PSEUDO-STATIC SEISMIC ANALYSIS OF …geoeng.ca/Directory/Bathurst/GI-V2-N5-Paper2.pdf · ·...

787GEOSYNTHETICS INTERNATIONAL S 1995, VOL. 2, NO. 5

Technical Paper by R.J. Bathurst and Z. Cai

PSEUDO-STATIC SEISMIC ANALYSIS OFGEOSYNTHETIC-REINFORCED SEGMENTALRETAINING WALLS

ABSTRACT: The paper examines seismic stability analyses of geosynthetic-rein-forced segmental retaining walls (modular block walls). Stability analyses are devel-oped within the framework of a pseudo-static approach that gives factors of safetyagainst collapse mechanisms or rupture of component materials. The Mononobe-Okabe method is used to estimate dynamic earth pressures. Parametric analyses offorces and factors of safety related to external, internal and facing failure modes forwalls constructed on competent foundations are presented. Shear interfaces betweenfacing units are considered as possible planes of failure in facing stability analyses. Thepotential for local toppling of the facing column is also investigated. The results of anal-yses demonstrate that there is a limiting value of the horizontal seismic coefficientabove which the margin of safety against base sliding and overturning may be unaccept-ably low during a seismic event for segmental retaining walls designed to just satisfyminimum factors of safety under static loading conditions. Pseudo-static seismic analy-ses of the performance of two geosynthetic-reinforced segmental walls during theNorthridge Earthquake in Los Angeles in 1994 are demonstrated to be consistent withvisual observation of tension cracks in the soil backfill. Limitations of pseudo-staticmethods are discussed and recommendations for further research are made.

KEYWORDS: Pseudo-static analysis, Segmental retaining walls, Geosynthetic rein-forcement, Seismic analysis, Modular block walls.

AUTHORS: R.J. Bathurst, Professor, and Z. Cai, Research Associate, Department ofCivil Engineering, Royal Military College of Canada, Kingston, Ontario, K7K 5L0,Canada, Telephone: 1/613-541-6000 Ext. 6479, Telefax: 1/613-541-6599, E-mail:[email protected].

PUBLICATION: Geosynthetics International is published by the Industrial FabricsAssociation International, 345 Cedar St., Suite 800, St. Paul, MN 55101, USA,Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics International isregistered under ISSN 1072-6349.

DATES: Original manuscript received 1 May 1995, revised manuscript received andaccepted 3 August 1995. Discussion open until 1 May 1996.

REFERENCE: Bathurst, R.J. and Cai, Z., 1995, “Pseudo-Static Seismic Analysis ofGeosynthetic-Reinforced Segmental Retaining Walls”, Geosynthetics International,Vol. 2, No. 5, pp. 787-830.

BATHURST AND CAI D Seismic Analysis of Reinforced Segmental Retaining Walls

788 GEOSYNTHETICS INTERNATIONAL S 1995, VOL. 2, NO. 5

1 INTRODUCTION

The use of segmental retaining walls that include dry-stacked concrete block unitsas the facia system together with extensible sheets of polymeric materials (geosynthet-ics) that internally reinforce the retained soils and anchor the facia hasgained wide pop-ularity in North America (Bathurst and Simac 1994). These structures have also beenreported in Europe, Scandinavia and Australia in recent years (Cazzuffi and Rimoldi1994; Gourc et al. 1990; Knutson 1990; Won 1994). An example of a reinforced seg-mental retaining wall structure is illustrated in Figure 1. The distinguishing feature ofthese structures is the facing column that is constructed using mortarless modular con-crete block units that are stacked to form a wall batter into the retained soils (typically3 to 15_ from vertical). The economic benefits of these systems over conventional rein-forced concrete gravity wall structures and mechanically stabilized soil retaining wallsthat use inextensible (steel) reinforcement and select backfills have been demonstratedin several of the references cited in an earlier paper by Bathurst and Simac (1994).

Figure 1. Typical geosynthetic-reinforced soil segmental retaining wall cross-section(after Simac et al. 1991).

Drainage collection pipegravity flow to outlet

4.4 m

6.1 m

12kPa surcharge

1

20

Masonryconcretefacing units

Compacteddrainage fill

Compacted drainagefill with geotextile

Compacted native soilfor reinforced soil zone

1.2m long geogrid type l

Geogrid type ll

Geogrid type ll

Geogrid type ll

Geogrid type ll

Total geogrid embedment length (all layers)

5.5 m

0.3 m (typical)

Slope 1:6 (typical) Excavation limits



Stability analyses for geosynthetic-reinforced soil walls under static loading condi-tions (including segmental retaining wall systems) involve separate calculations to es-tablish factors of safety against external modes of failure and internal modes of failure(Figure 2). External stability calculations consider the reinforced soil zone and the fac-ing column as a monolithic gravity structure. The evaluation of factors of safety againstbase sliding, overturning about the toe, and foundation bearing capacity is analyticallyidentical to that used for conventional gravity structures. Internal stability analyses forgeosynthetic-reinforced soil walls are carried out to ensure that the structural integrityof the geosynthetic-reinforced soil mass is preserved with respect to reinforcementover-stressing and pullout of geosynthetic reinforcement layers from the anchoragezone.

A comprehensive design methodology has been recently proposed by the NationalConcrete Masonry Association (NCMA) for the static analysis of segmental retainingwalls (Simac et al. 1993; Bathurst et al. 1993). The NCMA guidelines address potential

Figure 2. Modes of failure: external (top row); internal (middle row); and facing (bottomrow) (Simac et al. 1993).

(a) base sliding (b) overturning (c) bearing capacity(excessive settlement)

(e) pullout (f) internal sliding

(h) connection failure(g) shear failure(bulging)

(i) local overturning(toppling)

(d) tensile over-stress



failure mechanisms not found in other geosynthetic-reinforced soil wall systems as il-lustrated in Figures 2f, 2g, 2h and 2i. The dry-stacked (mortarless) concrete blocks arediscrete units that transmit shear through concrete keys, interface friction, mechanicalconnectors, or a combination of these methods. The stacked facing units result in poten-tial failure planes through the facing column and this requires that additional stabilitycalculations be carried out to estimate interface shear forces and to compare theseforces with available shear capacity. In addition, the connection between the reinforce-ment layers and the facia is typically formed by extending the reinforcing layers alongthe interface between facing units to the front of the wall. The connection detail mustalso be evaluated for satisfactory design capacity (Bathurst and Simac 1993).

The NCMA method proposes a consistent approach to calculate earth pressures forboth external and internal stability calculations that is based on Coulomb earth pressuretheory. The advantage of Coulomb earth pressure theory over Rankine theory, whichhas been adopted in earlier guidelines for geosynthetic-reinforced soil walls (AASHTO1990, 1992; Christopher et al. (FHWA) 1989), is that the former explicitly accounts forthe influence of wall batter and wall-soil friction on the development of earth pressuresand hence is less conservative. A review of the essential features of the Coulomb earthpressure approach as it applies to segmental retaining wall structures and comparisonsof the NCMAdesign methodology with earlier limit-equilibrium methodsof design canbe found in the paper by Bathurst et al. (1993).

The scope of the NCMA guidelines is currently restricted to design of routine seg-mental retaining walls under static loading conditions. Questions related to the perfor-mance of the discrete facia system and the connections between the facia units and geo-synthetic reinforcement layers during a seismic event have been raised (Allen 1993).Nevertheless, the satisfactory performance of a number of geosynthetic-reinforced seg-mental walls during the Loma Prieta Earthquake of 1989 (Eliahu and Watt 1991; Collinet al. 1992) and the Northridge Earthquake of 1994 in California (Sandri 1994) hasbeenqualitatively demonstrated.

The present paper investigates the stability of geosynthetic-reinforced segmental re-taining walls under dynamic loading due to seismic excitation (earthquake). The studyis restricted to structures built on competent foundations for which foundation collapseor excessive settlement is not a potential source of instability. A pseudo-static rigidbody approach that uses the well-known Mononobe-Okabe (M-O) method to calculatedynamic earth forces (Okabe 1924) is outlined in this study. The method is restrictedto a limit-equilibrium approach in which factors of safety against collapse or rupturemechanisms are calculated. The method developed in the current study should not beconfused with displacement methods which have been used for the design of conven-tional walls and explicitly incorporate permanent displacement criteria in stabilityanalyses (Richards and Elms 1979; Whitman 1990). The current approach is a logicalextension of the Coulomb wedge theory adopted by the NCMAguidelines for structuresunder static loading. The M-O method has been used to calculate earth forces for seis-mic stability analyses of conventional gravity wall structures (Seed and Whitman 1970;Richards and Elms 1979). Some of the questions raised in earlier papers concerned withthe implementation of pseudo-static methods for conventional gravity retaining wallsmust also be addressed for the special class of structure that is the focus of this paper.



2 MONONOBE-OKABE EARTH PRESSURE THEORY

2.1 Calculation of Dynamic Earth Force

The Mononobe-Okabe method is used to calculate dynamic active earth forces actingon a planar surface that is inclined at an angle, ψ, into an unsaturated, homogeneous,cohesionless soil mass (Figure 3). In Figure 3, W refers to the static weight of the activewedge of soil acting behind the wall and Ww refers to the static weight of the facing col-umn. Quantities kh and kv are horizontal and vertical seismic coefficients, respectively,expressed as fractions of the gravitational constant, g . In the current study, horizontalinertial forces are assumed to act outwards (+kh) to be consistent with the notion of ac-tive earth pressure conditions. The convention adopted in this paper is that a positivevertical seismic coefficient, +kv , corresponds to a seismic inertial force that acts down-ward and a negative seismic coefficient, --kv , corresponds to a seismic inertial force thatacts upward. The total dynamic active earth force, PAE , imparted by the backfill soilis calculated as (Seed and Whitman 1970):

PAE= 12 (1 kv)KAEγH2 (1)

where: γ= unit weight of the soil; and H= height of the wall. The dynamic earth pressurecoefficient, KAE , can be calculated as follows:

Figure 3. Forces and geometry used in pseudo-static seismic analysis.

