sie nl walls on line

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [Technical University of Crete]On: 21 July 2010Access details: Access Details: [subscription number 912914330]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Structure and Infrastructure EngineeringPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713683556

Effects of soil non-linearity on the seismic response of restrained retainingwallsProdromos N. Psarropoulosa; Yiannis Tsompanakisb; George Papazafeiropoulosba Department of Infrastructure Engineering, Hellenic Air-Force Academy, Greece b Division ofMechanics, Department of Applied Sciences, Technical University of Crete, University Campus,Chania, Greece

First published on: 21 December 2009

To cite this Article Psarropoulos, Prodromos N. , Tsompanakis, Yiannis and Papazafeiropoulos, George(2009) 'Effects ofsoil non-linearity on the seismic response of restrained retaining walls', Structure and Infrastructure Engineering,, Firstpublished on: 21 December 2009 (iFirst)To link to this Article: DOI: 10.1080/15732470903419677URL: http://dx.doi.org/10.1080/15732470903419677

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

Eects of soil non-linearity on the seismic response of restrained retaining walls

Prodromos N. Psarropoulosa, Yiannis Tsompanakisb* and George Papazafeiropoulosb

aDepartment of Infrastructure Engineering, Hellenic Air-Force Academy, Greece; bDivision of Mechanics, Department of AppliedSciences, Technical University of Crete, University Campus, Chania 73100, Greece

(Received 5 May 2009; nal version received 16 October 2009)

Retaining structures characterised by high rigidity and various kinematic constraints, such as bridge abutments andbasement walls, do not permit limit-equilibrium conditions to be developed. Therefore, according to contemporarynorms and geotechnical design practice worldwide, they are most frequently designed utilising the elasticity-basedmethods, which usually lead to substantially increased dynamic earth pressures. The present paper aims to examinehow and to what extent the potentially developed non-linearity of the retained soil may aect: (a) the dynamicdistress of a rigid xed-base retaining wall, and (b) the seismic response of the retained soil layer. For this purpose, aparametric study is conducted which is based on two-dimensional dynamic nite-element analyses using variousidealised or real seismic excitations scaled to several intensity levels. Soil non-linearity is realistically taken intoaccount via the commonly used equivalent-linear approach. The results of the present study demonstrate thatpotential non-linearity of a soil layer retained by a rigid xed-base wall alters the soil amplication pattern behindthe wall and leads to dynamic earth pressures usually lower than those proposed by seismic norms.

Keywords: retaining walls; dynamic wall-soil interaction; soil non-linearity; amplication; earthquake-inducedpressures

1. Introduction

Bridge abutment walls, basement walls, or harbourquay walls are characteristic cases in which a rigidgravity retaining wall or a exible cantilever retainingwall is constructed. More complicated retaining wallstructures are reinforced soil walls, anchored bulk-heads, or tieback walls. The compliance of eachretaining wall type depends on its structural exibility,and the kinematic constraints imposed by otherstructural elements on various locations along itsheight. Experience accumulated from failures occurredin recent earthquakes has shown that in many cases theseismic performance of retaining walls has not met therequirements of an operational retaining structure.There are many examples of retaining walls (primarilyharbour quay walls) that have failed during a seismicevent (Kobe, Japan 1995 and Lefkada, Greece 2003).Retaining wall failures can cause great economicallosses for a local region, especially when they are usedto support vital structures and infrastructures. Forinstance, severe damages took place in the Kobe portarea during the 1995 earthquake. In addition, manycases of bridge failures have been reported duringrecent earthquakes due to excessive abutment displace-ment. Seismic vulnerability of retaining systems isusually related to the strength degradation of saturated

cohesionless soils in the backll and the foundation.A very limited number of cases of high earth pressureshave been reported. Nevertheless, constrained rigidretaining systems seem to be designed rather conserva-tively. In any case, as it will be shown in the sequence,a more accurate evaluation of dynamic earth pressurescan reduce either the cost or the risk, depending on thecircumstances of each problem and the performanceobjectives of the design.

Despite its structural simplicity, the seismic re-sponse of a wall (retaining even a single soil layer) is arather complicated problem. The dynamic interactionbetween the wall and the retained soil is the primaryfactor that increases the complexity of the problem,especially when material and/or geometry non-linear-ities are considered (Kramer 1996, Wu and Finn 1999,Al-Homoud and Whitman 1999, PIANC 2001, Greenand Ebeling 2002, Moayyedian et al. 2008). Seismicresponse of various types of walls that support a singlesoil layer has been examined by a number ofresearchers in the past either experimentally, analyti-cally, or numerically (e.g. Veletsos and Younan 1997,Iai 1998, Wu 1999, Psarropoulos et al. 2005).

Depending on the material behaviour of the retainedsoil and the possible mode of the wall displacement,there exist two main categories of analytical methodsused in the seismic design of retaining walls:

*Corresponding author. Email: [email protected]

Structure and Infrastructure Engineering

2009, 112, iFirst article

ISSN 1573-2479 print/ISSN 1744-8980 online

2009 Taylor & FrancisDOI: 10.1080/15732470903419677

http://www.informaworld.com

Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

Pseudo-static methods incorporating the limit-equili-brium concept (Mononobe-Okabe type solutions),which assume yielding walls and rigid-perfectly plasticbehaviour of the retained soil (Okabe 1926, Mononobeand Matsuo 1929, Seed and Whitman 1970).

