Dynamic Response of Cantilever Retaining Walls Considering Soil Non-Linearity

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1 Missouri University of Science and Technoloy Scholars' Mine International Conferences on Recent Advances in Geotechnical Earthquake Enineerin and Soil Dynamics 2 - Fifth International Conference on Recent Advances in Geotechnical Earthquake Enineerin and Soil Dynamics May 24th - May 29th Dynamic Response of Cantilever Retainin Walls Considerin Soil Non-Linearity Leuzzi Francesco Politecnico di Torino, Italy S. Foti Politecnico di Torino, Italy R. Lancellotta Politecnico di Torino, Italy G. Mylonakis University of Patras, Greece Follow this and additional works at: Part of the Geotechnical Enineerin Commons Recommended Citation Francesco, Leuzzi; Foti, S.; Lancellotta, R.; and Mylonakis, G., "Dynamic Response of Cantilever Retainin Walls Considerin Soil Non-Linearity" (2). International Conferences on Recent Advances in Geotechnical Earthquake Enineerin and Soil Dynamics This Article - Conference proceedins is brouht to you for free and open access by Scholars' Mine. It has been accepted for inclusion in International Conferences on Recent Advances in Geotechnical Earthquake Enineerin and Soil Dynamics by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyriht Law. Unauthorized use includin reproduction for redistribution requires the permission of the copyriht holder. For more information, please contact scholarsmine@mst.edu.

2 DYNAMIC RESPONSE OF CANTILEVER RETAINING WALLS CONSIDERING SOIL NON-LINEARITY Leuzzi Francesco Foti S. *, Lancellotta R. *, Mylonakis G. ** * Politecnico di Torino, 24 Corso Duca deli Abruzzi Politecnico di Torino, 24 Corso Duca deli Abruzzi 29 Torino, ITALY 29 Torino, ITALY ** University of Patras, 26 Rion (Patras), GREECE ABSTRACT For many decades the analysis of earth retainin structures under dynamic or seismic conditions has been carried out by means of standard limit equilibrium (Coulomb, M-O) or elastic methods (Wood, Veletsos and Younan). These approaches are simplified, as they make use of considerable approximations which are often applicable only under particular conditions. A different and perhaps more realistic approach is possible usin established computer codes, which interate numerically the overnin equations of the soil and wall media. Since these problems may involve sinificant levels of strain in the backfill, material non-linearity should be taken into account to realistically simulate the response of the system. In the herein-reported study, a parametric analysis is carried out throuh the finite-difference code FLAC.. Startin from simple cases involvin elastic response, and movin radually towards more realistic conditions, salient features of the dynamic wall-soil interaction problem are addressed. The case of non-linear hysteretic behaviour of soil and flexibility of wall is considered at a second stae. Results indicate that with increasin levels of acceleration, there is a clear transition from elastic behaviour (in which the aforementioned V-Y type methods are applicable), to plastic behaviour in which M-O methods are thouht to be more suitable under pseudo-static conditions. The results of the parametric analyses are reported in terms of pertinent normalized parameters, to provide a eneral framework for the assessment of wall-soil dynamic interaction under stron seismic excitation. INTRODUCTION Analysis and desin of earth retainin structures under seismic conditions poses a challenin problem, as the mechanics of wall-soil interaction are not well understood. Dynamic soilstructure interaction effects are dominant and should be taken into account in the assessment and quantification of the problem. Analysis procedures can be rouhly classified into two main roups: pseudo-static limit equilibrium approaches based on the well-known Mononobe and Okabe solution (Mononobe and Matsuo [929], Okabe [926]) and its variants, and elastic approaches followin the seminal works by Wood [973], Arias et al. [98] and, more recently, Veletsos and Younan [994, 997]. The first roup of methods is essentially an extension of Coulomb s classical solution, includin additional inertial forces due to seismic shakin. The method has been extensively studied and modified (Seed and Whitman [97], Richards and Elms [979]), and it is widely used for the analysis and desin of both riid and flexible structures, reardless of the actual amount of wall displacement