Safety, Reliability, Risk and Life-Cycle Performance ofStructures & Infrastructures – Deodatis, Ellingwood & Frangopol (Eds)

© 2013 Taylor & Francis Group, London, ISBN 978-1-138-00086-5

Performance-based seismic design of flexible earth-retaining structures

P. Franchin & F. CavalieriDepartment of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

ABSTRACT: Performance-based design differs from conventional design in that target performance is clearlystated and explicitly checked. A faithful application of this design criterion requires refined analysis methodsto quantify performance and explicit consideration of all relevant uncertainties. The paper presents a designprocedure that satisfies these requirements. The procedure is presented with reference to flexible earth-retainingdiaphragm walls, which represent a challenging test for such a procedure given the influence of dynamic soil-structure interaction on the response in terms of internal forces and deformations. In order to remain affordableand feasible in practice, yet accounting for the aforementioned sources of uncertainties and the important physicalaspects of the system response, the procedure is based on small-sample Monte Carlo simulation and a nonlineardynamic model of such soil-wall systems developed recently. The latter strikes a balance between model setupand analysis time on one hand and accuracy on the other, being capable of re-producing the cumulative buildup of soil pressure in cyclic deformation and the accumulation of permanent displacement of the wall with anoverall modeling and analysis effort in the order of few tens of minutes. Performance is measured in terms ofmean annual rate of exceedance curves from which design values are obtained.

1 INTRODUCTION

Current formulations of seismic performance-baseddesign (PBD) specify performance requirements as anumber of target performance levels associated withpredefined mean return period of the seismic action.These formulations actually do not provide a reliablequantitative measure of the attainment of the targetperformances. Given the many uncertainties involved,on both action and system, the only option to do so is toaccount for them explicitly and provide a probabilisticmeasure of performance.

The research of the recent past has made thisan affordable task, simplifying to the extent of thepossible reliability methods so that they can nowbe applied in practice with confidence and a back-ground and computational resources within reach ofthe professional engineer (Cornell, 1996) (Cornell andKrawinkler, 2000).

This paper presents an application of these meth-ods, which rely on so-called small-sample simulation,conditional on a ground motion intensity measure(IM), to the PBD of flexible earth-retaining struc-tures, and in particular to anchored diaphragm walls.These structures represent a challenging test case sincesoil-structure interaction (SSI) phenomena and lack ofsymmetry play an important role in their response forboth static and seismic conditions. A crucial aspectis the so-called ratcheting (England et al, 2000),or progressive build-up of soil pressures behind thewall with associated increase in internal forces anddisplacements.

Results are provided in terms of mean annual fre-quency (MAF) of exceedance of predefined responsethresholds. From these curves design values arederived which duly account for all uncertainties, inaction and system. The usually time-consuming taskof deriving these curves through inelastic responsehistory analysis (IRHA) is made affordable employ-ing an inelastic dynamic model of these soil-structuresystems developed previously (Franchin et al 2007),which describes all relevant physical aspects of theirresponse.

2 MODEL OF THE WALL-SOIL SYSTEM

2.1 Introduction

Seismic analysis of earth-retaining structures can becarried out in principle with satisfactory accuracywith inelastic large-displacement finite element ordifference formulations (implemented e.g. in codeslike DYNAFLOW or FLAC). The drawbacks of thesetools is that they require advanced competence andtime for both model setup and analysis (Green andEbeling, 2002) (Gazetas et al, 2004). At the otherside of the spectrum, state of the practice has notevolved much from the pseudo-static methods devisedby Mononobe-Okabe (Mononobe and Matsuo, 1929)(Okabe, 1926), as modified e.g. by Seed and Whitman(1970), to approximately account for wall flexibil-ity. Elastic or visco-elastic solutions have been pro-posed (Wood, 1975) (Younan and Veletsos, 2000),

1665

which albeit idelized have shed light on importantaspects of the dynamic response such as the soildynamic response and the relative flexibility of soiland wall.

For flexible diaphragm walls SSI becomes evenmore important, making the application of pseudo-static methods less reliable. Alternatives that are mid-way between FEM-FDM and limit analysis have foundtheir way in professional practice, as reviewed e.g. inFaccioli and Paolucci (1996): in these approaches usu-ally the seismic action is described as a predefineddistribution of static forces and interaction is modeledby the subgrade reaction method (SRM).