Elasticity-based solutions that regard the retained soil asa linear (visco-) elastic continuum (Scott 1973, Wood1975, Veletsos and Younan 1997).

According to an ecient simplication of theMononobe-Okabe (M-O) method, developed by Seedand Whitman (1970), the maximum dynamic activeearth force imposed on the wall is given by:

DPAE 0:4ArH2 1

where A is the peak base acceleration, r is the soil massdensity, and H is the wall height. In contrast, theelastic solutions developed by Scott (1973) or Wood(1975) suggest that for the case of low-frequency(quasi-static) base motions, which refers to manypractical problems, the dynamic active earth forcedeveloped on a rigid xed-base wall is equal to:

DPAE ArH2 2

Dynamic response of rigid bridge abutments isconsidered as a very important design problem inseismic regions, therefore many studies have beenfocused on this topic over the last years (Siddharthanet al. 1994, Fishman and Richards 1995, 1997, Imbsenet al. 1997, Al-Homoud and Whitman 1999, MCEER2001, Munaf et al. 2003, Basha and Babu 2009, amongothers). Seismic norms usually correlate the intensityand the distribution of the dynamic earth pressureswith the ability of the wall to move and/or deform.In the case of a deformable wall, pseudo-static

methods are adopted by the majority of seismic norms,whereas in cases of walls constrained against deforma-tion and movement, elasticity-based solutions aremost frequently adopted. The above are shown inmore detail in Figure 1, where provisions of the GreekRegulatory Guide E39/99 (1999) for the seismicanalysis of bridge abutments are given. The dynamicpressure distribution depends on the ratio of thehorizontal displacement of the top of the wall to thewall height u/H. For the case of a very exible wall(u/H 0.10%), the Greek Regulatory Guide adoptsthe Seed & Whitman method and suggests that themaximum dynamic earth force is equal to:

DPAE 0:375ArH2 0:4ArH2 3

and acts at a height of 0.6H above the wall base.For a moderate exibility of the wall (0.05% u/H 0.10%) then the maximum dynamic earth force isderived by:

DPAE 0:75ArH2 4

acting at a height of 0.5H above the wall base. Finally,for a rather rigid wall (u/H 0.05%) the GreekRegulatory Guide, following the Woods solution,proposes that the maximum force is given by Equation(2) and acts at a height of 0.58H above the wall base.

According to contemporary seismic provisions(NAVFAC DM7.02 1986, ATC32-1 1996, EC8 2003,FEMA450 2003), the method applied for the deter-mination of the proper seismic coecient for theseismic design of retaining systems is the well-knownMononobe-Okabe method, modied accordingly toincorporate the behaviour of the retaining wall. InEurocode 8 (EC8 Part 5) three basic parameters are

Figure 1. Provisions of the Greek Regulatory Guide E39/99 for the seismic analysis of bridge abutments.

2 P.N. Psarropoulos et al.

Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

considered in order to determine the seismic coecient:the design acceleration, the soil factor (which takesinto account the ground type) and the reduction factorto account for the expected or acceptable displacementof the wall during earthquake loading. In any case,seismic norms propose that the seismic earth pressuresacting upon a rigid retaining wall are more than twicethose acting upon a deformable retaining wall, giventhat the wall dimensions, soil properties and peakground acceleration remain constant. As will be shownin the sequence, this is not always true. The twoaforementioned modes of wall-soil system behaviourare rather extreme, and in many cases fail to berealistic due to their exaggerated assumptions. Thelimit-equilibrium solutions imply the capability of thesystem to develop relatively large displacements (geo-metric non-linearity) together with the formation ofplastic zones (material non-linearity). In contrast, theelasticity-based solutions may take into account onlythe wall exibility and/or the wall foundation com-pliance (Veletsos and Younan 1997).

There exist many cases, such as bridge abutments,basement walls, or braced excavations, in which theexistence of kinematic constraints on the wall move-ment is incompatible with the limit-equilibrium con-cept, while on the other hand, the available elasticity-based solutions overlook the potential non-linearbehaviour of the retained soil, leading thus to eitherunsafe or over-conservative solutions. It has to bestressed that restrained basement walls, bridge abut-ments and other rigid retaining walls subjected todynamic loading are frequently encountered in geo-technical engineering practice. A basement wall or abridge abutment (such as those shown in Figure 2)is generally a non-compliant retaining structure.Dierent reasons in each case prevent the retaining

wall from yielding. In the case of a restrained basementwall, the rigidity is induced by the existence of concreteslabs at its top and its bottom which conne itsdisplacements. Similarly, a bridge abutment, the struc-ture is conned against displacement at its top by thebridge deck and usually at its bottom by deep piles.

Apart from the dynamic earth pressures developedon the non-compliant retaining walls considered in thisstudy, emphasis is also given on the soil amplicationof the base acceleration. Note that seismic norms(such as the Eurocode 8 (EC8 2003), or the GreekSeismic Code (EAK 2000)), being based on the limit-equilibrium methods, underestimate the role of thepotential soil amplication behind a retaining wall.For this reason, the objective of the present study isto examine thoroughly the inuence of material non-linearity not only on the dynamic distress of bridgeabutments and similar types of non-deformableretaining walls in general, but also on the soil ampli-cation of the base acceleration. For this purpose, two-dimensional numerical simulations are performed,utilising the nite-element method in order to investi-gate some of the most important aspects of thecomplex phenomenon of dynamic non-linear wall-soilinteraction.