associated with the formation of a failure surface. Ways to account for wall flexibility have been developed in the realm of the second roup of methods, leadin to certain closed-form solutions for visco-elastic soil (Wood [973], Arias et al. [98]). These solutions are often considered unduly conservative, as the kinematical constraint associated with a riid wall enerates pressures twice as lare as those predicted by the M-O formula. Recent work by Veletsos and Younan [994, 997] has shown a sinificant reduction of pressures with increasin wall flexibility and/or base compliance. Notwithstandin the theoretical sinificance and practical appeal of the above methods, they can both be criticized for makin use of considerable approximations, which are applicable only under particular conditions. A different and perhaps more realistic approach is possible usin established numerical methods, which aim at interatin in space and time the overnin differential equations of the soil and wall media. Since these problems may involve sinificant levels of strain in the backfill, material non-linearity should be taken into account to realistically simulate the response of the system. In the Paper No. 6.8a

3 herein-reported study, a parametric analysis is carried out by means of the finite-difference code FLAC. Startin from simple elastic analyses and proressin to more realistic cases, salient features of dynamic wall-soil interaction problem are addressed. Two basic systems have been analyzed under both pseudo-static and dynamic conditions: a homoenous elastic backfill retained by a vertical cantilever wall, and a vertically inhomoeneous backfill, characterized by a parabolic or exponential variation of soil stiffness with depth. In the first case, walls varyin from riid to flexible, with or without rotational base compliance, are simulated. In the second case, the effect of soil inhomoeneity is investiated for a wide class of walls and soil confiurations. Movin towards more realistic conditions, the case of flexible vertical cantilever wall retainin a horizontal layer of inhomoeneous backfill, obeyin an elastic-plastic Mohr- Coulomb criterion, is analyzed. The excitation consists of a sinle-cycle sinusoidal pulse with a iven amplitude and frequency applied at the bottom of the model. The results of the parametric analyses are reported in terms of pertinent normalized parameters, to provide a eneral framework for the assessment of soil-wall dynamic interaction under stron seismic excitation. ELASTIC PROBLEM In this section, the models used for the simulation of the elastic problem for both homoeneous and inhomoeneous soil, and correspondin results are reported. Results have been compared, whenever possible, with available closed-form solutions. Homoeneous Case The system consists of a semi-infinite layer of visco-elastic material of heiht H retained alon one of its vertical boundaries by a uniform flexible cantilever retainin wall restrained by a rotational sprin of stiffness R θ at its base. The system is excited by a harmonic horizontal motion, as shown in fi.. The properties of the soil are: Mass density: ρ Shear modulus: G Poisson s ratio: ν Material dampin factor: β The properties of the wall are: Thickness: t w Mass per unit surface area: µ w Youn s modulus: E w Poisson s ratio ν w Material dampin factor: β w y D w R θ X Fi.. Elastic model The relevant parameters for a parametric analysis are the relative flexibility between wall and retained soil d w, defined by: GH d w= () D where D w is the flexural riidity of the wall: w 3 3 Ew tw Dw= 2 (2) 2 ( ν ) and the relative flexibility between rotational base constraint and retained soil d θ, defined by: θ 2 w GH d θ= (3) R The elastic soil is characterized by uniform shear wave velocity V s = m/s and mass density ρ =.8 M/m 3, therefore G = 8 MPa. Furthermore, Poisson s ratio is assumed to be ν = /3 and the material dampin factor β = %. The rid dimensions are ( x m). The heiht of the wall is H = 8 m and it is discretized by means of beam elements of unit lonitudinal dimension and thickness t w = m. The mass per unit of surface area is assumed µ w = 2. M/m 2. It s worthwhile to mention that V-Y solution was obtained based on the assumption of a massless wall. The Youn s modulus of the wall can be linked to the other parameters usin equations () and (2) (as already done by Psarropoulos et al. [2]): 2 ( ν ) 3 3 w tw ( ) X&& ( t) = && A(t) sin ω t V s, ρ, β 2 GH w E w= (4) d The values of d w assumed here are (for riid wall),, and 4; the values of d θ are (for a fixed base wall),, and. The interface between the soil and the wall was selected as bonded in order to permit a comparative study aainst available closed-form solutions (see fi. 2). H x Paper No. 6.8a 2