+ψ

β

+khW

(1¦ kv)W

δ

PAE

αAE

khWw

(1¦ kv)Ww

H



KAE=cos2(Ô+ ψ− θ)

cosθ cos2ψ cos(δ− ψ+ θ)1+ sin(Ô+δ)sin(Ô−β−θ)cos(δ−ψ+θ)cos(ψ+β) 2 (2)

where: Ô = peak soil friction angle; ψ = total wall inclination (positive in a clockwisedirection from the vertical); δ = mobilized interface friction angle assumed to act at theback of the wall; β = backslope angle (from horizontal); and θ = seismic inertia anglegiven by:

θ= tan−1 kh

1 kv (3)

The seismic inertia angle represents the angle through which the vectorial resultantof the gravity force and the inertial forces (both horizontal and vertical) is rotated fromvertical. Equations 1 to 3 are an exact analytical solution to the classical Coulombwedge problem that is modified to include the inertial forces khW and kvW. Examinationof the trigonometric terms in Equation 2 shows that solutions are only possible forθ≤Ô—β. Hence, the maximum value of horizontal seismic coefficient for which thereare solutions to Equation 2 is restricted to kh≤ (1kv)tan(Ô--β) .

Equations 1 and 2 can be modified to account for additional loads due to a uniformlydistributed surcharge acting behind the wall (Okabe 1924; Motta 1994). However, theinfluence of any surface distributed surcharge loading on the stability of segmental re-taining walls is not investigated in the current study. A closed-form solution for the cal-culation of dynamic earth force for c--Ô soils in retaining wall design is reported by Pra-kash (1981); however, this solution is restricted to the special case of β = 0 and kv = 0.For more complicated wall geometries and cases with surface loadings, trial single fail-ure plane geometries, or two-part wedge failure plane geometries, can be evaluated tofind the critical geometry giving the maximum value ofPAE . However, while these solu-tions are more general, they do not offer the designer the convenience of the closed-form solutions adopted in the current study.

In the discussions to follow, it is convenient to decompose the total dynamic activeearth force, PAE , calculated according to Equations 1 and 2 into two components repre-senting the static earth force component, PA , and the incremental dynamic earth forcedue to inertial seismic effects, ∆Pdyn (Seed and Whitman 1970). Hence:

PAE= PA+ ∆Pdyn (4)

or

(1 kv)KAE= KA+ ∆Kdyn (5)

where: KA = static active earth pressure coefficient; and ∆Kdyn = incremental dynamicactive earth pressure coefficient. For brevity in the following text, the quantity PAE willbe called the dynamic earth force.



2.2 Distribution of Dynamic Lateral Earth Pressures and Point of Application

The position of the dynamic earth force, PAE , acting against gravity retaining wallshas been shown to be variable and to depend on the magnitude of ground acceleration.A general range for the point of application of the dynamic force increment (∆Pdyn inEquation 4) has been reported to be ηH=0.4H to 0.7H above the toe of the wall (Seedand Whitman 1970). (η is the distance of the dynamic load increment above the toe ofthe wall normalized with respect to wall height, H.) Seed and Whitman suggest that avalue of η= 0.6 is reasonable for practical design purposes and this value is consistentwith the results of small-scale shake table tests reported by Ishibashi and Fang (1987).Based on experience with conventional gravity retaining walls the earth pressure dis-tributions illustrated in Figure 4 for η= 0.6 are assumed to be applicable to segmentalretaining wall structures in this paper. The parameter m in Figure 4 denotes the normal-

δ—ψ

KAγH

+

0.8∆KdynγH

0.2∆KdynγH

∆Pdyn

(a) static component (b) dynamic increment

PA

0.8∆KdynγH

(KA+0.2∆Kdyn)γH

=

(c) dynamic (total) pressuredistribution

Figure 4. Calculation of dynamic earth pressure distribution due to soil self-weight.

+ψ

δ—ψ δ—ψ

δ—ψ

(Note: η=0.6 .)

H

H/3

ηH

mH

PAE = PA + ∆Pdyn



ized point of application of the dynamic earth force and is limited to the range1/3≤m≤0.6. This range compares favorably to measured valuesof m ranging from 0.3to 0.5 reported by Ichihara and Matsuzawa (1973) from the results of shake table testson small-scale gravity wall models. The distributions for static and dynamic incrementof active earth pressure illustrated in Figure 4 are also identical to those recommendedfor the design of flexible anchored sheet pile walls (Ebling and Morrison 1993).

Finally, to simplify all stability calculations in this paper, and to be consistent withthe convention adopted in the NCMA guidelines, only the horizontal component of PAE

is used in stability calculations, i.e. PAEcos(δ--ψ). This assumption results in a conserva-tive (i.e. safe) design by ignoring the stabilizing benefit of the vertical component ofPAE.

2.3 Orientation of Active Failure Plane

Closed-form solutions for the orientation of the critical planar surface from the hori-zontal, αAE , have been reported by Okabe (1924) and Zarrabi (1979). These solutionsare rewritten here as:

aAE = Ô − θ + tan−1− AAE + DAE

EAE (6)

where

CAE = tan(δ+ θ− ψ)

BAE = 1∕tan(Ô− θ+ ψ)

DAE= AAEAAE+ BAE

BAECAE+ 1

AAE = tan(Ô− θ− β)

EAE = 1+ CAEAAE + BAE

Equation 6 can be used to calculate the orientation of the assumed active failure planewithin the reinforced soil mass and in the retained soil.

2.4 Selection of Parameter Values

2.4.1 Soil and Interface Friction Angles

In the theoretical developments and parametric analyses to follow, the friction angle,Ô, of the cohesionless backfill soils is assumed to be the peak value determined fromconventional laboratory practice and its magnitude is assumed to be unchanged underseismic excitation due to an earthquake. The choice of peak friction angle for seismic



design is consistent with AASHTO (1990, 1992), FHWA (Christopher et al. 1989) andNCMA (Simac et al. 1993) guidelines for static design of geosynthetic-reinforced soilwalls. Under rapid loading conditions the strength of compacted unsaturated cohesion-less backfills can be expected to be at least as great as the static value.

The admissible range of the interface friction angle, δ, is 0≤δ≤Ô in Coulomb wedgeanalyses. In static stability analyses, δ is assumed to be equal to 2Ô/3 for internal stabil-ity analyses (facing column-reinforced soil interface) and δ= Ô for external stabilityanalyses (reinforced soil-retained soil interface). A value of 2Ô/3 has been shown to beapplicable for wall-soil interface friction based on small-scale shake table tests of con-ventional gravity wall structures (Ishibashi and Fang 1987) and is assumed to be alsoapplicable for geosynthetic-reinforced retaining wall structures in the current study.Parametric analyses are restricted to the case δ≥ψ to avoid the complication that re-sults from vertical components of earth forces that act upward. The condition δ≥ψ isvalid for typical segmental retaining walls since fully-mobilized interface frictionangles, δ, taken at the facing column-reinforced soil interface and the reinforced soil-retained soil interface are generally greater than the wall inclination angle, ψ .

2.4.2 Seismic Coefficients

The general solution to the M-O method of analysis admits both vertical and horizon-tal components of seismic-induced inertial forces. The range of horizontal seismic co-efficients used in the parametric analyses to follow is restricted to kh < 0.5 .

The choice of positive or negative kv values will influence the magnitude of dynamicearth forces calculated using Equations 1 and 2. In addition, the resistance terms in fac-tor of safety expressions introduced later in the paper will be influenced by the choiceof sign for kv . An implicit assumption in many of the papers on pseudo-static designof conventional gravity wall structures reviewed by the writers is that the vertical com-ponent of seismic body forces acts upward. However, the designer must evaluate bothpositive and negative values of kv to ensure that the most critical condition is consideredin dynamic stability analyses if non-zero values of kv are assumed to apply. For example,Fang and Chen (1995) have demonstrated in a series of example calculations that themagnitude of PAE may be 12% higher for the case when the vertical seismic force actsdownward (+kv) compared to the case when it acts upward (--kv). Nevertheless, selectionof a non-zero value of kv implies that peak horizontal and vertical accelerations are timecoincident which is an unlikely occurrence in practice. The assumption that peak verti-cal accelerations do not occur simultaneously with peak horizontal accelerations ismade in the current FHWA guidelines for the seismic design of mechanically stabilizedsoil retaining walls (Christopher et al. 1989). Indeed, Seed and Whitman (1970) havesuggested that kv = 0 is a reasonable assumption for practical design of conventionalgravity structures using pseudo-static methods. Wolfe et al. (1978) studied the effectof combined horizontal and vertical ground acceleration on the seismic stability of re-duced-scale model Reinforced Earth walls using shake table tests. They concluded thatthe vertical component of the seismic motion may be disregarded in terms of practicalseismic stability design. Their conclusion can also be argued to apply to geosynthetic-reinforced segmental walls. Nevertheless, significant vertical accelerations may occurat sites located at short epicentral distances and engineering judgement must be exer-



cised in the selection of vertical and horizontal seismic coefficients to be used in pseu-do-static seismic analyses.