A parametric study has been performed in order toexamine how the level of applied acceleration mayaect: (a) the dynamic earth pressures induced on suchretaining structures, and (b) the soil amplicationbehind a retaining wall. Dynamic response of anysystem depends on the seismic excitation character-istics (both in the time and in the frequency domain).In order to examine more thoroughly the aforemen-tioned complex phenomena the excitations utilised inthis study, include harmonic and simple pulses and realexcitations. Material non-linearity is taken into

Figure 2. Two cases of restrained retaining walls: (a) a basement wall conned against displacement at its top and its bottomby concrete slabs, and (b) a bridge abutment conned against displacement at its top by the bridge deck and at its bottom bypiles.

Structure and Infrastructure Engineering 3

Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

account in a simplistic, yet ecient way, throughthe use of an iterative equivalent-linear procedure inwhich strain-compatible shear modulus, G, and criti-cal damping ratio, x, are used to describe the soilbehaviour within each iteration.

Results provide a clear indication of the degree towhich both dynamic response of restrained walls andsoil amplication of ground acceleration are aectedby soil non-linearity, a crucial parameter which hasbeen ignored by the seismic norms as far as rigidretaining walls are concerned. Soil non-linearity seemsto increase the degree of complexity, being eitherbenecial or detrimental for the distress of a rigidretaining system, depending on the circumstances. Thisfact justies the necessity for a serious consideration ofpotential soil non-linearity by the seismic normsapplied worldwide.

2. Numerical modelling

In order to examine the non-linear dynamic distress ofrestrained walls and the non-linear soil amplication,two-dimensional (2D) numerical simulations of theretaining system depicted in Figure 3 were conducted.The simulations were performed utilising the popularQUAD4M nite-element code developed by Hudsonet al. (1993), which performs dynamic non-linearanalyses incorporating the well-known iterativeequivalent-linear procedure. Each iteration includes:

. A linear direct-integration dynamic analysis ofthe model.

. The calculation of the maximum eective shearstrain, ge, for each element (calculated as apercentage of the maximum strain).

. The calculation of the strain-compatible shearmodulus G(ge) and critical damping ratio x(ge)to be used in the next iteration, by means ofG/Gmax 7 g and x 7 g curves.

The procedure is terminated when convergencein the values of G and x occurs. The G/Gmax 7 g andx 7 g curves used (see Figure 4) are characteristic of

sandy soil material (Seed and Idriss 1970, Idriss 1990),modied accordingly to provide critical damping ratioequal to 5% in the low shear strain range.

As the wall exibility is examined in relation to soilstiness and the earth pressures are normalised withrespect to r and H, the soil material properties and thewall height values do not aect the dynamic pressureson the wall (Veletsos and Younan 1997). This was alsoveried in a recent numerical study by Psarropouloset al. (2005) where linear analyses of the examinedmodel were performed utilising ABAQUS software.With respect to the inhomogeneity of the soil, it hasalso been shown in the aforementioned paper that it isbenecial for the walls in the elastic range, since it wasproven that the dynamic earth thrust is substantiallylower compared to the one developed in the caseof homogeneous soil. Therefore, a single homogenoussoil layer and the parameters used in that studyhave also been applied in the current non-linearinvestigation.

All equivalent-linear analyses in QUAD4M wereperformed considering an 8-m-high wall and theretained soil layer is characterised by a relatively lowsmall-strain shear-wave velocity VS equal to 100 m/sand a mass density of 1.8 tn/m3. The discretisationof the retained soil was performed by four-nodedplane-strain quadrilateral elements. The model wasadequately elongated so as to reproduce adequatelythe free-eld conditions at its right-hand side. In orderto simulate the dynamic response of restrained walls,the wall is considered to be rigid and xed at its base.The rigid wall was simulated by an extremely sticolumn with linear elastic behaviour. The simplifyingassumption of no de-bonding or relative slip at thewall-soil interface was used.

The base of the wall and the soil stratum wereconsidered to be excited by a horizontal motion,assuming an equivalent force-excited system. Initially,the model was subjected to harmonic and Rickerpulses which allow for a better understanding andinterpretation of the results. Furthermore, the resultsof harmonic excitations can be easily generalised forany real earthquake excitations via Fourier transfor-mations. Subsequently, two characteristic real earth-quake records were used in the numerical analysis:the Shin-Kobe record from the 1995 Kobe (Japan)earthquake and the record from the 1995 Aegion(Greece) earthquake. Both real records used in thisstudy dier substantially with respect to their fre-quency content, while they are characterised by peakground acceleration (PGA) values close to the max-imum intensity level considered in this work (0.50 g).More specically, the record from Aegion earthquakehas PGA equal to 0.54 g, while Shinkobe record hasPGA equal to 0.52 g. The acceleration time-history of

Figure 3. The retaining system examined in the presentstudy: a rigid xed-base wall retaining a single soil layer withstrain-dependent material behavior, both excited by anacceleration time-history A(t).


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

the Ricker pulse together with its Fourier spectrum aregiven in Figure 5, while the acceleration time-historiesand the corresponding response spectra (for 5%damping) of the earthquake records of Aegion andShin-Kobe are shown in Figure 6. Note that plottedtime-histories are scaled to 0.50 g, while their spectraare scaled to 0.01 g.