4 When the frequency of excitation, ω, is very low compared to the fundamental frequency of the soil layer, ω, the excitation will be referred to as static, a term which should not be confused with that normally used to represent the effects of ravity forces. To avoid possible confusion, in the ensuin the term pseudo-static loadin will be used. In this case it s more convenient to replace the excitation with a set of horizontal ρ X & ( t) = ρ A applied to all rid body forces equal to ( ) nodes, as shown in fi. 2. y Beam Bonded interface H max Finite difference rid Fi. 2. Pseudo-Static seismic loadin via equivalent body forces in wall and soil To achieve converence of the model, the pseudo-static load was applied radually in several steps. The results are normalized accordin to the followin factors: Horizontal pressures: x ρ X&& (t) Wall displacements: w G α γh 2 Notice that the results reported in the ensuin encompass exclusively horizontal shakin, i.e. to determine the complete state of stress, the initial stresses due to ravity loads should be added. Fiure 3 shows the heihtwise distribution of the horizontal pressures on the wall, determined by V-Y, as function of relative wall flexibility (fi. 3a), and relative base flexibility (fi. 3b) (η = y / H is the normalized heiht). Results of analyses with FLAC have been compared with same simulations with the FEM code PLAXIS. Fiures 4 and show, respectively, the case of fixed base wall and riid wall; PLAXIS results are displayed with solid line, whereas FLAC results are depicted with dots. The numerical results are in a very ood areement with the results obtained by V-Y, except from the upper part of the wall where some discrepancies are visible. Similar features were also observed by Wood [973] and Psarropoulos et al. [2], and are probably to the assumption of complete bondin, as well as that of soil homoeneity, which implies finite soil stiffness not to vanish near the top of wall. (9) σ α γh st () where α = is the maximum horizontal round A / acceleration expressed in, γ = ρ is the unit weiht of the soil; Base shear: Base moment: V α γh b 2 (6) Fi. 3. Wall pressures for different wall and base flexibilities accordin to Veletsos and Younan [997] Effective heiht: M b 3 α γh M H V b (7) h = b (8) H which defines the point of application of the resultant force, measured from the base of the wall; It s observed that both for lare relative wall and base flexibilities, tensile stresses are developin near the top of the wall, clearly stated as a limit of elastic solutions by V-Y, since these tensile stresses can cause de-bondin in case they exceed the value of initial compressive stresses due to ravity. Fiures from 6 to 8 show the variation of normalized base shear (V b ), effective heiht (h) and top displacements (w) for different d w and d θ, whereas solid lines indicate the correspondin closed-form solution results. An excellent areement is observed, except for the case of riid walls, where the aforementioned numerical problems play a slihtly more important role. Paper No. 6.8a 3

5 Normalized heiht, y/h FLAC Plaxis dw = Normalized horizontal pressures, σ st / α γ H Fi. 4. Wall pressures for a fixed base wall (d θ = ) Normalized heiht, y/h FLAC Plaxis dθ = Normalized horizontal pressures, σ st / α γ H Fi.. Wall pressures for a riid wall (d w = ) Note, however, the lare reduction in terms of base shear due to both wall and base flexibility, as analytically proved by V-Y, and numerically confirmed by these analyses. It can be also observed that for the effective heiht a very ood areement with Seed and Whitman [97] modification of M-O method ( H) is visible for riid walls, whereas for flexible walls only a reasonable areement can be found with the Rankine distribution (/3 H), the reason bein the lare tensile stresses near the top that are lowerin the effective heiht. The comparison for top displacements is once aain very ood but a little deviation from analytical results is observed, especially for flexible walls or walls with flexible base. Similar results are found for the dynamic case for which, as analytically shown by Veletsos and Younan [994, 997], and numerically