In this paper, in view of the small sample simula-tion procedure adopted to account for uncertainties,use is made of 1D inelastic dynamic model developedin (Franchin et al, 2007), herein extended to accountfor construction stages. The model, which captures allrelevant physical aspects of the response, is shown inFigure 1.

The model has four building blocks, described inthe following sections: a 1D inelastic shear soil col-umn to reproduce free-field motion traveling from thebedrock to the surface; a soil-wall interface throughwhich seismic motion is imposed to the structure;

Figure 1. 1D model of the wall-soil system.

structural elements for the wall and the tie-back(s);a compliant base where motion is input to the sys-tem allowing for energy radiation. The figure reportsnumerical values for some of the input data in theapplication presented in §4. The discretization shownis coarse for clarity of illustration; in the application itis much finer.

The model is developed within the open-sourceanalysis package Opensees (McKenna et al, 2000).

2.2 Soil: free-field motion (1D soil column)

A 1D soil column extends from the bedrock to thesurface. This shear column is the typical schemeemployed in 1D site response analysis. Current stateof the practice employs equivalent linear formula-tion in the frequency domain (Schnabel et al, 1972)where soil stiffness and damping as a function of effec-tive maximum shear strain are described by so-calledmodulus-reduction curves, of experimental derivation(implemented in computer codes such as SHAKE orEERA).

On the other hand the limit of the equivalentlinear approach is particularly severe in the particu-lar case of interest here: residual displacements andprogressive build-up of earth pressures cannot be pre-dicted. For this reason an inelastic model in the timedomain is the only option. Hysteretic laws that can beemployed in this case can be as simple as a mono-tonic envelope, such as the hyperbolic model or apiece-wise model derived from modulus-degradationcurves, to which an unload-reload rule, such as e.g.the Masing criterion, is attached (Kramer 1996). Thisapproach, however, is well known to under-predictresponse at higher intensities, since it over-predictsenergy dissipation (hysteresis loops are “too full”) atmedium-large deformations. More refined models thatovercome this problems are by now available, such asthe multi-surface plasticity at the base of the Cyclic-1Dsoftware (Elgamal et al, 1998) or a modified ver-sion of the Bouc-Wen (BW) law (Bouc, 1971) (Wen,1976) employed in the code NL-DYAS (Gerolymosand Gazetas, 2005) (Drosos et al, 2012), which yieldsubstantially equivalent results (Drosos et al, 2012).

In this paper the soil shear column is modeledas a spring-mass system, with masses equal to m =ρA�z, where A is the column cross-section area and�z the layer thickness. Springs are described by thezeroLength element in Opensees (even though theyare not zero length but of finite length �z). Theforce-deformation relationship in the shear degreeof freedom is modeled with the plain BW law (notthe modified version by Gerolymos and Gazetas)implemented in the uniaxial material BoucWen:

1666

where f is the resisting force, product of the col-umn area and the shear stress, u is the deformationand ζ is an internal hysteretic variable governed byequation (2). The model has four parameters: initialstiffness k , hardening ratio α, yield deformation uy andthe exponent n which regulates the sharpness of thetransition between the elastic and post-elastic range.Equation (2) is obtained from the classical one:

by setting β = γ = 0.5, which corresponds to the Mas-ing unload-reload criterion, as shown in (Drosos et al,2012). The corresponding limitation (over-estimationof damping) will be easily removed by implementingthe modified version of Bouc-Wen by Gerolymos andGazetas (2005) within Opensees, but this is outsidethe scope of this contribution and does not influenceits generality.

Natural soil deposits exhibit variations of strengthand stiffness (which are related) with the confiningpressure σ ′

0(z), as a function of their plasticity index PI.Coarse grained soils with low PI are characterized bystrength and stiffness that increase with depth, whilefine grained ones with higher PI tend to have bothstrength and stiffness with or no lower variability withdepth (Ishibashi and Zang, 1993). The model thereforeallows for depth-dependent values in input for k and fy:

where G0 is the low-strain value of the soil shearmodulus and τy the shear strength of soil, all at depth z.

The stiffness G0 can be obtained either fro corre-lations with SPT/CPT data or as a function of massdensity and shear-wave velocity G0(z) = ρVs0(z)2 (inboth cases important model uncertainties in these rela-tions should be included in the simulation procedure,see §3).