Various characteristic levels of peak base accelera-tion (PGA) were used in this study, aiming at thedevelopment of dierent degrees of material non-linearity. The desired intensity levels were obtained by

scaling the aforementioned pulses and seismic excita-tions to specic peak base acceleration values. Asthe frequency content of the imposed acceleration time-histories varied substantially, their scaling was per-formed in terms of PGA values to maintain thefrequency content of each input motion almost un-changed and use only its PGA as the varying parameter.

It has to be noted at this point that, as it will befurther discussed in the numerical results, the use ofbase instead of surface peak values for scaling purposesis reasonable for the examined problems, since thesurface acceleration time-histories and their corre-sponding peak values vary substantially depending onthe distance of the surface backll point from the rigidwall. Moreover, due to non-linearities (which arerelated to base PGA levels) the eigenfrequency of thesoil layer is not only a function of its height as in thelinear case, thus, surface response and the resultingpeak values are dierent for each intensity level. Inaddition, having a unique reference acceleration valuefor each analysis is also more suitable for the normal-isation of the results. Nevertheless, regardless of thechoice of the scaling parameter, the rather minorscaling of the records that has been used in thisinvestigation does not induce any signicant bias on thenon-linear response of the system (Shome and Cornell1999).

3. Soil amplication of the retained backll

3.1. Linear response

3.1.1. Harmonic excitation

The dynamic linear response of a single soil layer withhorizontal stratigraphy, that actually resembles 1Dconditions, has been studied by many researchers, and

Figure 5. (a) Acceleration time-history (A is scaled to peak acceleration of 1m/s2), and (b) Fourier spectrum of the Ricker pulseexcitation (with central frequency fR 2 Hz).

Figure 4. The G/Gmax g and x g curves (modied by theauthors based on Seed and Idriss 1970; Idriss 1990) used todescribe the non-linear behavior of the retained soil.


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

analytical solutions for harmonic excitation can befound in the literature (Roesset 1977, Kramer 1996). Inthe case of harmonic excitation the response iscontrolled by the ratio f/fo, where f is the dominantperiod of the excitation, and fo the fundamental periodof the soil layer. In the model examined in this studythe fundamental eigenfrequency of the soil layer fo isalmost 3.1 Hz (or equivalently, the fundamentaleigenperiod of the soil layer To is 0.32s). The durationof the sinusoidal pulse was such that steady stateconditions were reached. In that case the maximumamplication factor (AF) for linear response is given bythe simple expression:

AF 2px

1

2n 1 5

where x is the critical damping ratio and n theeigenmode number. For rst mode (n 0) andx 5%, AF is approximately equal to 12.7.

In this study the response of the soil layer under 1Dconditions is compared with the corresponding re-sponse of the model shown in Figure 3. The presenceof a rigid retaining wall essentially imposes a verticalboundary condition, leading thus to a 2D model. Inaddition, this model has a fundamental low-straineigenfrequency slightly lower than the corresponding1D model, due to the fact that the existence of the rigidwall makes the model stier. However, the dierence inthe two values of eigenfrequency is considerednegligible as it is lower than 0.03 Hz. In addition, theresults of the equivalent linear analysis for very lowintensity levels (A 0.0001 g) are identical with theresults of linear analysis.

The distribution of the amplication factor (AF) onthe surface of the backll in the case of harmonicexcitation at resonance (f fo) is plotted in the uppercurve (for A 0.0001 g) in Figure 7. It is evident that,for the rigid xed-base wall examined, the motion inthe vicinity of the wall is practically induced by thewall itself, and therefore, no amplication is observed(AF 1). The amplication factor converges to itsmaximum value (AF 12.7) at a distance of 10H fromthe wall, since at that distance 1D conditions arepresent (free-eld motion). Note that this distance wasalso calculated by Wood (1975), as the minimumdistance needed to eliminate the eects of the wall onthe retained soil.

3.1.2. Ricker pulse excitation

As previously mentioned, apart from harmonic excita-tions a Ricker pulse with central frequency fR 2Hzhas also been used in the present study (Ricker 1960).Despite the simplicity of its waveform, this waveletcovers smoothly a broad range of frequencies up tonearly 3fR ( 6Hz). The acceleration time-history andthe corresponding Fourier spectrum of this pulse aregiven in Figure 5. Figure 8 depicts the waveforms ofthe acceleration time-histories on the surface of theretained soil layer for the rigid wall excited with theRicker pulse. It is obvious that in the vicinity ofthe rigid wall the amplication is approximately equalto unity, while in a similar distance ( 10H) as in thecase of the harmonic excitation, the response of the soilconverges to free-eld conditions.

3.2. Non-linear response

The aforementioned results referring to the case oflinear soil behaviour are valid for very low levels ofbase input acceleration, when the induced strains

Figure 6. The acceleration time-histories of the recordsfrom the 1995 Aegion, Greece, earthquake (top) and the 1995Kobe, Japan, earthquake (bottom), both scaled to peakground acceleration equal to 0.50 g. The correspondingresponse spectra, both scaled to 0.01 g, are also shown.