by Psarropoulos et al. [2], the dynamic quantities are the product of the correspondin static ones by an appropriate amplification or deamplification factor. Normalized Base Shear, Vb,st / α γ H 2 dθ =.8 Veletsos and Younan (997) Relative wall flexibility, d w Fi. 6. Normalized base shear for different d w and d θ.7 Inhomoeneous Case The assumption of soil homoeneity, which is practically unrealistic, is one of main limitations of the analyses discussed in the previous section. Indeed, in reality, the soil modulus is likely to increase with depth, that is with confinin pressure. Under this assumption V-Y have analyzed the case of a riid wall elastically constrained aainst rotation at the base, from a static point of view. As proposed by Veletsos and Younan [994] the inhomoeneity is expressed throuh a parabolic variation of shear modulus with depth (see fi. 9): ( 2 ) G ( η) = G η () in which G is the value of shear modulus at the base of the horizontal soil layer. Effective heiht, h / H Seed and Whitman (97) Mononobe and Okabe ( ).3 Veletsos and Younan (997) dθ = Relative wall flexibility, d w Fi. 7. Normalized effective heiht for different d w and d θ Paper No. 6.8a 4

6 ..9 Normalized top wall displacement.3 Veletsos and Younan (997) dθ = Relative wall flexibility, d w ( - ) Fi. 8. Wall displacements for different d w and d θ y / H ( - ) eqn. (2) n = n = n = n =.7 n = V&Y eqn. () The numerical analyses reported herein are limited to a wall with flexibility (d w ) fixed at the base (d θ = ), therefore only the case of riid wall can be compared with analytical results. The main parameter of the problem is, aain, the relative flexibility of the wall and retained soil, defined by: GavH d w= () D w where G av is the averae value of shear modulus, for this case equal to 2/3 G. The soil stratum considered has an averae shear wave velocity V s,av = m/s while the others soil parameters have been maintained to allow a comparative study with the case of homoeneous soil. The properties of the wall are the same, except its Youn s modulus, obtained usin relation (4), by substitutin G with G av. The values of d w taken into account are (riid wall),, and 4. A comparison of pressures for the case of riid wall is shown in fi. : ood areement is observed, except, aain for the upper part of the wall, where a value of pressure different from zero is observed, whereas the pressures should be null, as in the solution by V-Y. This is due to the shear modulus near the top, which in the numerical analysis assumes a finite value, as well as the assumption of complete bondin. In eneral, the assumption of soil inhomoeineity is responsible for a sensible reduction of stresses. This is clearly shown in fiures and 2, where pressures and bendin moments for different wall flexibilities are presented toether with correspondin values for the homoeneous case. The reduction is hiher for the case of a riid wall than for a flexible wall, especially in terms of bendin moments, and this is due to the fact that for flexible walls both compressive and tensile stresses are lower. The reduction of tensile stresses for flexible walls (d w = 4) is another important effect arisin from the assumption of soil inhomoeneity. Indeed now tensile stresses are practically not influent in the overall behaviour, Fi. 9. Different soil inhomoeneities taken into account = G / G ( - ) σ xx / γ α H Fi.. Parabolic law: comparison of wall pressures for a riid wall (d w = ) (modified from Veletsos and Younan [994]) and a more realistic simulation of the interaction phenomenon is achieved. In order to explore in more detail the role of soil inhomoeneity, an exponential variation of shear modulus with depth has been taken into account, accordin to the expression: n n ( η ) G ζ G η ) = G (2) ( = Paper No. 6.8a