Shear strength can be obtained again through cor-relation with field data or as the product G0(z)γy(z),where γy is the soil yield strain in shear. The latter istabulated e.g. as a function of σ ′

0 and PI in (Drosos et al,2012). Herein the simple Mohr-Coulomb criterion hasbeen employed:

where c and φ are the cohesion and internal frictionangle of the soil material.

Finally, as far as the other parameters of the BWmodel are concerned, in the application the constantdepth-independent values have been adopted: n = 2(a smooth transition) and α = 0 (no hardening), bothfalling within the usual intervals identified in (Drososet al, 2012): between 0 and 5 for n, 0.4 to 0.6 for β and0 to 2% for α.

Figure 2. Constitutive laws for the interaction spring (left)and the tie-back (right).

2.3 Soil: interface with diaphragm

The portion of soil in direct contact with the wall isdenoted here as interface, and it is modeled by meansof two sets of inelastic springs (Figure 1), one onthe left side, extending from z = 0 to z = −(H + D),and the other on the right extending from z = −H toz = −(H + D). Springs in the two sets have oppositepositive axes, always pointing against the wall, so thata wall movement towards them induces compression.The second set initially is symmetrical to the first one,but springs are removed from z = 0 to z = −H duringthe construction phase. All springs (left and right) areconnected to the column node at the same depth z,and thus they work in the relative motion betweenthe wall and the free field. These springs are alsomodeled with the zeroLength element, to which theinelastic law shown in Figure 2 (left) is assigned to theaxial degree of freedom. The force-deformation lawis a simple compression-only non symmetric elastic-plastic law, and has been implemented in Opensees(Franchin et al, 2007). It is similar to the constitutivelaw presented in Becci and Nova (1987) at the basisof the analysis package “Paratie”, widely employedin Italy for the Winkler-type analysis and design offlexible diaphragm walls.

Depth-dependent values are assigned also to thesesprings, with stiffness:

in which the Young modulus is given by E = 2G/(1 +ν) and the Poisson ratio ν = K0/(1 + K0) is a func-tion of the at rest pressure coefficient, Ac = 1 · �zis the contact area (for a unit width wall strip); L =(La + Lp)/2 is the average of the so-called activeand passive length employed for the same purpose ofaxial stiffness determination in the already cited code“Paratie” (basically a SRM solution, where seismicactions are represented by predefined forces, and thewall is attached to a fixed reference, while here thesupport moves and imposes the motion).

The active and passive strength are given by (wheresigns account for the fact that σv < 0):

1667

where Ka and Kp are the active and passive earth pres-sure coefficients.The following expressions have beenused in the application:

in which the last cos δ projects on the horizontalthe pressure which has a vertical component due tothe soil-wall friction δ. Finally, the at rest pressurecoefficient has been taken equal to:

2.4 Tie-backs

Tie-backs when present are modeled as additional hor-izontal springs on the left, up-hill side of the wall, inparallel with the soil interface spring at the same depth.Their constitutive law is shown in Figure 2 (right),where it is shown how the reaction of an inclined tie-back is projected on the horizontal. In approximation,the tie-back works in the relative motion between thewall and the soil column at the same depth of the wallconnection.

2.5 Diaphragm wall

The wall is modeled with elasticBeamColumn ele-ments, since they are required to remain elastic.

2.6 Construction stages

The model presented in Franchin et al (2007) hasbeen herein extended to account for construction instages: springs on the right, down-hill side of thewall are removed (remove command in Opensees) inN = H/�z steps, while at the same time the proper-ties (basically the vertical stress) of the remaining onesare changed (update command in Opensees) to reflectthe decreasing overburden pressure. The excavationprocess is thus modeled as non linear static analysis.

2.7 Base and input motion

The system is connected to the support at the baseof the soil column at z = −Hc. In order to modelthe absorbing boundary and allow for energy to exitfrom the model (radiation damping) the boundary ismodelled with a viscous damper of constant:

proportional to the bedrock mass density ρb and shearwave velocity Vsb (A is the column area). Correspond-ingly the motion is input to the system as a forceproportional through the damper constant in (12) tothe velocity time series at the model base (Lysmer andKuhlemeyer, 1969) (Kramer, 1996):

This velocity is obtained by deconvolution of a rockoutcropping signal to the model base, and integration.