However, in order to address specific concerns raised by Allen (1993) related to fac-ing stability of geosynthetic-reinforced segmental retaining walls during a seismicevent that includes vertical ground accelerations, parametric analyses were carried outin the current study for the range kv = --2kh/3 to + 2kh/3. The upper limit on the ratio kv

to kh is equal to the calculated ratio of peak vertical ground acceleration to peak horizon-tal ground acceleration from seismic data recorded in the Los Angeles area(UCB/EERC, 1994).

In conventional pseudo-static M-O methods of analysis the choice of horizontal seis-mic coefficient, kh , for design is related to a specified horizontal peak ground accelera-tion for the site, ah . On the west coast of British Columbia (the most seismically activearea of Canada), the typical maximum design horizontal ground acceleration on rock,based on a 10% probability of exceedance in 50 years, is ah =0.32g (CFEM 1992). Therelationship between peak ground acceleration for the site, ah , and a representative val-ue of kh is nevertheless complex and there does not appear to be a general consensusin the literature on how to relate these parameters. For example, Whitman (1990) re-ports that values of kh from 0.05 to 0.15 are typical values for the design of conventionalgravity wall structures and these values correspond to1/3 to 1/2 of the peak accelerationof the design earthquake. Bonaparte et al. (1986) used kh = 0.85ah/g to generate designcharts for geosynthetic-reinforced slopes under seismic loading using the M-O methodof analysis. However, the results of finite element (FE) modelling of reinforced soilwalls by Segrestin and Bastick (1988), Cai and Bathurst (1995) and limited 1/2 scaleexperimental work (Chida et al. 1982) has shown that the average acceleration of thecomposite soil mass may be equal to or greater than ah depending on a number of factorssuch as: magnitude of peak ground acceleration; predominant modal frequency ofground motion; duration of motion; height of wall; and stiffness of the composite mass.Current FHWA guidelines use an equation proposed by Segrestin and Bastick (1988)that relates kh to ah according to:

(7)kh= (1.45− ah∕g)(ah∕g)

This formula results in kh > ah/g for ah < 0.45g . However, as clearly stated by Segrestinand Bastick, their equation should be used with caution because it is based on the resultsof FE modeling of steel reinforced soil walls up to 10.5 m high that were subjected toground motionswith a very high predominant frequency of 8 Hz. The results of FE mod-eling reported by Cai and Bathurst (1995) for a 3.2 m high geosynthetic-reinforced seg-mental retaining wall with ah = 0.25g and a predominant frequency range of 0.5 to 2Hz gave a distribution of peak horizontal acceleration through the height of the compos-ite mass and retained soil that was for practical purposes uniform and equal to the basepeak input acceleration. These observations are consistent with the results of Chida etal. (1982) who constructed 4.4 m high steel reinforced soil wall models and showed thatthe average peak horizontal acceleration in the soil behind the walls was equal to thepeak ground acceleration for ground motion frequencies less than 3 Hz. In practice, thefinal choice of kh may be based on local experience, and/or prescribed by local buildingcodes or other regulations.



In the current analyses, kh and kv are assumed to be uniform and constant throughoutthe facing column, the reinforced soil mass and in the retained soils. This assumptionsimplifies the analysis for geosynthetic-reinforced soil walls but may not be true forwalls higher than (say) 7 m, or walls with complex geometries, surface loadings, and/ordifficult foundation conditions. For these structures, and/or structures subjected to highfrequency ground motions, more sophisticated analyses may be warranted.

2.4.3 Other

In order to simplify stability analyses in this study, the facing units and backfill soilsare assumed to have the same unit weight, γ . This assumption introduces negligibleerror in the example calculations to follow. It can be noted that the majority of the seg-mental retaining wall units on the market today are hollow soil infilled masonry units.For these systems the facing unit weights and typical backfill soil unit weights are verysimilar.

2.5 Parametric Analyses Related to Mononobe-Okabe Earth Pressure Theory

A distinguishing feature of the geometry of segmental retaining walls is that the sur-faces against which active earth pressures are assumed to act are oriented atψ> 0 fromthe vertical (Figure 3). Hence, the wall-soil interface and reinforced soil-retained soilinterface are rotated in the opposite direction to that of many conventional gravity wallstructures. Typically, ψ varies from 3 to 15_ from the vertical (Simac et al. 1993) de-pending on the setback of the stacked modular units and the initial base unit inclination.The influence of horizontal seismic coefficient, kh , and wall inclination angle, ψ , onthe dynamic earth force, PAE , is illustrated in Figure 5. The data shows that the effectof positive wall inclination is to reduce dynamic earth forces to levels less than thosedeveloped against conventional gravity wall structures of the same height and retainingthe same frictional soil.

The backslope angle, β, also influences the magnitude of dynamic earth force. As thebackslope angle becomes larger, the magnitude of dynamic earth force increases as il-lustrated in Figure 6. The effect of increasing positive wall inclination is seen to reducethe magnitude of dynamic earth force for a given backslope angle.

The influence of wall inclination angle on the orientation of the critical failure sur-face, αAE , through the reinforced soil mass is shown in Figure 7. The figure illustratesthat for a given wall inclination angle, the size of the active soil wedge behind the wallfacing increases with increasing magnitude of the horizontal seismic coefficient, kh . Asimilar result has been reported by Vrymoed (1989). The implication of the resultsshown in Figure 7 to internal stability design is that the length of reinforcement layers,particularly those near the top of the reinforced soil zone, may have to be extended inorder to capture potential failure surfaces propagating from the heel of the facing col-umn (inset diagram in Figure 7). A similar conclusion was made by Bonaparte et al.(1986) for the design of reinforced slopes under seismic loading. The same calculationforαAE can be carried out to determine the critical failure plane through the retained soilmass that is assumed to propagate from the heel of the reinforced soil mass. The im-plication of such calculations to external stability analysis is that the size of the failure



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.0 0.1 0.2 0.3 0.4 0.5

Conventional walls

kh

---15_

---10_

--- 5_

0_

+ 5_

+10_+15_

ψ

Segmental retaining walls

Figure 5. Influence of seismic coefficient, kh , and wall inclination angle, ψ, on dynamicearth force, PAE .

Ô = 35_δ = 2Ô/3β = 0_kv = 0

β

PAE

—ψ +

2PAEγH2

H

zone behind the reinforced soil zone will also increase with increasing horizontal accel-eration.

The combined influence of horizontal and vertical accelerations on dynamic earthforce, PAE , is illustrated in Figure 8 for two wall inclination values. For values of hori-zontal seismic coefficient, kh , less than about 0.35 and kv =2kh/3, downward verticalcomponents of seismic earth force (kv > 0) give the largest dynamic earth forces. How-ever, the value of PAE using kv = +2kh/3 is only 7% greater than the value calculated usingkv = 0 for kh < 0.35. Hence, over a wide range of horizontal seismic coefficient valuesthe assumption that kv = 0 is reasonably accurate and, in fact, results in a slightly moreconservative value of PAE than values calculated assuming that the vertical componentof seismic earth force acts upward (kv < 0).



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Figure6. Influenceof seismiccoefficient,kh , backslope angle,β, and wall inclination angle,ψ, on dynamic earth force, PAE .

kh

Ô = 35_δ = 2Ô/3kv = 0

β

PAE

—ψ +

0_

5_10_15_

ψ

0_

5_

10_15_

ψ

0_

5_

10_

15_

ψ β = 0_

β = 10_

β = 20_

2PAEγH2

H

The influence of backslope angle, β, and wall inclination angle, ψ, on calculated val-ues of dynamic earth force can be argued to have a more significant effect than the mag-nitude of kv in the example calculations. The requirement that peak horizontal and verti-cal accelerations must be time coincident in order to generate even the small calculateddifferences in dynamic earth forces shown here gives support to recommendations incurrent design guidelines for conventional gravity structures that vertical accelerationscan be safely ignored in many seismically active areas.



20

25

30

35

40

45

50

55

60

65

70

0.0 0.1 0.2 0.3 0.4 0.5

Segmentalretaining walls

Ô = 35_δ = 2Ô/3β = 0_kv = 0

---15_---10_--- 5_0_

+ 5_+10_+15_

ψα AE(degrees)

kh

β

αAE

—ψ +kh = 0

kh > 0

Figure 7. Influence of seismic coefficient, kh , and wall inclination angle, ψ, on orientationof internal failure plane, αAE .

Conventional walls

Reinforced soil mass for kh = 0

Increasedreinforced soil massfor kh > 0Increased reinforcementlengths

3 EXTERNAL STABILITY ANALYSES

3.1 General

Potential external modes of failure are illustrated in Figures 2a, 2b and 2c. The analy-ses described in this section assume that the foundation provides a competent base suchthat potential modes of failure are restricted to translational sliding along the base and



—ψ +

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

kv = 0kv = ---2kh/3

Ô = 35_δ = 2Ô/3β = 0_

kh

kv = 0

kv = ---2kh/3

ψ = 10_

ψ = 0_

Figure 8. Influence of seismic coefficients, kh and kv , and wall inclination angle, ψ, ondynamic earth force, PAE .

β

PAE

kv = +2kh/3

kv = +2kh/3

2PAEγH2

H

overturning about the toe of the gravity mass (reinforced soil zone plus facing column).The dynamic earth force, PAE , calculated according to Equation 1 is used in externalstability calculations to estimate destabilizing active earth forces. In the external stabil-ity analyses to follow, the dynamic earth force imparted by the cohesionless retainedsoil on the gravity mass is assumed to act along a surface that is parallel to the wall face(i.e. at angle ψ from the vertical) and at a constant distance L from the front face of thewall (i.e. L is the minimum width of the gravity mass). According to NCMA guidelines,



the minimum width of the gravity mass is L = 0.6H for critical structures and L = 0.5Hfor non-critical structures. Dimension L is used to locate the heel of the assumed mini-mum gravity mass for external stability calculations and to control the minimum lengthof all reinforcement layers (Figure 9).