7 dw = 4 Hom. dw = 4 y / H y / H.3 dw = 4 Hom. dw = σ xx / γ α H M / γ α H 3 Fi.. Parabolic law: wall pressures for different d w in which: G is aain the shear modulus at the base of the layer ζ = z / H = η is the normalized depth n is a soil inhomoeneity exponent (fi. 9): - n = represents the case of homoenous soil - n = the case of linear variation with depth. The exponent n has been assumed equal to,,.7 (fi. 9): particularly important is the case n = which reproduces the observed increase of G in loose coarse rained soils. Fiure 3 shows the distribution of horizontal pressures on the wall, for different derees of soil inhomoeneity (n) and wall flexibilities (d w ). The case of homoeneous soil as well as the parabolic case considered by Veletsos and Younan (V-Y), are also reported. Evidently as soil inhomoeneity increases, the horizontal stresses on the wall decrease, the reduction bein more important for riid walls. As in the case of parabolic variation, a reduction of tensile stresses on the upper part is observed, especially for hih values of n. The reduction of stresses transferred from the soil to the wall, results in a reduction of internal forces, as shown in fi. 4 for base shear. The continuous lines show closed-form solutions for the homoenous case. As can be seen, there is an almost constant reduction of base shear with d w, whereas a consistent reduction of base moment for riid walls can be shown, as already seen in fi. 2. In terms of effective heiht (fi. ) these different reductions produce a reduction for riid walls (around H, which is close to Seed and Whitman [97] suestion) whereas an increase is observed for flexible walls (clearly due to the reduction of tensile stresses on the upper part), movin towards the Rankine value (H / 3). Fi. 2. Parabolic law: bendin moments for different d w The simple assumption of soil inhomoeneity implies a reduction of internal forces in the wall, and therefore a more realistic simulation of the interaction between soil and retainin structure. Similar observations have been made by Psarropoulos et al. [2]. As for the homoeneous case, dynamic results are omitted, as they are similar to the correspondin static ones, since elastic behaviour has been assumed in this part of the analysis. In the followin a more realistic non-linear elastic-plastic interaction simulation is attempted. NON-LINEAR ELASTIC-PLASTIC PROBLEM Since soil-wall interaction may involve sinificant levels of strain in the backfill, material non-linearity should be accounted for to realistically simulate the response (Callisto and Soccodato [27]). Several studies have been carried out to assess, approximately, the main features of non-linear dynamic soil-wall interaction, but there is a lack of riorous analyses. In the followin, results of a series of parametric fully non-linear analyses will be presented, and a eneralization of the results will be made, to provide a eneral framework for understandin complexities associated with the problem. Description of the Model The proposed model (fi. 6) is essentially a box in which a pair of vertical flexible cantilever walls, at a relative distance of H, retains a horizontal layer of inhomoeneous soil. The material obeys an elastic-plastic Mohr-Coulomb criterion coupled with hysteretic dampin, resultin from a straindependent modulus decay law. Paper No. 6.8a 6

8 d w = d w = y / H y / H.3 n = n = n =.7 V-Y V & Y Homoeneous n = n = n =.7 V-Y V & Y Homoeneous d w = d w = y / H y / H.3 n = n = n =.7 V V-Y & Y Homoeneous Normalized wall pressures, σ xx / α γ H n = n = n =.7 V-Y V & Y Homoeneous Fi. 3. Effects of soil inhomoeneity on wall pressures Normalized base shear Vb,st / α γ H n =.7 V - Y homoeneous Veletsos and Younan (997) Effective heiht, h / H.7.3. Veletsos and Younan (997) Seed and Whitman (97) Mononobe and Okabe ( ) n =.7 V - Y homoeneous Relative wall flexibility, d w Relative wall flexibility, d w Fi. 4. Effects of soil inhomoeneity on base shear Fi.. Effects of soil inhomoeneity on effective heiht Paper No. 6.8a 7

9 y,η L = H η.2. Seed and Idriss (97) FLAC Default 7 6 D w φ, ρ, G(γ ), ν, β(γ ) D w H G / Gmax ( - ).8 Shear Modulus β (γ) (%) X&& Fi. 6. Model proposed for non-linear analyses The excitation ( X & = a ( t) = A sin ( ωt) G av ) is a full cycle sinusoidal acceleration pulse with iven amplitude and frequency applied at the bottom of the model. The fundamental parameters of the problem are: shearin resistance anle: φ = 3 relative soil-wall flexibility: d w = 4 peak base acceleration: A = (..3) normalized excitation frequency: ω /ω = 3 The soil inhomoeneity is expressed via the exponential law (eqn. (2)) with n =, G = 2. MPa (V s, = 2 m/s, ρ =.8 M/m 3, ν = /3). The modulus decay law G(γ) / G max adopted for the soil (fi. 7), simulates the sand (upper rane) law proposed by Seed and Idriss [97]: the application of Masin rules automatically defines hysteretic dampin of the soil β(γ) (where γ is the shear strain). As it may be seen, FLAC larely overestimates the dampin ratio after approximately γ =.3- %, but this was not the case in the performed analyses. A small amount of stiffness-proportional