3 PERFORMANCE-BASED DESIGN

As anticipated, the proposed procedure is an appli-cation of “IM-based” probabilistic demand analysiswhereby the response distribution is obtained via thetotal probability theorem by first finding the responsedistribution conditional an some local intensity mea-sure (IM) of motion and then combining it withthe corresponding hazard function from probabilisticseismic hazard analysis (PSHA, Cornell 1968). Thefinal result is the mean annual frequency (MAF) ofexceedance of chosen response thresholds. In the caseat hand the response distribution must be conditionedon a second parameter, i.e. the random initial (end ofexcavation) value of the considered response, due tothe lack of symmetry of the problem.

With reference for instance to the wall bendingmoment, its MAF is found as:

which states that for each intensity Sa = y, M exceedsx whenever the moment at the excavation end M0equals z and the seismic moment increment �Mexceeds the difference x−z. Functions G, f and λSaare the complementary cumulative distribution func-tion (CCDF) of �M conditional on the IM and theinitial moment M0, the probability density of the latter,and the MAF of the IM, or seismic hazard curve.

The CCDF of �M can be expressed, under theassumption that �M and M0 are joint lognormal, as:

The conditional log-mean and log-standard deviationof �M are expressed as a function of the marginalparameters and of the correlation coefficient as(Benjamin and Cornell, 1970):

The marginal mean of ln�M and the correlationcoefficient ρ are expressed as:

1668

In order to evaluate the previous expressions asmall-sample simulation consisting of IRHAs withnatural, unscaled recorded motions (what is termedin jargon a cloud analysis) selected according to bestpractice criteria, e.g. within magnitude and distancebins from hazard de-aggregation. Sample size canbe in the order of N = 20 ÷ 30. To account for sys-tem parameters uncertainty different approaches areavailable, from response surface (Pinto et al, 2004)(Schotanus et al, 2004), to Latin Hypercube Sam-pling (Helton and Davis, 2003) (Dolsek 2012), toFOSM (Baker and Cornell, 2008). Herein a simplerapproximate (no a posteriori control of the correla-tion structure of the small sample, as done for instancein stratified sampling) approach is taken, by generat-ing N samples of the uncertain system properties andassociating each sample to one of the N records. Byso doing the response sample obtained by simulationwill have a different median and dispersion (the latterreasonably larger). For each signal-system sample pairtwo non linear analysis are carried out in sequence, thefirst static to simulate the excavation process, and thesecond one dynamic for the earthquake, which yieldsan IM-response couple.

Median and standard deviation of M0 are then eval-uated based on the N values at the end of the firstanalysis. Parameters a and b of the power law (17) areestimated by ordinary least squares based on the NIM-response couples obtained from the second anal-ysis, while the marginal dispersion σln�M is assumedto be independent of seismic intensity (Cornell et al,2002).

The design value of the total bending moment isobtained by inverting (14) for an accepted value λ0of the MAF: Md = x(λ0). The mean flexural strength(ultimate moment evaluated with mean material prop-erties) can then be designed by amplifying Md ofthe factor exp (−0.5σ2

ln MRk/b) which accounts for theuncertainty in the flexural strength MR, through σln MR,as well as of the exponents in the power laws for haz-ard and demand vs IM, equation (17), (Cornell et al,2002).

4 EXAMPLE APPLICATION

The procedure is illustrated with reference to theanchored diaphragm wall shown in Figure 1. The wallretains a cohesion-less dry soil for an excavation heightH = 6 m. The wall is embedded D = 5 m in the soil andrestrained also by a tie-back at Ht = −2 m (fu = 75 kN).The figure reports data of the reference case, in whichsystem uncertainty is disregarded and the responseMAF is affected only by the uncertainty in the seis-mic action (record-to-record variability and hazard).

Figure 3. Soil profile uncertainty: shear wave velocity (left)and friction angle (right). Target median, sample median, onerealization.

The complete case includes instead strength and stiff-ness random variables for the soil deposit, sampled asexplained in the next section.

4.1 Uncertainty in the system parameters

Profiles of Vs and φ have been sampled in number N =20. Both are assumed to be lognormal random fields,the first with a depth-dependent median (170z0.25) andthe second with constant median (35◦), with equalstandard deviations of the logarithms σln Vs = σln φ =0.10. The Vs field has an autocorrelation functionρVs(�z) = exp (−|�z|/b) with b = 5 m, and correla-tion coefficient with the friction angle ρ = 0.8 (cor-relation between field values at the same depth).Fields have been sampled starting from a vector ofindependent standard normal variables u as:

where µln Vs = [ ln 170 + 0.25ln(zi)] is the vector oflog-mean values, D = Iσln Vs the diagonal matrix ofstandard deviations and L the lower triangular decom-position of the correlation matrix R = [ρVs(�zij)] (Liuand Der Kiureghian, 1986). The field φ has been sam-pled conditionally on Vs with expressions similar to(16) and (17). Figure 3 shows a sample of Vs and φwhere the auto-correlation and the cross-correlationbetween Vs and φ are clearly observed. The figurereports also the target and sample medians, in thickline, showing how the sample size (N = 20) is adequateto describe the median.