3.2 Base Sliding (Figure 2a)

The dynamic factor of safety against base sliding for purely frictional soils can be ex-pressed as:

0

1

2

3

4

5

6

7

8

9

10

11

12

0.0 0.1 0.2 0.3 0.4 0.5 0.6

λ = 0.6Ô = 35_δ = Ôψ = 0_L = 0.5H

kv=0

kv = ---kh/4

kv = ---2kh/3kv = ---kh/2kv = ---kh/4kv = 0

PAEcos(δ—ψ)

kv = ---2kh/3

kh

β = 0_

β = 20_

ψ

khλWR

WR(1¦kv)

R = WR(1¦kv)tanÔ

β

Figure 9. Static factor of safety against base sliding to give a minimum dynamic factor ofsafety of 1.125 against base sliding for a range of seismic coefficients, kh and kv , andbackslope angle, β.

kv = ---kh/2

0.06

1.5

ψ

(Notes: WR = weight of reinforced zone plus weight of facing column; and R = base sliding resistance.)

(static)

bsl

Lw

H

L

FS



FSbsl=L−Lw

H a2+Lw

H(1 kv) tanÔ

12 KAE(1 kv)a2

1 cos(δ− ψ)+ khλL−Lw

H a2+Lw

H (8)

where:

a1= 1+ L− Lw

H tan β

a2= 1+ L− Lw

2H tan β

Here, Lw is the width of the facing column. Equation 8 can be simplified by setting Lw

= 0 for Lw << L. This assumption introduces no error for β = 0 and negligible error formoderate values of β> 0. Parameter λ is an empirical constant that is used to artificiallyreduce the inertial force of the gravity mass and applies only to the inertial part of Equa-tion 8. A value of λ = 0.6 has been used for design purposes for both geosynthetic-rein-forced soil walls (Christopher et al. 1989) and for Reinforced Earth walls that use steelreinforcement strips (Segrestin and Bastick 1988). Parameter λ is assumed to be lessthan unity to account for the transient nature of the peak accelerations in the gravitymass and retained soils and the expectation that the inertial forces induced in the gravitymass and the retained soil zone will not reach peak values at the same time during aseismic event. The terms a1 and a2 in Equation 8 are geometric constants that accountfor the effect of the backslope angle on the calculation of the mass of the reinforced soilzone. The factor of safety for static loading conditions (FSbsl (static)) can be recoveredfrom Equation 8 by setting kh = kv = 0 .

The FHWA (Christopher et al. 1989) recommends that the minimum factor of safetyagainst base sliding under dynamic loading be no less than 75% of the minimum allow-able static value. If this rule is applied to a minimum allowable static factor of safetyagainst sliding of 1.5 then, the data in Figure 9 shows that the maximum permissibleground acceleration is 0.06g. Larger static factors of safety will be required to ensureFSbsl(dynamic)≥ (0.75)(1.5) = 1.125 as illustrated in the figure. Nevertheless, thereare limiting seismic coefficient values for any set of wall parameters for which thereis no solution using Equations 1 and 2 (i.e. Ô—β—θ< 0 ). The analytical results de-scribed here are consistent with conclusions made by Richards and Elms (1979) whoshowed that there is very little margin of safety against base sliding of gravity structuresdesigned to meet minimum conventional static factors of safety. Figure 9 also demon-strates that the required static factor of safety against base sliding for a structure witha horizontal backslope is relatively insensitive to the magnitude of vertical accelerationfor a horizontal seismic coefficient kh < 0.3. The vertical seismic coefficient used to gen-erate the curves in Figure 9 was taken as kv ≤ 0 since negative values of kv gave themost conservative results (i.e. largest required FSsld (static) values). The data in Figure9 do show that even a modest backslope angle will require large increases in the staticfactor of safety against base sliding to ensure an acceptable margin of safety under seis-mic loading.



3.3 Overturning (Figure 2b)

The moment arm, Ydyn , of the dynamic force normalized with respect to the wallheight (Figure 4) can be calculated as follows:

m=Ydyn

H =13 KA+ η[KAE(1 kv)− KA]

KAE(1 kv)(9)

The relationship between normalized moment arm, m, and horizontal seismic coeffi-cient, kh , is shown in Figure 10a for values of η ranging from 0.4 to 0.7 and δ = Ô andδ= 2Ô/3 . The location of the dynamic force moment arm, m, is sensitive to the assumedvalue of η but is relatively insensitive to the range of interface friction angle, δ. For η≤ 0.6, the point of application of the dynamic force is generally at 1/3 to 1/2 of the wall

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.0 0.1 0.2 0.3 0.4 0.5 0.60.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.0 0.1 0.2 0.3 0.4 0.5 0.6

δ = Ô

δ = 2Ô/3

Ô = 35_

0.7

0.6

0.5

0.4

η

m

15_0_

---15_

ψ

β = 0_

(a) ψ = 0_ (b) η = 0.6, δ = 2Ô/3

kv = 0

kh

Figure 10. Influence of seismic coefficient, kh, normalized dynamic force incrementlocation, η, wall inclination angle, ψ, and wall-soil interface friction angle, δ, on location ofnormalized dynamic moment arm, m.

kh



height above the toe. Figure 10b illustrates the influence of horizontal seismic coeffi-cient, kh , and magnitude of wall inclination angle, ψ, on parameter m for η = 0.6. Thepoint of application of the dynamic earth force is shown to be only slightly dependenton the magnitude of wall inclination angle associated with typical segmental retainingwalls.

The dynamic factor of safety against overturning about the toe of the free body com-prising the reinforced soil mass and the facing column can be calculated as follows:

FSbot=

L−Lw

H2b2+

2Lw

L−Lwa2+ Lw

H2(1 kv)

mKAE(1 kv)a31 cos(δ− ψ)+ kh λL−Lw

H b1+ Lw

H

(10)

where

b2= 1+ 23 (a1− 1)

b1= a1+ 13 (a1− 1)2

and terms a1 and a2 are defined in Equation 8. Equation 10 is applicable to the case ofa vertical wall (ψ= 0) and will give a slightly conservative estimate of FSbot for inclinedwall facing columns. The static factor of safety, FSbot (static), required to satisfy FSbot

(dynamic) = (0.75)(2.0) = 1.5 is plotted in Figure 11 for the case of two different rein-forcement length to wall height ratios, L/H . The vertical component of seismic forcehas been taken as upward (--kv) in order to calculate results for the most critical orienta-tion. If the conventional rule that the dynamic factor of safety be not less than 75% ofthe minimum allowable static factor of safety against overturning (i.e. 2) is applied,then the horizontal seismic coefficient is limited to kh = 0.04 in the example calcula-tions. The data in Figure 11 illustrates that even a modest backslope angle leads to adramatic increase in the required static factor of safety for a given design acceleration.Indeed, for the case of β = 20_ and kh > 0.25 an acceptable margin of safety against over-turning may not be possible. Figure 11 also shows that the effect of the L/H ratio on therequired value of FSbot (static) is relatively insignificant compared to the effect of back-slope inclination for typical structures designed for static loading environments.

4 INTERNAL STABILITY

4.1 General

The influence of the magnitude of seismic coefficients on lateral earth forces hasbeendemonstrated earlier in the paper. If the calculation of dynamic reinforcement loads iscarried out in the same manner as for conventional static wall design then, the effectof seismic loading can be shown to increase the magnitude of the net horizontal forcecarried by the geosynthetic reinforcement layers. In addition, the change in the distribu-



0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.0 0.1 0.2 0.3 0.4 0.5

PAEcos(δ—ψ)

λ = 0.6Ô = 35_δ = Ôψ = 0_η = 0.6

β = 0_

Figure 11. Minimum static factor of safety against overturning required to give a factor ofsafety of 1.5 against dynamic overturning for a range of seismic coefficients, kh and kv ,backslope angle, β, and length to height ratio, L/H.

kv = 0

kv = ---2kh/3

L/H= 0.5L/H= 0.7

ψ β

kh0.04

kv = ---2kh/3

kv = 0

khλWR

WR(1¦kv)

bot(static)

(Note: WR = weight of reinforced zone plus weight of facing column.)

Lw

β = 20_

H

L

ma2H

FS

tion of the lateral earth pressure (Figure 4) means that the percentage of total lateralforce to be carried by the reinforcing elements in the upper portions of the wall will in-crease. Finally, the influence of ground accelerations on the volume of the internal po-tential failure wedge leads to an increase in the length of the reinforcement layers asdiscussed in Section 2.5 and illustrated in Figure 7.



4.2 Over-stressing of Reinforcement (Figure 2d)

The contributory area approach used for the static stability analysis of segmental re-taining walls is extended to the dynamic loading case. In this method the reinforcementlayers are modelled as tie-backs with the dynamic tensile force, Fdyn , in each layer equalto the dynamic earth pressure integrated over the contributory area, Sv , at the back ofthe facing column plus the corresponding wall inertial force increment, kh∆Ww . Thecontributory area for the topmost reinforcement layer is taken from the top of the crestto mid-elevation between the first and second reinforcement layers from the crest. Forthe simple geometry illustrated in Figure 12, the dynamic factor of safety, FSos , againstover-stressing of a reinforcement layer at depth z below the crest of the wall is givenby:

=Tallow

0.8∆Kdyn cos(δ− ψ)+ (KA− 0.6∆Kdyn) cos(δ− ψ) zH+ kh

LW

H γHSv

FSos =Tallow

Fdyn(11)

Here, Tallow is the allowable tensile load for the reinforcement under seismic loading.

kh∆Ww

0.8∆KdynγHcos(δ—ψ)

(KA+0.2∆Kdyn )γHcos(δ—ψ)

Figure 12. Calculation of tensile load, Fdyn , in a reinforcement layer due to dynamic earthpressure and wall inertia.