Rayleih dampin (%) was added, since hysteretic dampin provides almost no enery dissipation at very low cyclic strain levels, which may be unrealistic. The wall heiht is H = 8 m and the wall is discretized by means of beam elements of unit lonitudinal dimension and thickness t w.73 m. The wall material is supposed to be concrete, with mass per unit of surface area µ w = 2. M/m 2, Youn modulus E w = 28. GPa and Poisson s ratio ν w =. The fundamental frequency of the soil is only an estimate, since there is no closed form solution for an inhomoeneous soil with an exponential increase of shear modulus with depth. Trials were conducted to identify this frequency in elastic domain, althouh for non-linear soil these values are valid only for small strain level. The computational steps were the followin:. model the ravitational (initial) condition 2. model the pseudo-static condition (ω ) and dynamic condition. In the first step the ravity was applied to the model and the initial stresses were imposed to be the eostatic: σ = ρ z, G G(η) yy x,ξ σ Dampin Ratio.... γ (%) Fi. 7. Comparison between FLAC and Seed and Idriss [97] modulus and dampin ratio decay law for sands = =, where K = sin φ (Jaky, 944) is the xx σ zz K σ yy coefficient of earth pressure at rest. Gravitational and Pseudo-Static Loadin Fiure 8 shows the distribution of horizontal pressures behind the wall after ravitational loadin, normalized to horizontal pressures at rest. Due to the lare wall flexibility (d w = 4), for a lare part the soil behind the wall is close to plastic conditions, where K a (active) conditions apply, whereas in the vicinity of the base (which is fixed aainst all movements) the pressures radually move towards K (at rest) conditions. y / Η ( - ) K a line..8.2 σ xx / K γ Η ( - ) d w = 4 K line Fi. 8. Wall pressures after ravitational loadin Paper No. 6.8a 8

10 . a =.. a =. a =. a = Normalized heiht, y / H Normalized wall pressures, σ xx / α γ H Elastic-Plastic Elastic M-O Fi. 9. Pseudo-static wall pressures for different acceleration levels Normalized base moment, Mb / γ α H 3 ( - ) FLAC elastic-plastic model. FLAC elastic-plastic model µw = Mononobe-Okabe FLAC linear elastic model. Normalized base acceleration, α ( - ) Fi. 2. Base moment due to pseudo-static loadin τ (kpa) E-7 4.E-7 2.E-7.E+ γ ( - ) -2.E-7-4.E-7 y = 3 m y = 6 m Fi. 2. Dynamic behaviour of soil at a =. -6.E-7 Followin the ravitational phase the pseudo-static loadin is applied. Fiure 9 shows the distribution of pseudo-static pressures behind the wall (after subtractin ravitational pressures), as well as linear analyses results, shown in solid line. Very ood areement with M-O solution is found for all levels of round acceleration taken into account. This is not surprisin, since the model is now elastic-plastic and the loadin is applied in a pseudo-static fashion. It s interestin to notice that for low a values there is a reasonable areement between elastic and elastic-plastic results, whereas with increasin a level this areement is radually lost (except for the lowest part of the wall where the fixity doesn t allow plastic deformation to occur). Therefore, a transition is observed from elastic behaviour, in which V-Y type methods seem to be applicable, to plastic behaviour in which M-O methods are thouht to be more suitable under pseudo-static conditions. This transition becomes more or less important dependin on the a imposed. In terms of bendin moments (fi. 2), a ood comparison between pseudo-static and application of M-O formula is observed, takin into account the inertial effect due to wall mass, that obviously cannot be directly accounted for into M-O formula. Indeed, simulations with zero wall mass (µ w = ) clearly point out this aspect. This was not so influent in elastic analyses because of the relatively small wall thickness t w, contrary to the present problem, for which a realistic concrete wall with a specific thickness has been simulated. Linear elastic results seem to provide a lower bound of elasticplastic results (nelectin wall mass contribution). This is attributed to the improved simulation of soil-wall interaction phenomenon. Indeed, in elastic analyses complete bondin between soil and wall was assumed, leadin to formation of tensile stresses near the top of wall. In the elastic-plastic case, Mohr-Coulomb criterion doesn t allow the tensile stresses to exceed the ravitational compressive stresses (see fi. 9, for a =., where tensile stresses do not exceed yet the ravitational compressive stresses, shown in fi. 8). As a result, an increase in stresses near the top of wall and in associated internal forces in the wall (shear and bendin moment) is observed. Therefore, the common thinkin that considerin elastic solutions is conservative does not apply for this case, mainly due to the lare wall flexibility assumed. Paper No. 6.8a 9