4.2 Seismic action

Twenty recorded motions have been selected from thePEER strong motion catalogue according to best prac-tice in the bins M = [5.5, 6.5] and R = [15 km, 60 km].Table 1 reports essential properties. Both site andmotions are the same as in Pinto and Franchin (2012),to which the reader is referred for details.

1669

Table 1. Selected ground motions.

Record M R (km) Sa(T1) (m/s2) IA

TCU067-N.at2 5.9 33.9 3.76 0.052TCU071-N.at2 5.9 23.1 3.39 0.053TCU075-N.at2 5.9 34.1 0.95 0.012TCU076-N.at2 5.9 34.2 2.29 0.036H-Z06000.at2 6.4 42.8 1.56 0.082H-Z09000.at2 6.4 41.2 4.02 0.066H-Z10000.at2 6.4 41.4 2.54 0.171L-EDE-NS.at2 6.1 32.9 0.76 0.218B-AUL000.at2 6.2 37.2 1.21 0.007B-BAG000.at2 6.2 22.3 1.51 0.029CLS220.at2 6.2 30.1 4.06 0.065SJL270.at2 6.2 57.7 2.37 0.184SIL000.at2 6.1 27.7 1.81 0.095C12050.at2 6.2 36.2 7.05 0.054TMB205.at2 6.2 40.3 2.66 0.451CAD250.at2 5.9 34.0 1.38 0.021chan1155.at2 5.6 18.7 2.67 0.405A-CHL030.at2 6.0 37.1 2.97 0.010A-MU2032.at2 6.0 31.1 3.76 0.077A-TUJ262.at2 6.0 29.5 3.39 0.103

Figure 4. Motion #1: acceleration time series (left) andresponse spectrum (right) for the outcrop and bedrock signals.

The table reports two IMs, i.e. the usual spectralacceleration at the fundamental period Sa(T1), and theArias intensity IA. The latter is included because giventhe nature of the system it could be better correlatedwith system response (duration effects on accumula-tion of stresses behind the wall), even if hazard wasnot available for IA. The fundamental period of thesoil-structure system (basically the soil column, sinceits mass is taken orders of magnitude larger than thatof the wall in order not to be influenced by the walland produce an undisturbed free-field motion) for themedian model is T1 = 0.288 s (in good agreement withthe theoretical value for a deposit with constant Vsequal to the average over 20 m of the 170z0.25 profile,equal to 4H/Vs = 0.28 s).

The IM values are those of the rock outcrop signals.Consistently hazard on rock is used in (14). Responseon the other hand includes the effect of amplifica-tion of the motion deconvoluted at the bedrock, dueto the response of the soil column. Figure 4 showsthe acceleration time series and response spectrum ofmotion #1, as recorded and at the bedrock: differencesin this case are very small since 20 m is a very shortdistance for the rock profile (a Vs = 800 m/s as for thebedrock has been used in the deconvolution).

Figure 5. Motion #1: wall total (gravity+earthquake) bend-ing moment (left) and deformation (right).

Figure 6. Motion #1: time series of input ground accel-eration (outcrop) maximum bending moment and wall topdisplacement.

4.3 Results

This section provides a sample of the results and thedesign values for the wall bending moment (the anal-ysis yields values for all other quantities, such as e.g.tie-back force, not shown). Figure 5 shows the wallbending moment and displaced shape for motion #1.

Figure 6 shows three time-series: the input motion(rock outcrop) and the maximum moment and top dis-placement. The lower two exhibit clearly the cyclicaccumulation due to system lack of symmetry.

Samples of N = 20 values of M0, �M = M − M0and D, collected from results such as those shownin the previous figures, are used to estimate µln M0,σln M0, ρln M0 ln� M and the coefficients in the regres-sion (18). Results in terms of �M and D are shownin Figure 7 and Figure 8 for the median (blue) and thecomplete model (red), respectively.