Sv

Dynamicearth pressuredistribution

Reinforcementlayer (typical)

LW

z

+ = Fdyn

(Note: η = 0.6.)

H



The influence of seismic coefficient values on the magnitude of tensile reinforcementloads can be explored by computing a magnification factor, rF , that is the ratio of dy-namic tensile force, Fdyn , to static tensile force, Fsta , for a reinforcement layer at depthz below the wall crest. Results of this calculation for reinforcement layers at five differ-ent depths below the wall crest are presented in Figure 13. The data illustrates that thelargest increases in reinforcement force occur in the shallowest layers in a reinforcedsoil wall. This result is not unexpected due to the change in the active earth pressuredistribution that results from dynamic loading as discussed in Section 2.2 and illustrated

0

1

2

3

4

5

6

7

8

9

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6

z/H

0.2

0.4

0.6

0.8

0.3

ψ = 0_Ô = 35_

δ = 2Ô/3

β = 0_

η = 0.6

rF

LW/H= 0.1

khFigure 13. Influence of seismic coefficients, kh and kv , and normalized depth below crestof wall, z/H, on dynamic reinforcement force amplification factor, rF .

kv = 0

kv = ---2kh/3

kv = +2kh/3

rF=FdynFsta



in Figures 4 and 12. The plotted data in Figure 13 also reveals that rF is sensibly indepen-dent of the magnitude of kv for kh≲ 0.35, and hence solutions using kv = 0 are sufficient-ly accurate for design over this range and even slightly conservative at the shallowestdepth investigated (z/H=0.2).

An implication of these results to design is that the number of reinforcement layersmay have to be increased at the top of the wall in order to keep tensile loads within al-lowable limits. However, it should be noted that the allowable design tensile load underdynamic loading is routinely taken as a greater percentage of the index strength of thereinforcement than the percentage used for static loading design because of the shortduration of peak tensile loading during a seismic event. AASHTO(1992) guidelines canbe interpreted to permit the value of Tallow used for static loading designs to be increasedby 33% for the seismic loading condition. Rapid in-isolation wide-width strip tensileloading of a typical high density polyethylene (HDPE) geogrid reinforcement reportedby Bathurst and Cai (1994) has demonstrated a potentially large increase in reinforce-ment stiffness of the material when compared to conventional rates of loading. This ob-servation suggests that HDPE geogrids may be designed for much greater strengths un-der seismic loading than those values that result from the interpretation of AASHTOrecommendations.

4.3 Reinforcement Anchorage (Figure 2e)

The dynamic reinforcement tensile load must be carried by the reinforcement anchor-age length which is located between the internal active failure plane (oriented at αAE

from horizontal) and the reinforcement free end (Figure 7). A common approach foranchorage capacity design is to use a simple Coulomb-type interface model in whichanchorage capacity is linearly proportional to anchorage length, overburden pressureand soil shear strength (AASHTO 1990). An implication of this model for dynamic an-chorage design is that the required anchorage length will increase in proportion to themagnification factor, rF , introduced in the previous section. Owing to the short durationof the anchorage force during a seismic event, the factor of safety against anchoragepullout may be taken as 75% of the static value of 1.5 according to AASHTO (1992).However, the principal effect of dynamic loading on anchorage design is the require-ment to increase the length of reinforcement layers in order to capture the larger activewedge of soil that occurs as a result of seismic loading, i.e. the internal failure planeangle,αAE , decreases as kh increases. This effect is illustrated in Figure 14. For example,this figure shows that for a horizontal seismic coefficient, kh = 0.25, reinforcementlengths would have to be increased by 60 to 100% of the lengths required under staticloading conditions. The requirement that the lengths of the uppermost reinforcementlayers may need to be increased for reinforced slopes subject to seismic loading hasbeen noted by Bonaparte et al. (1986) and is based on similar arguments.

4.4 Internal Sliding (Figure 2f)

The modular facing construction in geosynthetic-reinforced segmental retainingwalls requires that the designer check for internal sliding along horizontal planes thatpass along the reinforcement-soil interface and through the facing column between fac-ing units. The results of large-scale shear tests reported by Bathurst and Simac (1994)



1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 0.1 0.2 0.3 0.4 0.5

30_

35_

40_

45_

30_ 35_ 40_ 45_Ô =

kv = 0

kv = ---2kh/3

ψ = 0_

β = 0_

δ = 2Ô/3

—ψ +

β

LdynLsta

αAE

Ldyn

Lsta

khFigure 14. Influence of seismic coefficients, kh and kv, and soil friction angle, Ô, on ratio ofminimum reinforcement lengths, Ldyn/Lsta , to capture the internal failure wedge inpseudo-static Coulomb wedge analyses.

1.6

0.25

kh = 0kh > 0

have shown that the static shearing resistance, Vu , available at a horizontal interfacein the facing column can be described by a Coulomb-type failure law. This failure crite-rion can be modified to account for the dynamic loading case as follows:



Vu = au+Ww(1 kv) tan λu (12)

Parameters, au and λu represent minimum interface shear capacity and equivalent inter-face friction angle, respectively, and are not expected to vary between static and seismicloading conditions. The analysis required to calculate the factor of safety against inter-nal sliding is similar to that described for external sliding. The dynamic factor of safetyagainst internal sliding along a horizontal surface at depth z below the crest of the wallis calculated as:

FSisl=

Vu

γ z2+L−Lwz a2 (1 kv) tanÔds

12 KAE(1 kv)a2

1 cos(δ− ψ) + kh λL−Lwz a2+

Lwz

(13)

Coefficients a1 and a2 are the same as those reported for Equation 8 substituting H = z.Equation 13 assumes that the critical internal sliding mass is initiated at the free endof the reinforcement layer. Parameter Ôds is the interface (direct sliding) friction anglebetween the geosynthetic-reinforcement and the cohesionless reinforced soil. In gener-al, Ôds < Ô but the reduction in sliding resistance is typically more than compensatedfor by the large value of shear interlock that is available in many block systems. Howev-er, the combined effect of a low interface friction value, Ôds , and facing units with lowshear capacity, Vu , can result in unacceptably low factors of safety against interfaceshear and this failure mechanism must be checked as a matter of routine.

Aparametric analysis involving Equation 13 wasnot performed. However, the resultsplotted in Figure 9 are applicable for internal sliding for the case au = 0, λu = Ôds = Ô andsetting H = z.

5 FACING STABILITY

5.1 Facing Failure Mechanisms

The following potential failure mechanisms must be examined in pseudo-static seis-mic analysis of the facing stability of segmental retaining walls: interface shear failure;connection failure; toppling (local overturning) (Figure 2).

5.2 Interface Shear (Figure 2g)

The influence of interface shear transmission on facing column stability can be ana-lyzed by treating the facing column as a beam in which the integrated lateral pressure(i.e. distributed load) must equal the sum of the reactions (forces in reinforcement lay-ers). The calculation of interface shear force under dynamic loading must include theeffect of wall facing inertia. The general approach is illustrated in Figure 15. The totalforce carried by reinforcement layers located above facing unit j is calculated as the areaABDC of the lateral earth pressure distribution, plus the facing column inertial force



Figure 15. Calculation of dynamic interface shear force acting at a reinforcementelevation.

Ni+1F idyn

A B

C D

F idynS jdyn EF

khM

j+1

∆Wjw

kh∆Wjw

S i+1

Horizontalcomponent ofdynamicearth pressuredistribution

S jdyn = kh∆Wjw+ AREACDEF

z

(Notes: N = total number of reinforcement layers; and M = total number of facing units.)

H

Siv

v

over the same height. The out-of-balance force to be carried through shear at the bottomof facing unit j is simply the sum of the incremental column inertial force kh∆Wj

w plusthe force due to area CDEF in the figure. The partitioning of forces illustrated in thefigure is a direct result of the contributory area approach introduced earlier to assigntensile loads to reinforcement layers. The locally maximum interface shear forces willoccur at reinforcement elevations. A general expression for the factor of safety againstdynamic interface shear failure (FSsc (dynamic)) at a reinforcement layer is:

FSsc=Vu

Sdyn

= Vu

0.8∆Kdyn cos(δ− ψ)+ (KA− 0.6∆Kdyn) cos(δ− ψ) zH− Sv

4H + kh

LW

H γHSv

2

(14)

where: Sdyn = interface shear force; and Vu = shear capacity. For facing columnsconstructed at large inclination angles the magnitude of the normal force, Ww , trans-mitted between facing units may be less than the sum of the weights of the individual



facing units above the interface elevation. The loss of normal load is due to the effectof the facing column units leaning into the reinforced soil mass. The maximum heightof column units that will transmit all of the facing weight to a target interface is calledthe “hinge height”. Its calculation is described in detail by Bathurst et al. (1993) andSimac et al. (1993).