11 - a =. -3 a = τ Shear stress, τ (kpa) 2.E E-4.E+ γ ( - ) a = -.E-4 y = 3 m y = 6 m E-4 6.E E-4 2.E-4.E+ γ ( - ) a =.3-2.E-4 y = 3 m y = 6 m -4.E-4-6.E E-3 8E-4 6E-4 4E-4 2E-4 γ ( - ) E+ -2E-4 y = 3 m y = 6 m -4E-4-6E E-3 Shear strain, γ ( - ).E-3.E-3.E-4 γ.e+ -.E-4 y = 3 m y = 6 m -.E-3 Fi. 22. Dynamic behaviour of soil for different acceleration levels Dynamic Loadin Some preliminary analyses were conducted to check the response of the hysteretic model in FLAC. For this purpose, the dynamic behaviour of two soil elements located at ξ = x / L = (i.e. on the centerline of the model), respectively at 3 and 6 m from the model base, was monitored. Sinle pulses with frequency equal to the fundamental, and acceleration levels from a =. to.3 were applied, includin a special case of a =., representative of elastic conditions. The results are presented in the form of τ γ curves in fiures 2 and 22. At a =. (fi. 22), the behaviour is elastic, but some hysteretic dampin is appearin, and the two soil elements show different dampin since are subjected to different shear strain. When the acceleration is larer this dampin is clearly increasin. When the base acceleration level becomes very hih (i.e. above ) plastic effects are observed, coherently with the hypothesis of elastic-plastic model: for and.3 lare plastic deformations are observed, and this plastic behaviour is also takin part in enery dissipation addin up extra dampin to the hysteretic component. Normalized pressures due to horizontal shakin decrease for increasin excitation, as shown in fi. 23, where correspondin elastic results are reported with a black solid line. This results in a reduction of internal forces actin on the structure, as shown in fi. 24. It s clearly visible that for the elastic-plastic model there is a decay in terms of normalized bendin moment with acceleration level. The elastic case is below the elastic-plastic case for the reasons discussed previously. Concernin the M-O formula, which cannot allow for any kind of dynamic effect, the results are well below elastic-plastic results. Total (ravitational + dynamic) base moments M b,tot for all frequencies and acceleration levels are plotted in fi. 2. Horizontal line indicates the value of M b,rav, that is the value of base moment for ravitational loadin. Paper No. 6.8a

12 Elastic a / =..3 Normalized heiht, y / H ( - ).7.3 a / Mb,tot (knm / m) M b,rav: Mb,rav: ravitational only only Normalized base moment, Mb / γ α H 3 ( - ) Normalized pressures, σ xx xx / / γ α H (--) ) Fi. 23. Dynamic wall pressures at ω = ω Mononobe-Okabe FLAC elastic-plastic model FLAC linear elastic model. Normalized base acceleration, α ( - ) Fi. 24. Base moment due to dynamic loadin (ω = ω ) It is clear that with increasin acceleration level, the base moment is increasin, but soil non-linearity effects are clearly visible: as acceleration level increases the fundamental frequency of the system is decreasin, due to softenin of the soil material. Furthermore, for lare acceleration levels (a / =.3) the curve is not so reular, which can be attributed to larer plasticity effects in the soil. As frequency increases, base moment tends to move towards the ravitational value, since dynamic effects become less important. It is instructive to introduce an amplification factor (fi. 26): M b, tot M b, rav M b AF( M b) = = (3) M M M M b, pst b, rav b, pst b, rav where M b,pst is the total base moment for pseudo-static loadin. AF Mb ( - ) ω / ω ( - ) Fi. 2. Total base moment for different a / and ω / ω a / = ω / ω ( - ) Fi. 26. Amplification factor for different a / and ω / ω Analysin fi. 26 the followin conclusions can be drawn: - as acceleration level is increasin, AF(M b ) is decreasin due to hysteretic dampin and plasticity effects; - as acceleration level is increasin, the fundamental frequency of the system is decreasin, due to soil softenin; - as excitation frequency is increasin there is deamplification (dynamic bendin moment is lower than pseudo-static moment). The effective heiht of the resultant dynamic force is located in the reion of (.3 ) H as shown in fi. 27, hence in areement with Rankine theory (/3 H), and below EC8 part [24] suestion ( H) or Seed and Whitman [97] suestion ( H)..3 Paper No. 6.8a