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

remain small (g 5 0.005%). However, when themaximum acceleration acting on the soil mass takesmore realistic values, the induced strains are substan-tially greater, and thus, the impact of material non-linearity (expressed by the G/Gmax 7 g and x 7 gcurves) becomes more evident. The equivalent linearapproach (in which as previously described a few linearanalyses are performed in each cycle), is consideredaccurate enough for shear strains lower than 1% (see

Kramer 1996). In the current study the only case inwhich the induced seismic shear strains increasesubstantially (approaching values in the range of 1%)is the case of the highest acceleration level and highmean period earthquake (i.e. the Kobe excitation withPGA 0.50 g). Therefore, the results of the per-formed dynamic nite-element analyses can be re-garded as accurate.

The wall-soil system behaviour strongly depends,not only on the level of applied acceleration, but alsoon the f/fo ratio, as it is justied by the subsequentresults. The distribution of the amplication factor onthe surface of the backll in the case of the harmonicexcitation at resonance (f fo) is plotted in Figure 7,for ve levels of peak base acceleration: 0.0001 g(corresponding practically to a linear soil behaviour),0.12 g, 0.24 g, 0.36 g, and 0.50 g, covering a broadrange of the induced dynamic strains. Note that in therange of small shear strains the critical damping ratio,x, was set equal to 5%, instead of the much lowervalues of the curves proposed by Seed and Idriss(1970), in order to ensure that the theoreticalamplication (AF 12.7) for linear conditions is alsonumerically achieved for the lowest peak base accel-eration case (0.0001 g). As it was expected, increasingthe degree of material non-linearity makes the systemmore exible, thus decreases its fundamental Eigenfrequency and leads to detuning. This phenomenoncan be easily observed in Figure 7 by noticing thesubstantially reduced values of AF for all levels (0.12 gto 0.50 g) of non-linear behaviour.

4. Dynamic distress of restrained walls

4.1. Harmonic excitation

4.1.1. Linear response

In order to calculate the linear elastic dynamic earthpressures which develop behind a rigidly restrainedretaining wall (bridge abutment or basement wall), twoharmonic excitations were employed, which had twocharacteristic frequencies: the rst was set equal to thelow-strain fundamental eigenfrequency of the soil layer(f fo), and the second pulse had frequency six timeslower (f fo/6), approximating a quasi-static excita-tion. The curves for A 0.0001 g in Figure 9 andFigure 10 presents the height-wise distribution of thenormalised induced linear elastic dynamic earthpressures for the two harmonic excitations examined.

According to common practice (Veletsos andYounan 1997, Psarropoulos et al. 2005), all dynamicdistress results presented here are normalised withrespect to basic mechanical and geometrical para-meters of the model. In particular, the dynamic earthpressures are normalised with respect to rHA and the

Figure 8. Linear analysis results: waveforms of theacceleration time-histories along the surface of the retainedsoil layer excited with the Ricker pulse. It is evident that inthe vicinity of the rigid wall there exists no amplication.

Figure 7. Non-linear analysis results: distribution of the soilamplication factor (AF) along the surface of the backll inthe case of harmonic excitation with frequency f equal to thelow-strain rst Eigen frequency fo.


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

dynamic earth thrust is normalised in terms of rH2A.Therefore, if any of the three basic parameters, namely,the soil density r, the soil layer height H andacceleration amplitude A change, new estimates ofthe dynamic distress measures can be calculated fromtheir normalised values. For instance, a change in Hwill aect only the eigenfrequencies of the retainingsystem (the higher the soil layer thickness, the lowerthe eigenfrequencies). An increase in the soil density rwill make the system stier, thus, the degree of non-linearity will be reduced.

4.1.2. Non-linear response

Figure 9 depicts the height-wise distribution of thenormalised induced dynamic earth pressures for theve levels of peak base acceleration examined inthe case of the harmonic excitation with f fo.As the level of applied acceleration increases thedynamic earth pressures decrease. In particular, thenormalised dynamic earth force on the wall reduces to

values ranging from 0.60 to 0.90 (corresponding toA 0.50 g and A 0.12 g levels, respectively) com-pared to the previously calculated value of 3.00 forthe linear soil behaviour case. It is evident that forexcitations having dominant frequency close to thelow-strain fundamental eigenfrequency of the retainedsoil layer, the material non-linearity seems to act in abenecial way.

The case of quasi-static excitation is of greaterinterest. In Figure 10 the height-wise distribution ofthe normalised induced dynamic earth pressures isplotted for the case of the low-frequency harmonicexcitation (f fo/6) for the ve levels of peak baseacceleration examined. Apparently, for higher levels ofpeak base acceleration, which incur non-linear responseof the retained system, the normalised dynamic earthpressures are always higher than those correspondingto the linear elastic response. The two envelope curvesfor the harmonic excitations shown in Figure 11 (whichdepicts the cumulative results for all the examinedexcitations) stem from the results shown in Figures 9and 10, as it presents DPAE as a function of peak baseacceleration A. Taking also into consideration thevalues obtained using Equations (1) and (2) (i.e. 0.4 and1.0, respectively), one could easily notice that when thefundamental frequency of the input motion f ap-proaches that of the retained soil layer fo, the dynamicearth force DPAE is almost three times greater in thecase of resonance, compared with the correspondingvalue in the linear (A 0.0001 g) case of quasi-staticexcitation, which is almost equal to unity.