The calculation of factor of safety against interface shear failure under static loadingconditions is carried out using Equation 14 with kv = kh = 0 . Figure 16 shows the ratio

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

kv = 0kv = ---2kh/3

Sv/H = 0.2

kh

ψ = 0_Ô = 35_

δ = 2Ô/3

β = 0_

η = 0.6

LW/H = 0.1

Figure 16. Influence of seismic coefficients, kh and kv, and normalized depth below crestof wall, z/H, on the ratio of dynamic to static interface shear factor of safety.

(dynamic)

sc(static)

0.50.3

0.9

0.7

0.1

z/H

0.1

0.3

0.5

0.9

0.7

/sc

FSFS



of dynamic factor of safety to static factor of safety against interface shear failure fora range of horizontal and vertical seismic coefficient values applied to an example wallwith five evenly spaced reinforcement layers (i.e. Sv/H=0.2). The data shows that thepotential for interface shear failure under seismic loading increases with proximity ofthe shear interface to the crest of the wall. However, the effect of vertical accelerationon calculated dynamic factors of safety diminisheswith height of interface for the worstcase situation of vertical seismic force components acting upward. The curves forz/H=0.1 correspond to sliding stability of the top unreinforced portion of the wall fac-ing. These curves appear to support the argument that narrow unreinforced heights ofsegmental facing units are susceptible to sliding failure. However, in practice, large val-ues of interface shear capacity are possible with many modular block systems that areconstructed with shear keys, or other forms of positive interlock. These systems, as op-posed to systems that rely solely on frictional sliding resistance, are the preferred choicein order to achieve an adequate margin of safety against interface sliding. Furthermore,Bathurst and Simac (1994) have demonstrated that the shear capacity at an interfacelayer may be substantially reduced by the presence of a reinforcement inclusion if inter-face shear capacity is developed primarily through frictional resistance.

A minimum factor of safety against interface shear failure, for critical structures un-der static loading, is 1.5 according to NCMA guidelines. Reducing this criterion by 25%for the dynamic loading condition is recommended in order to be consistent with theapproach adopted for base sliding.

5.3 Connections (Figure 2h)

The influence of increased dynamic forces on connection load is identical to the anal-ysis described for reinforcement over-stressing. Peak connection load capacities understatic loading conditions have been described using bi-linear failure envelopes basedon the results of full-scale connection tests carried out at rates of loading matching the10% strain/min rate used in the ASTM D 4595 method of test (Bathurst and Simac1993). A Coulomb-type law with a maximum connection load cut-off has been used bythe first writer and co-workers to characterize a large number of (static) test results. Mo-dified for the dynamic loading condition, the peak connection load envelope becomes:

Fc= acs+Ww (1 kv) tan λcs≤ Fc(max) (15)

where the parameters acs and λcs represent the minimum connection capacity and theslope of the connection strength envelope, respectively. The dynamic factor of safetyagainst connection failure, FScn , is expressed by Equation 11 by replacing Tallow withFc . Depending on the connection type, it is possible that the maximum reinforcementload, Fdyn , may be limited by the facing connection capacity. The results of connectiontests carried out at different rates of loading have demonstrated that peak connectioncapacities may be sensitive to rate of loading (Bathurst and Simac 1993). At the timeof writing there is no data available that can be used to quantify changes in connectioncapacity that may develop as a result of repeated application of load and the rapid load-ing rates anticipated during a seismic event.



5.4 Toppling (Figure 2i)

The distribution of internal moments at interface elevations can also be calculatedusing the beam analog described in Section 5.2 for interface shear stability calculations.Internal moments that cause net outward moment at the toe of a facing unit provide apossible failure mechanism for which an adequate factor of safety should be checked.Local peak destabilizing moments will occur at reinforcement elevations. The factorof safety expression adopted by the NCMA (Simac et al. 1993), for local overturningat a reinforcement layer i under static loading conditions, can be modified for the dy-namic loading case as follows:

FSlot=

MR(1 kv)+N

i+1

Fic Y i

c

16 Ka cos(δ− ψ) z

H+ (0.4− 0.1 zH)∆Kdyn cos(δ− ψ)+ 1

2 khLw

H γHz2

(16)

where: MR = resistance to static overturning due to facing column self-weight above thetoe of the target facing unit; and N = number of reinforcement layers. The summationwith Fic Yic terms denotes the resisting moment due to the connection capacities of rein-forcement layers, Fi

c , and their corresponding moment arms, Yic , from the target point

of rotation.The ratio of the dynamic to static factor of safety against local overturning is plotted

in Figure 17 for five different interface elevations ranging from z/H= 0.2 to 1. To simpli-fy example calculations the connection capacities,Fi

c , have been assumed to be purelyfrictional (acs = 0). The largest reductions in factor of safety under seismic loading werefound to occur when the vertical component of the seismic force acts upward (kv =--2kh/3). The figure clearly shows that for the seismic coefficient values investigated,the shallow interface layers (small z/H values) require a higher static factor of safetyagainst overturning to maintain a dynamic factor of safety equal to or greater than unity.The unsupported height of the facing column at the top of the structure is the most criti-cal portion of the wall in these calculations (i.e. z/H=0.2 in this example). However, theeffect of the magnitude and orientation of the vertical component of seismic force onthe upper portions of the wall (z/H < 0.4) in the example used to produce Figure 17 isonly significant for horizontal seismic coefficient values greater than 0.3.

The experience of the writers is that designers typically try to maximize the unrein-forced height of wall at the crest in order to reduce reinforcement quantities in seg-mental retaining wall structures. This strategy will result in unacceptably low marginsof safety against toppling at the top of the structure under dynamic loading conditionsas illustrated in Figure 17. The only strategy to minimize the potential for this failuremechanism to occur is to introduce reinforcement layers close to the wall crest and toensure that these layers have adequate facing connection capacity. Although the resultspresented in Figure 17 suggest that toppling of the facing column is a potential problem,the results of the analyses presented in the next section for two walls that survived theNorthridge Earthquake in 1994 show that this problem did not develop in practice be-cause reinforcement layerswere placed close to the top of the structure and static factorsof safety against local overturning were very high.



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

PAEcos(δ—ψ)

0.2

0.6

1.00.8

0.4

β = 0_

kv = ---2kh/3

z/H

0.2

0.4

0.6

1.0

LW/H = 0.1

kv = 0

khWw

kh

β

LW

Figure 17. Influence of seismic coefficients, kh and kv, and normalized depth below crest ofwall, z/H, on the ratio of dynamic to static local overturning factor of safety.

MR

Fi+1c

Layer i

(1kv )Ww

ψ = 0_Ô = 35_

δ = 2Ô/3η = 0.6

ψ

0.8

lot

(dynamic)

lot

(static)

/

FNc

H

z

FSFS

6 CASE STUDIES

6.1 Survey of Segmental Retaining Walls after the Northridge Earthquake of1994

Sandri (1994) conducted a survey of reinforced segmental retaining walls greaterthan 4.5 m in height in the Los Angeles area immediately after the Northridge Earth-



quake of 17 January 1994 (moment magnitude = 6.7). The results of the survey showedno evidence of visual damage to 9 of 11 structures located within 23 to 113 km of theearthquake epicenter. Two structures (Valencia and Gould Walls) showed tensioncracks within and behind the reinforced soil mass that were clearly attributable to theresults of seismic loading. Data supplied by Sandri and from other sources can be usedto estimate factors of safety against external, internal and facing stability modes of fail-ure for these two structures. The results of these analyses allow a preliminary assess-ment to be made of the applicability of the pseudo-static approach described in this pa-per to actual field performance.

6.2 Valencia Wall

The Valencia wall has a maximum height of 6.5 m and is located at a distance of 23km from the epicenter of the Northridge Earthquake. This wall had the smallest epicen-tral distance of all 11 structures surveyed by Sandri. Three landslides were reported tohave occurred in the general area of the Valencia wall as a result of the earthquake. Thefoundation soils at the Valencia wall site are composed of a deep deposit of silty sandand clay. A cross-section of the wall is shown in Figure 18. The width of the reinforcedmass measured from the toe of the wall is 5.5 m with the exception of the top 2.2 m ofthe structure where the reinforcement lengths were shortened to facilitate placement ofsubsurface utilities. Hence, this upper portion of the wall has a reinforced mass that is1.8 m wide measured from the wall face. Design data for the wall is limited but it ap-pears that the wall was designed for kh = 0.3 and kv = 0 and the effect of the horizontalacceleration was treated as an additional uniform horizontal earth pressure distributionequal to 104 kPa. No data is available to show how this distribution was used (if at all)in stability calculations related to internal and local facing modes of failure.

The estimated peak horizontal ground accelerations at this site range from 0.19g and0.5g based on data from UCB/EERC (1994). The lower value is based on a mean esti-mate taken from a peak horizontal ground acceleration-epicentral distance attenuationcurve for the Northridge earthquake. The maximum value is based on peak ground ac-celeration contours reported for the Northridge earthquake in the Los Angeles area. Asimplifying assumption made by the writers is that the range of peak ground accelera-tions estimated for the site can be used as a conservative estimate of the range of kh val-ues for the backfill soils (i.e. would be conservative for design). This range of valuesalso includes the value of kh that was used in the original design. Application of Equa-tion 7 to this range of peak ground acceleration values according to current FHWAguidelines would result in a mean value of kh = 0.23 and a maximum value of kh = 0.48.However, these adjustments do not influence the general conclusions that follow, andto simplify the analyses, values of kh = 0.19 and kh = 0.5 are used for demonstration pur-poses only. No site-specific vertical acceleration data is available and kv = 0 was as-sumed by the writers. Soil and reinforcement properties used in the stability analysisof the structure are given in Table 1.