13 Effective heiht, h / H ( - ) Seed and Whitman (97) Eurocode 8- (24) Mononobe and Okabe ( ) a / =..3 REFERENCES Arias, A., Sanchez-Sesma, F.J., Osvaldo-Shelley, E. [98], A simplified elastic model for seismic analysis of earth-retainin structures with limited displacements, Proceedins of the International Conference on Recent Advances in Geotechnical Earthquake Enineerin and Soil Dynamics, University of Missouri, Rolla, MO, I, pp Callisto, L., and Soccodato F.M. [27], Seismic analysis of an embedded retainin structure in coarse-rained soils, 4th International Conference on Earthquake Geotechnical Enineerin, June 2-28, Thessaloniki, Greece ω / ω ( - ) Fi. 27. Effective heiht for different a / and ω / ω FINAL REMARKS Results of a series of numerical analyses with the finitedifference code FLAC presented in the present paper have demonstrated limits of applicability of both limit equilibrium and elastic methods in the computation of pressures behind riid and flexible structures. The assumption of soil nonlinearity has shown aspects that cannot be accounted for by means of these simplified approaches. Improvements in elastic analyses have been achieved by means of the simple assumption of soil inhomoeneity, as already noted by Veletsos and Younan [994, 997], and Psarropoulos et al. [2]: () a consistent reduction of internal forces actin on the structure is observed for riid walls, and (2) tensile stresses near the top of wall for lare wall flexibility or base compliance are reduced. Elastic-plastic analyses for flexible walls have shown several important features: () dynamic amplification is important for excitation frequency between one and two times the fundamental of the soil layer, therefore the simple pseudostatic assumption of M-O method results in a underestimation of effects; (2) non-linearity implies decrease of the resonant frequency of the layer, as well as the important effect of reduction of dynamic amplification effects with increasin acceleration level, a beneficial effect which is impossible to take into account by means of M-O and V-Y methods. In eneral, with the obtained informations non-linear dynamic soil-wall interaction effects for flexible walls are clarified, and a more comprehensive and innovative approach for analysis and desin of earth retainin structures can be developed. Current research is devoted to comparison of outcomes with available experimental findins, analysis of effects of real accelerorams, as well as the assumption of a plastic hine at wall base, possibly leadin to development of a performance based approach for the analysis of cantilever retainin walls. Comité Européen de Normalisation (CEN), [24], EN 998- :24 Eurocode 8: Desin of structures for earthquake resistance - Part : Foundations, retainin structures and eotechnical aspects. Mononobe, N. and Matsuo, H. [929], On the determination of earth pressures durin earthquakes, Proceedins of the World Enineerin Conress, vol. 9, pp Okabe, S. [926], General theory of earth pressures, Journal of the Japan Society of Civil Enineerin, Tokyo, vol. 2, n.. Psarropoulos, P.N., Klonaris, G., Gazetas, G. [2], Seismic earth pressures on riid and flexible retainin structures, Soil Dynamics and Earthquake Enineerin, Vol. 2, pp Richards, J. and Elms, D.G. [979], Seismic behaviour of ravity retainin walls, Journal of Geotechnical and Geoenvironmental Division, ASCE, (4), Seed, H.B., and Idriss, I. M. [97], Soil Moduli and Dampin Factors for Dynamic Response Analysis, Earthquake Enineerin Research Center, University of California, Berkeley, Report No. UCB/EERC-7/, p. 48, Dec. 97. Seed, H.B., and Whitman, R.V. [97], Desin of earth retainin structures for dynamic loads, Proceedins of the special conference on lateral stresses in the round and desin of earth retainin structures, pp Veletsos, A.S. and Younan, A.H. [994], Dynamic modelin and response of soil-wall systems, Journal of Geotechnical and Geoenvironmental Enineerin, ASCE, 2 (2), pp Veletsos, A.S. and Younan, A.H. [997], Dynamic response of cantilever retainin walls, Journal of Geotechnical and Geoenvironmental Enineerin, ASCE, 23 (2), pp Wood, J. [973], Earthquake-induced soil pressures on structures, Report EERL 73-, California Institute of Technoloy, Pasadena, California, 3 pp. Paper No. 6.8a 2