This seeming discrepancy can be attributed to thefact that at certain increased levels of induced strain,the shear modulus of the retained soil becomes lower,according to the G/Gmax 7 g curves proposed by Seedand Idriss (1970). Therefore, this softening of the wall-soil system shifts its fundamental Eigen frequency tolower values, closer to resonance with the applied inputexcitation, and thus, leads to an increased distress.For higher levels of induced strain (A 0.50 g), theresonance is avoided since the excitation frequencyexceeds the fundamental low-strain eigenfrequency ofthe retained system. Therefore, for the case of low-frequency harmonic excitation, the eects of materialnon-linearity may be benecial or detrimental depend-ing on the circumstances.

4.2. Ricker pulse excitation

The height-wise distribution of the normalised induceddynamic earth pressures in the case of the Ricker pulseexcitation is plotted in Figure 12, for all ve levels ofpeak base acceleration examined. The pattern revealedin Figure 9 for the harmonic excitation with f fo isrepeated in this case, due to the fact that the excitation

Figure 10. Non-linear analysis results: height-wisedistribution of the normalised induced dynamic earthpressures for the low-frequency harmonic excitation(f fo/6).

Figure 9. Non-linear analysis results: height-wisedistribution of the normalized induced dynamic earthpressures for the harmonic excitation with frequency fequal to the low-strain rst eigenfrequency fo.


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

pulse includes a broad range of frequencies closeto the fundamental frequency of the retained soil layer(fo 3 Hz), as shown in the Fourier spectrum of thepulse shown in Figure 5(b). As a result, despite thelower levels of linear elastic earth pressures in this case,the system response is quite similar to that caused bythe harmonic excitation at resonance.

As this Ricker pulse covers smoothly the range offrequencies between 1 and 5 Hz, it provides an ecientway to comprehend the eect of material non-linearityon the wall distress in the frequency domain as well.Figure 13 presents the variation of the pressureamplication factor (PAF) as a function of frequency.This amplication parameter is dened as:

PAF FFT DPAE t

FFT A t 6

where FFT DPAE t

is the Fourier spectrum of thenormalised induced dynamic earth force time-history

DPAE t , and FFT[A(t)] is the Fourier spectrum of theacceleration time-history of a Ricker pulse excitationwith unit peak value (see Figure 5).

It is evident that in the case of linear soil behaviour,PAF reaches its maximum value frequencies close tothe fundamental frequency of the retained soil layer.This result matches the value calculated previously inthe case of linear harmonic response at resonance.Additionally, for low-frequency excitations, the valueof PAF converges to that proposed by Scott (1973) andWood (1975) as previously calculated. By examiningFigure 13 it is obvious that for increased levels of peakbase acceleration, the development of material non-linearity not only aects the maximum value of PAF,but also shifts the range of its maximum valuestowards lower frequencies. This phenomenon can beeither benecial or detrimental, depending on thepredominant frequency of the input motion.

4.3. Seismic excitations

As it was previously mentioned, two real earthquakerecords are used in the present study. The rst recordwas derived from the Shin-Kobe station during theearthquake of Kobe in Japan in 1995, and the second istaken from the OTE Building in Aegion, during the1995 Aegion earthquake in Greece. Figure 6 shows theacceleration time-histories of these earthquake records,both scaled to peak acceleration of 0.50 g. The height-wise distribution of the normalised induced dynamicearth pressures in the cases of the Aegion and the Shin-Kobe records is plotted in Figure 14, for ve levels ofpeak base acceleration. The maximum pressure dis-tribution occurs for very low acceleration amplitudeimposed by the seismic excitation, for both earth-quakes. Furthermore, the pressure distributions in thecase of linear elastic response (0.0001 g) are nearly the

Figure 12. Non-linear analysis results: height-wisedistribution of the normalised induced dynamic earthpressures for the case of the Ricker pulse excitation.

Figure 13. Non-linear analysis results: pressureamplication factor (PAF) values for the case of the Rickerpulse excitation.

Figure 11. Non-linear analysis results: maximumnormalised dynamic earth force as a function of peak baseacceleration A, for all the excitations examined. Graph alsoincludes the proposals made by Wood and Seed andWhitman (which are also adopted by the Greek RegulatoryGuide E39/99).


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

same for both earthquakes. Moreover, it is observedthat the pattern of the pressure distributions is thesame with that in the case of Ricker excitation, and itdiers essentially from the corresponding one for thequasi-static harmonic excitation.

4.4. Discussion of the results

As mentioned previously, Figure 11 presents themaximum normalised dynamic active earth force as afunction of peak base acceleration A for the veexcitations examined in this study. In the same plot,the values of DPAE proposed by Wood (1975) andby Seed and Whitman (1970) are also included forreference. As it has been mentioned in the introductorysection of this study, Woods solution is identical to theguidelines of the Greek Regulatory Guide E 39/99 forrigid bridge abutments, whereas the dynamic earthforce proposed by Seed and Whitman is identical tothe provisions for exible walls.

It can be observed that in the case of linearresponse the wall distress is dominated by the

frequency content of the excitation. More specically,DPAE varies between the values of 1.0 and 3.0, beingthus always higher than the standard bounding valuesadopted by the seismic norms (noted as Wood 1975and Seed and Whitman 1970 (M-O)). It has to bementioned that, the divergence is maximum when thefundamental frequency of the base excitation equalsthat of the retained soil layer under 1D linear elasticconditions, i.e.:

fo VS4H

Gmax r

p4H

7

where VS is the shear-wave velocity of the soil, andGmax is the corresponding small-strain shear modulus.Nevertheless, as the degree of non-linearity increasesthe induced dynamic active earth force decreasessubstantially, ranging between the aforementionedWood and M-O bounds in the cases of the harmonicresonant motion, the Ricker pulse and the two recordsfrom Kobe and Aegion earthquakes.