The Coulomb failure wedge geometries calculated using Equation 6 for static, meanand maximum kh values are illustrated in Figure 18. Due to the shortened length of rein-forcement at the top portion of the wall, the top 2.2 m of wall was analyzed as a separatestructure (Figure 18a). The results of static and dynamic stability analyses are given inTable 2 for the top 2.2 m height of wall and Table 3 for the entire wall. All of the calcu-



0.500.190.0H=2.2 m

H=6.5 m

Type 1 geogrid

Type 2

Type 3

Type 4

L=5.5 m

MeanMax

48_26_

kh

23

1 2

Static 56.6_1

3

1 2 3

L=1.8 m

Wedge

Observed 6 mm surface cracks


Figure 18. Cross-section view of Valencia Wall showing location and orientation ofinternal failure planes under static and dynamic loading conditions: (a) top portion of wall;(b) entire wall.

(a)

(b)

123456

789

1011

12

Layer number

αAE

αAE

10

11

12

lated factors of safety for the static loading condition are larger than minimum recom-mended values reported in the NCMA guidelines for critical structures (Simac et al.1993).

Analytical results in Table 2 corresponding to mean and maximum kh values show thatfactors of safety for the top 2.2 m of wall are within acceptable limits (i.e. >0.75× minimum allowable static values) with the exception of base overturning (1.18)and reinforcement over-stressing (0.75). However, over-stressing may not be a problemsince the peak seismic loading is transient and the long-term design strength of the rein-forcement based on conventional static design is very conservative for seismic design.A value of 1.18 for dynamic overturning is still well above unity and in the opinion ofthe writers appears unacceptably low only because the default minimum static factorof safety (2.0) is unreasonably high (i.e. FSdyn (dynamic) = (0.75)(2.0) = 1.5). The pre-



dicted dynamic failure wedges are seen to extend well beyond the topmost reinforce-ment layer. The shortening of these layers over the top section of the wall is a designshortcoming and is consistent with visual evidence of distress to this structure reportedby Sandri (i.e. 50 mm wide surface cracks directly at the back of the shortened rein-forcement length portion).

Table 1. Material properties for Valencia Wall and Gould Wall.

PropertiesValues

PropertiesValencia Wall Gould Wall

Soil1

Ô , Friction angle (o) 33 33

γ , Unit weight (kN/m3) 19.8 19.8

Geosynthetic properties2

Index strength (ASTM D 4595) (kN/m)

Type 1 125 --

Type 2 100 --

Type 3 49.5 --

Type 4 35.5 35.5

Design strength (Tallow) (kN/m)

Type 1 and 2 27 --

Type 3 and 4 8.3 8.3

Segmental block properties

Width (toe to heel) (mm) 600 600

Height (mm) 200 200

Length (mm) 450 450

Infilled block weight (kg) 117 117

Interface shear3

au (kN/m) 32.7 32.7

λu (o) 15 15

Connection strength4

Type 1 and 2

acs (kN/m) 23.5 --

λcs (o) 32 --

Fc(max) (kN/m) -- --

Type 3 and 4

acs (kN/m) 17 17

λcs (o) 0 0

Fc(max) (kN/m) 17 17

Notes: (1) Sandri (1994); (2) Manufacturer’s recommended values; (3) unpublished data; and (4) GeoSyntecConsultants (1991).



Table 2. Stability analysis results of top 2.2 m of Valencia Wall.

Static Dynamic

MechanismFactor

of Default Calculated DefaultCalculated minimum

Mechanism ofsafety

Defaultminimum(NCMA)

Calculatedminimum

Defaultminimum kh = 0.19

(mean)kh = 0.5(max)

Base sliding FSbsl 1.5 5.21 1.125 2.94 1.16

Baseoverturning FSbot 2.0 11.30 1.5 2.68 1.18

Reinforcementover-stressing FSos 1.2 1.26 (10) 1 1.14 (10) 0.75 (10)*

Pullout FSpo 1.5 1.75 (12) 1.125 X X

Internal sliding FSisl 1.5 11.52 (10) 1.125 4.15 (10) 1.63 (10)

Localoverturning FSlot 2.0 8.05 (9) 1.5 4.94 (12) 1.81 (12)

Facing shear FSsc 1.5 25.47 (11) 1.125 13.8 (11) 5.94 (11)

Connection FScn 1.5 2.64 (10) 1.125 2.38 (10) 1.57 (10)

Notes: Values in parentheses () refer to reinforcement layer number counted from the bottom of the wall; Xdenotes internal failure plane extends beyond free end of reinforcement; and * factor of safety less than unity.

Table 3. Stability analysis results of entire Valencia Wall.

Static Dynamic

MechanismFactor


Mechanism ofsafety


Calculatedminimum


(mean)kh = 0.5(max)

Base sliding FSbsl 1.5 5.30 1.125 2.07 1.01*



Pullout FSpo 1.5 17.88 (8) 1.125 15.70 (8) X

Internal sliding FSisl 1.5 5.26 (1) 1.125 2.15 (1) 1.02 (1)*

Localoverturning FSlot 2.0 6.42 (1) 1.5 3.80 (8) 1.61 (8)


Connection FScn 1.5 4.78 (5) 1.125 4.20 (5) 2.50 (5)

Notes: Values in parentheses () refer to reinforcement layer number counted from the bottom of the wall; Xdenotes internal failure plane extends beyond free end of reinforcement; and * marginal factor of safety.



Analytical results summarized in Table 3 for the entire structure (i.e. assuming a rein-forced soil mass with a base width of 5.5 m) show that all dynamic factors of safety aregreater than unity. However, the dynamic factors of safety against base sliding (1.01)and internal sliding (1.02) are marginal with respect to collapse for kh = 0.5. The pre-dicted internal Coulomb wedge failure planes plotted in Figure 18b can be seen to inter-sect the soil surface beyond the back of the reinforced soil zone for both kh = 0.19 andkh = 0.5 and is consistent with surface cracking reported by Sandri.

Inspection of dried mud that had accumulated on the face of the retaining wall priorto the seismic event was observed to be undisturbed and intact at the time of post-earth-quake inspection supporting the analytical results that predict no facing instability.

6.3 Gould Wall

The Gould Wall has a maximum height of 4.6 m and is located at a distance of 35 kmfrom the epicenter of the Northridge Earthquake. The structure is founded on rock. Across-section of the wall is shown in Figure 19. Asingle geogrid reinforcement type wasused in this structure and each layer was extended to a uniform length of 3.6 m fromthe face of the wall. Soil and reinforcement properties assumed for the structure are giv-en in Table 1. The wall was not designed for seismic loading. The peak mean and maxi-

kh

0.120.30

0.0H=4.6 m

αAEL=3.6 m

Figure 19. Cross-section view of Gould Wall showing location and orientation of internalfailure planes under static and dynamic loading conditions.


MeanMax

αAE

53_42_

23

Static 56.6_1Wedge

1 2 3

12

34

56789

10

11 Layer number



mum horizontal ground acceleration values at this site are estimated by the writers torange from 0.12g (mean) and 0.3g (maximum), based on interpretation of data fromUCB/EERC (1994) as described for the Valencia Wall. The vertical seismic coefficientvalue, kv , has been taken as zero. As in the Valencia Wall case, the writers have assumedthat peak ground accelerations estimated at the site can be used as a first approximationof the range of kh for demonstration purposes. The results of stability analyses understatic and dynamic loading conditions are given in Table 4. The data in Table 4 for theGould Wall show that factors of safety for the static loading condition are sufficientlygreat that reduced factors of safety using kh = 0.3 at the site do not fall below minimumacceptable levels (i.e. 0.75× minimum allowable static values) with the exception ofreinforcement over-stressing (0.69) and local overturning at the bottom of the unrein-forced portion of the wall (1.35). Comparison of factors of safety for internal and facingmodes of failure for static and dynamic loading conditions show clearly how the mostcritical elevations are higher in the wall under seismic loading conditions than understatic loading conditions. The internal failure plane which is approximately at the loca-tion of the back of the reinforced soil mass under static conditions can be seen to extendbeyond the reinforced soil mass under dynamic loading. This analytical result is consis-tent with the observed surface cracks in this zone. Close inspection of the wall face didnot reveal any evidence of relative movement of the facing units which is consistentwith estimated minimum values of dynamic factors of safety against facing modes offailure which are well above unity (Table 4).

Table 4. Stability analysis results of Gould Wall.

Static Dynamic

MechanismFactor


Mechanism ofsafety


Calculatedminimum


(mean)kh = 0.3(max)

Base sliding FSbsl 1.5 4.85 1.125 2.55 1.38



Pullout FSpo 1.5 3.63 (11) 1.125 2.02 (11) X

Internal sliding FSisl 1.5 5.13 (1) 1.125 2.76 (1) 1.49 (1)

Localoverturning FSlot 2.0 4.58 [ 1.5 2.96 (11) 1.35 (11)


Connection FScn 1.5 2.95 (1) 1.125 2.56 (11) 1.48 (11)

Notes: Values in parentheses () refer to reinforcement layer number counted from the bottom of the wall; Xdenotes internal failure plane extends beyond free end of reinforcement; [ overturning about the toe of thebottom facing unit; and * factor of safety less than unity.



6.4 Discussion of Pseudo-Static Analyses of Valencia and Gould Walls

Seismic stability analyses of the Valencia Wall and Gould Wall using the pseudo-stat-ic approach introduced in this paper are based on a number of assumptions. The writershave used properties for backfill soils reported in the original designs and reinforcementproperties suggested by the manufacturer of the reinforcement materials for static load-ing conditions. In addition, connection performance and interface sliding data has beeninferred from laboratory test results under static loading conditions.

