W

wall movement

Retained Soil:(γt , φ')

wall movement

ψ

θd = θ + ψ

Hd =H⋅cos(θ+ψ)

cos(θ)

βd = β + ψ

δ

PAE W⋅(1-kv)

W⋅kh

ψ

αAE

γtd =γt⋅(1-kv)cos(ψ)

, φ'

Retained Soil:


Paper Number 87:

a)

b)

Figure 2. Passive earth pressures for a) static conditions and b) analogous staticconditions for the dynamic case.

β

θ αP

δPP

H

W

wall movement

Retained Soil(γt , φ')

wall movement

ψθd = θ - ψ

Hd =H⋅cos(θ-ψ)

cos(θ)

W⋅kh

W⋅(1-kv)ψ

βd = β - ψ

δPPE

αPE

γtd =γt⋅(1-kv)cos(ψ)

, φ'

Retained Soil:


Paper Number 87:

Figure 3. a) Mononobe-Okabe lateral earth pressure coefficients for β = θ = δ = 0°and φ' = 35°. b) Comparison of active lateral earth pressures (KFLAC) back-calculated from FLAC results with values computed using the Mononobe-Okabe expressions.

a) b)

Figure 4. Cantilever retaining wall analyzed in FLAC a) geometry and b) materialproperties.

kh

0.0 0.2 0.4 0.6 0.8 1.0

KPE

0

1

2

3

4La

tera

l Ear

th P

ress

ure

Coe

ffic

ient

(K)

KP

KA

KAE

a)

kh

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

Ko

b)

B = 4 m

0.6 m

H =

6 m

Bh = 2.4 m0.5 m

0.2 m

Bt = 0.9 m

HeelToe

Backfill

Stem

Base

Backfill:medium-dense cohesionlesscompacted fill(γm = 19.6 kN/m3, φ = 35°)Foundation:natural deposit of densecohesionless soil(γm = 19.6 kN/m3, φ = 40° )Reinforced concrete:γ = 23.6 kN/m3, f'c = 27.6 MPa,fy = 413.4 MPaHydraulic factor: 1.3


Paper Number 87:

Figure 5. Annotated deformed mesh from one of the FLAC analyses; deformationsmagnified by a factor of 3. (Note: Toe of wall not initially embedded.)

Figure 6. Time history of the horizontal inertial coefficient (kh) at approximately thecenter of the structural wedge, and the time history of the resultant of theimposed stresses (PFLAC) on the stem of the cantilever retaining wall.

0 10 20 30 40-0.4-0.20.00.20.4

k hP F

LAC (k

N)

17.835.653.471.289.0

0.0

Stem

0 10 20 30 40

Koconditions

KA conditions

Time (sec)

graben

mononobe okabe

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