In contrast, the distress in the case of low-frequency (quasi-static) harmonic excitation and non-linear response is always higher than the upper bound(Woods solution) for all levels of peak base accelera-tion, and is approximately 50% greater than the valueof Woods solution. An important conclusion resultingfrom Figure 11 is that, for high values of the imposedbase acceleration, the resulting force approximatesin general the proposal of Seed and Whitman, eventhough the limit-equilibrium conditions (imposedby Coulombs static theory, or its pseudo-staticextension of M-O) are not valid in the specic retainingsystem. In other words, the force acting on the back ofa yielding retaining wall (resulting from the weight of arigid wedge of soil above a planar failure surface,according to M-O theory) coincides with the force thatacts on the back of a restrained wall (resulting from theearth pressures of a yielding soil material).

In current engineering practice, the dynamic earthpressures depend on the PGA and the compliance ofthe wall and/or its foundation. The current studyhas shown that the potential soil non-linearity mayaect substantially the distribution of earth pressures,usually leading to a decrease in the wall distress.Therefore, the following equation that contains areduction factor l could be adopted in seismic norms:

DPAE lArH2 8

According to the cumulative results and the relateddiscussion of Figure 11 the reduction factor l inEquation (8) will be a function of PGA levels. Morespecically, for small intensity levels (A 5 0.1 g),no reduction factor is required, thus l 1; for

Figure 14. Non-linear analysis results: height-wisedistribution of the normalized induced dynamic earthpressures in the cases of Aegion record (top) and Shin-Kobe record (bottom).


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

medium intensity levels (A 0.1 g to 0.3 g), it will beequal to l 0.75; nally for greater intensity levels(A 4 0.3 g) then it will be set as l 0.5, actuallyconverging to M-O solution.

5. Conclusions

The present study has examined how and to whatextent the potential soil non-linearity that a retainedsoil layer exhibits under moderate or severe seismicexcitations can possibly aect: (a) the seismic responseof the retained soil layer itself, and (b) the dynamicdistress of restrained walls. It was found that soil non-linearity reduces in general the soil amplication of theretained soil and the dynamic earth pressures, leadingthus to lower wall distress. However, as soil non-linearity alters the eigenfrequencies of the wall-soilsystem, there is (under certain circumstances) thepossibility that increased non-linearity may lead toan amplied response. This phenomenon is moreprobable to occur when the frequency content of theexcitation is narrow and concentrated around afundamental frequency that is lower than the lineareigenfrequency of the soil layer.

A major conclusion of this study is the fact thatthe seismic active earth force induced on restrainedwalls is not necessarily much larger than the corres-ponding force induced on yielding retaining walls.It depends on the potential non-linearity of theretained soil and the relationship between the funda-mental frequency of the imposed excitation and thefundamental large-strain eigenfrequency of the retain-ing system. Consequently, the existence of a rigidretaining wall, such as a bridge abutment or abasement wall, does not necessarily entail that itsdistress will be much greater than the distress inducedon the corresponding yielding wall, as it is statedin the majority of seismic provisions. Therefore,these provisions have to be modied accordingly, assuggested by the ndings of this study, to include theeects of soil non-linearity on the seismic distress ofrestrained walls.

Acknowledgements

The authors would like to acknowledge the assistance of V.Zania and S. Tsimpourakis and to thank the anonymousreviewers for their constructive comments.

References

Al-Homoud, A.S. and Whitman, R.V., 1999. Seismicanalysis and design of rigid bridge abutments consideringrotation and sliding incorporating non-linear soil beha-viour. Soil Dynamics and Earthquake Engineering, 18,247277.

ATC32-1, 1996. Improved seismic design criteria for Califor-nia bridges: Resource document. USA: Applied Technol-ogy Council.

Basha, B.M. and Babu, G.L.S., 2009. Computation of slidingdisplacements of bridge abutments by pseudo-dynamicmethod. Soil Dynamics and Earthquake Engineering, 29,103120.

E39/99, 1999. Greek regulatory guide for the seismic analysisof bridges. Athens, Greece: Ministry of Public Works (inGreek).

EAK, 2000. Greek seismic code. Athens, Greece: GreekMinistry of Public Works (in Greek).

EC8, 2003. Eurocode 8: Design of structures forearthquake resistance. European Standard CEN-ENV-1998-1. Brussels: European Committee forStandardisation.

FEMA 450, 2003. NEHRP2003: Recommended provisions forseismic regulations for new buildings and other structures.Part 2: Commentary Chapter 7: Commentary foundationdesign requirements. USA: Building Seismic SafetyCouncil (BSSC).

Fishman, K.L. and Richards, R.J., 1995. Seismic vulner-ability of existing bridge abutments [online]. NCEERBulletin, 9 (3). Available from: http://mceer.bualo.edu/research/HighwayPrj/bullarticles/Jul95Vol9No3Pg8.asp[Accessed November 2008].

Fishman, K.L. and Richards, R.J., 1997. Seismic analysisand design of bridge abutments considering sliding androtation. Technical Report NCEER-97-0009. NCEER,Bualo, NY.