The main observations are that the displacementD is usually more dispersed than the moment incre-ment, and that the use of IA leads to a much reduceddispersion than Sa, making it the best choice werethe corresponding hazard available (in this case Sais used). Further, dispersion in the complete modelare consistently larger than that of the median model,even though the value are not indicative since the inputvalues for σln Vs and σln φ are purely illustrative.

Finally Figure 9 shows the MAF of the IM (seis-mic hazard) and of the moment and displacementresponses (demand hazards), obtained through theintegral in (14). Design values for the demands are

1670

Figure 7. Mean model: IM-response relations (momentincrement on the left, permanent displacement on the right,versus spectral acceleration above andArias intensity below).

Figure 8. Complete model: IM-response relations (momentincrement on the left, permanent displacement on the right,versus spectral acceleration above andArias intensity below).

obtained by setting an acceptable frequency thresh-old (1/950 years in the example) and inverting thesecurves; if the displacement D is not acceptable thedesign is revised and the analysis rerun, otherwise thebending moment value is used for strength design ofthe wall.

The figure reports for reference a curve denotedas “code-value” obtained by assuming a deterministicrelation between input and response (demand disper-sion set to zero, as required by most codes whichrequire the use of the mean or median over the analy-ses as design value). Comparison with this term allowsappreciating the importance of uncertainties whichlead to an increase in the design moment of 20% and29% for the median and complete model, respectively.

Figure 9. Mean annual frequency of Sa(T1) (top), and ofwall moment (bottom left) and residual displacement (bottomright).

5 CONCLUSIONS

Seismic analysis and design of flexible earth-retainingstructures is made difficult by the role played bySSI and the lack of symmetry in the problem. Thispaper introduces an iterative procedure based on sim-plified and computationally effective yet physicallyrealistic model of the problem to perform fully proba-bilistic performance-based design of these structures,accounting for uncertainty in the action as well as in thesystem. The procedure requires a small-sample simu-lation involving IRHAs but due to the effectivenessof the underlying response model is still affordablein practice (the full simulation, including model setupand post-processing is a matter of few tens of minutes).The real effort is to be put in the motions selection,a task that, however, is more and more accessibleto the designer, due to software aids and increasingdiffusion of the necessary background.

The example shows how design values can easily be30% higher than those determined by current codes.

REFERENCES

Baker, J.W., Cornell, C.A. 2008. Uncertainty propagation inprobabilistic seismic loss estimation. Structural Safety,30(3), 236–252.

Becci, B., Nova, R. 1987. A method for analysis and designof flexible sheetpiles (in Italian), Rivista Italiana diGeotecnica, 1/87, 33–47.

Benjamin, J.R., Cornell, C.A. 1970. Probability, statistics,and decision for civil engineers, NewYork: McGraw-Hill.

Bouc, R. 1971. Modèle matematique d’hysteresis, Acustica,21, 16–25.

CEN European Committee for Standardization, 2004.EN1998-1 Design of Structures for Earthquake resis-tance Part 1 General rules, seismic actions and rules forbuildings, Brussels, Belgium

1671

Cornell, C.A. 1968. Engineering Seismic Risk Analysis.Bulletin Seism. Society Am., 58(5), 1583–1606.

Cornell, C.A. 1996. Reliability-based earthquake-resistantdesign; the future (paper 2166). 11th World Conferenceon Earthquake Engineering, Acapulco, Mexico.

Cornell, C.A., Krawinkler, H. 2000. Progress and challengesin seismic performance assessment. PEER News, April.

Cornell, C.A., Jalayer, F., Hamburger, R.O., Foutch, D.A.2002. The probabilistic basis for the 2000 SAC/FEMAsteel moment frame guidelines. ASCE J. Struct. Eng.,128(4), 526–533.

Dolšek, M. 2012. Simplified Method for Seismic RiskAssessment of Buildings with Consideration of Aleatoryand Epistemic Uncertainty. Structure and InfrastructureEngineering, 8(10), 939–953.

Drosos, V.A., Gerolymos, N., Gazetas, G. 2012. Constitutivemodel for soil amplification of ground shaking: parametercalibration, comparisons, validation, Soil Dynamics andEarthquake Engineering, 42, 255–274.