Perhaps the most important assumptions are related to the selection of seismic coeffi-cient values that are implemented in the stability calculations. As discussed in Section2.4.2, strategies to calculate a representative seismic coefficient for wall design basedon site peak ground acceleration values vary widely. Nevertheless, the calculations con-sistently show that for the range of kh values assumed, the inadequacy of reinforcementlengths in the upper portions of each wall is apparent. The design methodology pro-posed herein would have led to their increase. It is interesting to note that tilting of ageogrid-reinforced soil wall that was observed after the Great Hanshin Earthquake of17 January 1995 in Japan has been attributed to inadequate reinforcement lengths at thetop of the structure (Tatsuoka et al. 1995).

Finally, with respect to Valencia and Gould Walls, the writers wish to emphasize thatwhile tension cracks were observed in these structures the function of the structureswasnot compromised in any practical way. In fact, these structures can be judged to haveperformed satisfactorily despite potentially large dynamic loadings.

7 CONCLUSIONS

The paper has presented a pseudo-static Mononobe-Okabe (M-O) approach for limitequilibrium stability analysis of geosynthetic-reinforced segmental retaining walls.The approach extends the Coulomb wedge method that is currently recommended forthe static stability analysis of these types of structures. Calculations are reasonably sim-ple and are framed within the conventional limit-equilibrium approach used by geo-technical engineers for geosynthetic-reinforced soil walls.

The following implications to geosynthetic-reinforced segmental retaining wall de-sign can be made based on the results of a number of parametric analyses presented inthe paper:

1. The method proposed in this paper to calculate internal dynamic earth pressure dis-tributions results in a redistribution of tensile load to reinforcement layers and fac-ing connections located close to the top of the wall. The number of reinforcementlayers at the top of the wall may have to be increased to compensate for the com-bined effect of larger earth forces and redistribution of earth pressures.

2. The progressive inclination of the internal active failure plane with increasing mag-nitude of horizontal seismic coefficient can lead to the requirement for greater rein-forcement lengths at the top of geosynthetic-reinforced soil walls than those lengthscalculated based on static loading conditions.

3. Example calculations for base sliding and overturning about the toe of the gravitymass assumed in geosynthetic-reinforced soil wall design demonstrate that there is



little margin of safety against these modes of failure under seismic loading condi-tions for walls designed to just satisfy minimum factors of safety under static load-ing conditions.

4. The unreinforced portion of the facing column above the uppermost reinforcementlayer was demonstrated to be the most critical portion of a geosynthetic-reinforcedsegmental wall with respect to local shear failure and toppling. However, reductionsin factors of safety against local shear and toppling failure were relatively insensi-tive to the magnitude of vertical seismic coefficient assumed in the example cal-culationswhen compared to the influence of the magnitude of the horizontal seismiccoefficient. Hence, for the unreinforced top portions of these walls, negligible errorresults from assuming kv = 0.

5. Minimizing the height of the top unreinforced portion of the wall is an importantstrategy to ensure adequate factors of safety against local shear failure and topplingof the facing column under seismic loading. Segmental facing units that have posi-tive shear interlock in the form of concrete keys, pins, or other forms of mechanicalconnectors are the preferred choice in segmental retaining wall design to ensure thatthese systems have adequate interface shear capacity.

6. Pseudo-static seismic analysis of two walls that experienced significant ground ac-celerations during the 1994 Northridge Earthquake predicted failure surfaces exit-ing beyond the reinforced soil zone. Observed cracks in the backfill soils can be con-sidered to be consistent with the predicted range of internal failure planegeometries.

7. The observed good performance of the facing column of two walls during the North-ridge Earthquake is predicted by pseudo-static seismic analysis and demonstratesthat geosynthetic-reinforced segmental retaining walls can be designed to withstanda significant earthquake event provided that facing units are able to develop ade-quate interface shear capacity and reinforcement layers are placed close to the crestof the wall.

8 RECOMMENDATIONS FOR FURTHER RESEARCH

The results of this investigation have identified the following research needs relatedto stability analyses of geosynthetic-reinforced soil retaining walls in general and seg-mental retaining walls in particular:

1. The properties of geosynthetic reinforcement materials under rapid loading are notwell understood and new methods to select allowable design loads under seismicloading are required to reduce likely conservativeness in current seismic designmethods.

2. Laboratory testing of the connection formed between the reinforcement and modu-lar facing units is required to provide connection capacity data that is applicable toseismic loading conditions.



3. The effect of rapid cyclic loading on load transfer in the reinforcement anchoragezone is not well understood. Current anchorage models for static loading conditionsshould be investigated and modified to account for the seismic loading condition.

A shortcoming of the pseudo-static method outlined in this paper is that there appearsto be very little guidance on how to select seismic coefficient values based on site-spe-cific ground motion data. Recommendations that are available in the literature varywidely. Another shortcoming of the pseudo-static seismic method of design proposedby the writers is that it can only provide the designer with an estimate of the marginsof safety against collapse of segmental retaining walls, or failure of their components,and doesnot provide any direct estimate of anticipated wall deformations. This is a defi-ciency that is common to all limit-equilibrium methods of design in geotechnical engi-neering. In practice, geosynthetic-reinforced segmental retaining walls may fail be-cause of unacceptable deformations.

More sophisticated analytical techniques are available to the designer that can beused to predict the time-deformation response of these systems. For example, dynamicfinite element analyses have been carried out on reinforced soil structures (Yogendra-kumar et al. 1992; Bachus et al. 1993; Cai and Bathurst 1995). However, the experienceof the writers is that finite element model techniques require material properties thatare seldom available to designers and the interpretation of results by inexperienced us-ers of finite element programs is always a concern.

An alternative strategy for the design of gravity retaining walls is a “displacementmethod” approach (Richards and Elms 1979; Whitman 1990) which can explicitly in-corporate horizontal wall movements in stability analyses. This approach, adapted toreinforced segmental retaining walls, is currently under development by the writers andwill be described in a forthcoming publication.

ACKNOWLEDGEMENTS

The writers would like to thank Mr. M.R. Simac of Earth Improvement Technologiesand Mr. D. Sandri with the Nicolon Mirafi Group for their review of the original manu-script. The writers also acknowledge the efforts of two anonymous reviewers whosecomments materially improved the revised manuscript. The funding for the work re-ported in the paper was provided by the Department of National Defence (DND, Cana-da) through an Academic Research Program (ARP) grant to the senior writer and by theDirector of Architecture (DArch) (DND, Canada).

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NOTATIONS

Basic SI units are given in parentheses.

acs = minimum connection strength (N/m)ah = peak horizontal ground acceleration (m/s2)au = minimum interface shear strength (N/m)av = peak vertical ground acceleration (m/s2)Fc = wall/reinforcement connection capacity (N/m)Fc(max) = maximum wall/reinforcement connection capacity (N/m)Fdyn = dynamic reinforcement force (N/m)Fsta = static reinforcement force (N/m)FSFScn = factor of safety against connection failure between facing units

(dimensionless)FSbot = factor of safety against base overturning (dimensionless)FSlot = factor of safety against local overturning (dimensionless)FSpo = factor of safety against pullout (dimensionless)FSsc = factor of safety against interface shear failure (dimensionless)FSbsl = factor of safety against base sliding (dimensionless)FSisl = factor of safety against internal sliding (dimensionless)g = gravitational constant (m/s2)H = wall height (m)KA = static earth pressure coefficient calculated using Coulomb earth pressure

theory (dimensionless)KAE = dynamic earth pressure coefficient calculated using Mononobe-Okabe

method (dimensionless)kh = horizontal seismic coefficient (dimensionless)kv = vertical seismic coefficient (dimensionless)L = base width of reinforced soil zone plus facing column (m)Lw = width of facing column (m)MR = resisting moment due to weight of facing column (N-m/m)m = ratio of moment arm of dynamic active earth force to wall height

(dimensionless)PA = static active earth force (N/m)PAE = dynamic active earth force (N/m)rF = dynamic reinforcement force magnification factor

= ratio of dynamic reinforcement force to static reinforcement force(dimensionless)

Sdyn = dynamic interface shear force (N/m)Sv = contributory area (m2/m)



Tallow = allowable reinforcement design load (N/m)Yc = moment arm of reinforcement force (m)Ydyn = moment arm of dynamic force from wall base (m)Ww = weight of facing column (N/m)z = depth from crest of wall (m)Vu = interface shear capacity (N/m)αAE = orientation of active failure plane from horizontal under dynamic loading

(_)β = wall backslope angle (_)∆Kdyn = dynamic earth pressure coefficient increment (dimensionless)∆Pdyn = dynamic force increment (N/m)∆Ww = incremental weight of facing column (N/m)δ = interface friction angle (_)Ô = peak friction angle of soil (_)Ôds = angle of geosynthetic-soil interface friction (_)γ = soil or facing column unit weight (N/m3)η = ratio of moment arm of dynamic force increment to wall height

(dimensionless)λ = inertial force reduction factor for gravity mass in external stability

calculations (dimensionless)λcs = slope of connection strength failure envelope (_)λu = interface friction angle between facing units (_)θ = inertia angle (_)ψ = wall inclination angle from vertical (_)

ABBREVIATIONS

AASHTO-AGC-ARTBA:American Association of State Highway and Transportation Officials-Association of General Contractors-American Road and TransportationBuilders Association

CFEM: Canadian Foundation Engineering ManualFHWA: Federal Highway AdministrationNCMA: National Concrete Masonry Association

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