Green, R.A. and Ebeling, R.M., 2002. Seismic analysis ofcantilever retaining walls. Phase I Report ERDC/ITLTR-02-3. Washington, DC: US Army Corps ofEngineers.

Hudson, M., Idriss, I.M., and Beikae, M., 1994. Usersmanual for QUAD4M. Davis, USA: Center for Geotech-nical Modeling, Department of Civil and EnvironmentalEngineering, University of California.

Iai, S., 1998. Seismic analysis and performance of retain-ing structures. In: P. Dakoulas, M. Yegian, andR.D. Holtz, eds. Proceedings of geotechnical earth-quake engineering and soil dynamics III, GeotechnicalSpecial Publication No. 75. Reston, VA: ASCE, 10201044.

Idriss, I.M., 1990. Response of soft soil sites duringearthquakes. In: J.M. Duncan, ed. Proceedings ofH. Bolton Seed Memorial Symposium, Vol. 2. May,Berkeley, CA. Vancouver, B.C., Canada: BiTech Publish-ers Ltd., 273289.

Imbsen, R.A., et al., 1997. Structural details to accommodateseismic movements of highway bridges and retainingwalls. Technical Report NCEER-97-0007, NCEER,Bualo, NY.

Kramer, S.L., 1996. Geotechnical earthquake engineering.New Jersey: Prentice-Hall.

MCEER, 2001. Recommended LRFD guidelines for theseismic design of highway bridges. NCHRP Project12-49. Bualo, NY: MCEER.

Moayyedian, M., Moslem, K., and Shooshtari, A., 2008.Proposed dynamic soil pressure diagram on rigid walls.IJE Transactions A: Basics, 21 (3), 213224.

Mononobe, N. and Matsuo, H., 1929. On the determinationof earth pressures during earthquakes. In: Proceedings ofthe World Engineering Congress, Vol. 9, Paper 388,OctoberNovember, Tokyo, Japan. Tokyo: JSCE,177185.


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

Munaf, Y., Prakash, S., and Fennessey, T., 2003. Geotech-nical seismic evaluation of bridge abutments in south-east Missouri. In: Proceedings of the 2003 PacicConference on Earthquake Engineering, 1315 February,Christchurch, New Zealand. Christchurch, NZ:Department of Civil Engineering, University of Canter-bury. CD ROM Paper No 5.

NAVFAC DM7.02, 1986. Foundations and earth retainingstructures design manual. Alexandria, Virginia: NavalFacilities Engineering Command.

Okabe, S., 1926. General theory of earth pressures. Journal ofthe Japanese Society of Civil Engineering, 12 (1), 311.

PIANC, 2001. International Navigation Association(PIANC): Seismic design guidelines for port structures.Rotterdam, The Netherlands: Balkema Publishers.

Psarropoulos, P.N., Klonaris, G., and Gazetas, G., 2005.Seismic earth pressures on rigid and exible retainingwalls. Soil Dynamics and Earthquake Engineering, 25 (710), 795809.

Ricker, N., 1960. The form and laws of propagation ofseismic wavelets. Geophysics, 18, 1040.

Roesset, J.M., 1977. Soil amplication of earthquakes. In:C.S. Desai and J.T. Christian, eds. Numerical methods ingeotechnical engineering. New York, NY: McGraw HillBook Co., 639682.

Scott, R.F., 1973. Earthquake-induced pressures on retainingwalls. In: Proceedings of the 5th World Conference onEarthquake Engineering, Vol. 2, 2529 June, Rome, Italy.Rome: Ministry of Public Works, 16111620.

Seed, H.B. and Idriss, I.M., 1970. Soil moduli and dampingfactors for dynamic response analyses. Report EERC 7010, Earthquake Engineering Research Center, Universityof California, Berkeley, CA.

Seed, H.B. and Whitman, R.V., 1970. Design of earthretaining structures for dynamic loads. In: Proceedings ofthe special conference on lateral stresses in the ground anddesign of earth retaining structures, 2224 June, Ithaca,NY, New York, NY: ASCE, 103147.

Shome, N., and Cornell, C.A., 1999. Probabilistic seismicdemand analysis of nonlinear structures. Report No.RMS-35, RMS Program, Stanford University, Palo Alto,CA.

Siddharthan, R., El-Gamai, M., and Maragakis, E.A., 1994.Investigation of performance of bridge abutments inseismic regions. ASCE Journal of Structural Engineering,120 (4), 13271346.

Veletsos, A.S. and Younan, A.H., 1997. Dynamic responseof cantilever retaining walls. ASCE Journal of Geotech-nical and Geoenvironmental Engineering, 123 (2), 161172.

Wood, J.H., 1975. Earthquake-induced pressures on a rigidwall structure. Bulletin of New Zealand National Earth-quake Engineering, 8, 175186.

Wu, G. and Finn, W.D.L., 1999. Seismic lateral pressures fordesign of rigid walls. Canadian Geotechnical Journal, 36(3), 509522.

Wu, Y., 1999. Displacement-based analysis and design of rigidretaining walls during earthquake. PhD Dissertation.University of Missouri-Rolla, USA.


Downlo

aded

By:

[Te

chni

cal

Univ

ersi

ty o

f Cr

ete]

At:

09:

15 2

1 Ju

ly 2

010

sie nl walls on line

Documents