Elgamal, A.W., Yang, Z., Parra, E., Ragheb, A. 1998. Cyclic1D: an internet-based finite-element computer programfor one-dimensional site amplification analysis using anincremental plasticity coupled solid-fluid formulation,University of California San Diego.

England G.L., Tsang, N.C.M., Bush, D.I. 2000. Integralbridges: a fundamental approach to the time-temperatureloading problem, London: Thomas Telford.

Faccioli, E., Paolucci, R. 1996. Seismic behaviour anddesign of foundations and retaining structures, Tech. Rep.,European Commission – Human Capital and MobilityProgram: prenormative research in support of Eurocode8 (ECOEST-PREC8).

Franchin, P., Pinto, P.E., Noto, F. 2007. A Simpli-fied Nonlinear Dynamic Model For Seismic AnalysisOf Earth-Retaining Diaphragm-Walls (paper 1220), InK. Pitilakis (ed.) 4th International Conference on Earth-quake Geotechnical Engineering, June, Thessaloniki,Greece.

Gazetas, G., Psarropoulos, P.N., Anastasopoulos, I.,Gerolymos, N. 2004. Seismic behaviour of flexible retain-ing systems subjected to short-duration moderately strongexcitation, Soil Dynamics & Earthquake Engng, 24:537–550.

Gerolymos, N., Gazetas, G. 2005. Constitutive model for 1-Dcyclic soil behaviour applied to seismic analysis of layereddeposits, Soils and Foundations, 45(3), 147–159.

Green, R.A., Ebeling, R.M. 2002. Seismic analysis of can-tilever retaining walls, phase I. Tech. Rep. ERDC/ITLTR-02-3, US Army Corps of Engineers – EngineerResearch and Developmnt Center.

Helton, J.C., Davis, F.J. 2003. Latin hypercube sampling andthe propagation of uncertainty in analyses of complexsystems. Reliability Engng System Safety, 81, 23–69.

Ishibashi, I., Zhang, X. 1993. Unified dynamic shear mod-uli and damping ratios of sand and clay, Soils andFoundations, 33(1), 182–191.

Kramer, S.L. 1996. Geotechnical Earthquake Engineering.Upper Saddle River: Prentice-Hall.

Liu, P., Der Kiureghian, A. 1986. Multivariate distributionmodels with prescribed marginals and covariances, Prob.Eng. Mech., 1(2):105–112.

Lysmer, J., Kuhlemeyer, R.L. 1969. Finite dynamic modelfor infinite media, ASCE Jnl Eng. Mech. Div., 4,859–977.

McKenna, F., Fenves, G.L., Scott, M.H. 2000. Open sys-tem for earthquake engineering simulation, University ofCalifornia, Berkeley, CA.

Mononobe, N., Matsuo, H. 1929. On the determination ofearth pressures during earthquakes, World EngineeringCongress (volume 9), Tokyo, Japan.

Okabe, S. 1926. General theory of earth pressures, Jnl Japan.Soc. Civil Eng., 12.

Pinto, P.E., Franchin, P. 2012. Performance-based design ofintegral abutment bridges, 2nd International Conf. onPerformance-based design in Earthquake GeotechnicalEngineering, May 28–30, Taormina, Italy.

Pinto, P.E., Giannini, R., Franchin, P. 2004. Seismic Reliabil-ity Analysis of Structures, Pavia: IUSS Press.

Schnabel, P.B., Lysmer, J., Seed, H.B. 1972. SHAKE: a com-puter program for earthquake response analysis of hor-izontally layered sites, Report EERC 72-12, EarthquakeEngineering Research Center, University of California,Berkeley.

Schotanus, M.IJ., Franchin, P., Lupoi, A., Pinto, P.E. 2004.Seismic fragility analysis of 3D structures, Structuralsafety, 26(4), 421–441.

Seed, H.B., Whitman, RV. 1970. Retaining structures fordynamic loads, ASCE Specialty Conf. of Lateral stressesin the ground and the design of earth-retaining structures,Cornell University, Ithaca, NY.

Wood, J.H. 1975. Earthquake-induced pressures on a rigidwall structure, Bull. NZ Natl Earthquake Engng, 8,175–186.

Wen, Y.K. 1976. Method for random vibration of hys-teretic systems, Jnl Engng Mech. Div., ASCE, 102(2),249–263.

Younan, A.H., Veletsos, A.S. 2000. Dynamic response offlexible retaining walls, Earth. Eng. Struct. Dyn., 29,1815–1844.

1672