Influence of damping and soil model on the computed seismic response of flexible retaining structures

Effets du coefficient d'amortissement et du model constitutif du sol sur la réponse sismique d’écrans de soutènement souples

S. Aversa & R.M.S. Maiorano Università di Napoli “Parthenope”, Naples, Italy

C. Tamagnini

Università di Perugia, Perugia, Italy

ABSTRACT

The main goal of this paper is to explore the potentialities offered by a commercially available FE code, ex-plicitly developed for geotechnical engineering applications, in the analysis of the seismic response of an “ideal” retaining wall (a cantilevered RC diaphragm wall) in a dry granular soil. Particular attention is given to the sensitivity of the predicted response to such key factors in the analysis as the amount of the material damping included in the analysis, the constitutive model adopted and the choice of the parameters controlling soil stiffness upon first loading and unloading/reloading.

RÉSUMÉ

Le but principal de cet article est d’enquérir les possibilités offertes par un programme FE commercial, développé spécifiquement pour application de mécanique du sol, dans l’analyse de la réponse sismique d’un écran de soutènement idéal (un écran de parois moulées) réalisé à l’intérieur du un terrain granulé sec. Une particulière attention est donnée à la dépendance de la réponse par quelques facteurs clés comme par exemple le coefficient d'amortissement du matériel, le model constitutif adopté et le choix des paramètres qui contrôlent la rigidité des terrains à la première charge et dans les cycles décharge/recharge.

Keywords: flexible retaining walls, seismic response, damping, constitutive models

1 INTRODUCTION

The dynamic interaction between an embedded retaining wall and the surrounding soil under seismic conditions can be evaluated using several procedures at different levels of complexity. The simplest pro-cedures (pseudostatic approach) are relevant for very simple retaining structures. Active and passive earth pressures are evaluated using the classical ap-proaches of Mononobe and Okabe (Mononobe & Matsuo, 1929; Okabe, 1924) or using more complex solutions obtained via limit analysis (e.g., Chang 1981). The main difficulty of such an approach lies in the correct evaluation of the equivalent seismic coefficient which is related to the characteristics of the seismic input, to the properties of both the soil and the wall, and to the required level of perform-ance of the structure under the seismic action. The second group of methods (simplified dynamic analy-ses) considers separately the free-field propagation of the seismic action trough the ground and the soil-structure interaction process between wall and soil. The latter is analyzed by means of a visco-elastoplastic Winkler subgrade reaction model

(Richards et al., 1999). The main problems with such kind of approach are the proper definition of the impedance (dynamic stiffness) of the springs, and the lack of commercial computer codes imple-menting the full procedure, which forces to resort to different computational platforms for free-field propagation and soil-structure interaction analyses.

A third, and more promising, strategy to analyze the seismic response of flexible retaining structures is to perform a complete soil-structure interaction analysis, based on one of the several FE or FD com-puter codes specifically developed for geotechnical engineering applications nowadays available on the market at a relatively low price, and thus easily ac-cessible to practitioners.

However, using such tools to perform a complete soil-structure interaction analysis is a very complex task which requires careful consideration of a num-ber of important issues which may have a dramatic impact on the computed results.

From the numerical standpoint, two major issues are the modelling of the radiation condition via proper boundary conditions and the choice of the in-tegration method used to advance the solution in

time. From the mechanical standpoint, careful con-sideration should be given to; i) the appropriate definition of the seismic input; ii) the modelling of soil/structure interface behavior; and, most impor-tant, iii) the constitutive modelling of the cy-clic/dynamic behavior of the soil.

About this last point, it is worth noting that one of the major drawback of commercially available FE or FD codes is the lack of any constitutive model capa-ble to simulate with sufficient accuracy the cyclic and dynamic response of natural soils in their mate-rial library, most likely due to the inherent complex-ity of such models and the large number of experi-mental data required for their calibration.

The main goal of this paper is to explore the po-tentialities offered by one of such codes – the FE code PLAXIS

® v8.5 – in the analysis of an “ideal” re-taining wall (a cantilevered RC diaphragm wall) in a dry granular soil. Particular attention is given to the sensitivity of the predicted response to such key fac-tors in the analysis as the amount of the material damping adopted in the analysis and the constitutive model adopted for the soil layer. The research pre-sented in this paper is part of a wider research pro-ject on the seismic behaviour of tunnels and dia-phragm walls under seismic actions. In this framework Callisto and Soccodato (2007) and Car-ruba and Brusarosco (2007) published papers on similar topics.

2 THE PROBLEM EXAMINED

Figure 1 illustrates the particular problem considered in this study. A 9.0 m long and 0.6 m thick cantile-vered RC retaining wall supporting a 4.0 m deep ex-cavation in a dry sand layer is subject to earthquake loading from a bedrock located at a depth of 30 m.

The sand layer is characterized by a constant fric-tion angle ϕ’ = 35°, an average density ρ = 2.04 kN*s2/m4, and a stress-dependent dynamic shear stiffness G0 defined by the following empirical power law:

( ) ( )0.5

0 010000 and in kPaG p G p′ ′= × (1)

where p’ is the mean effective stress. The in-situ ini-tial profiles of the shear wave velocity Vs resulting from Eq. (1) and of the small strain shear modulus G0 are shown in Figure 2. An angle of friction δ = 20° has been considered at the soil-wall inter-faces. The embedded length of the wall, equal to 5.0 m, was computed by means of the classical Blum method, in which a horizontal seismic coefficient kh = 0.1 was used and a global factor of safety was imposed to the resultant of the passive earth pres-sures. In the present study, three different accelera-tion time histories have been adopted as seismic in-put at the base of the sand layer.

These are taken from a database of Italian seismic events collected by Scasserra et al. (2006). The sig-nals have been scaled to values of ag

equal to 0.35g

according to the seismic zonation specified by the OPCM 3274 Italian building code (2003).

Figure 1. Geometry of the problem.

Figure 2. In-situ profiles of G0 and Vs.

Table 1 summarizes the main characteristics of the three earthquakes, in terms of peak ground accelera-tion (PGA), predominant period of the event (T) and the fundamental circular frequency ω1N.

Station Name Event PGA (g)

T (s)

ωN1

(rad/s)

Tolmezzo A-TMZ000 Friuli, 1976 0.357 0.50 12.7

Sturno A-STU270 Irpinia, 1980 0.320 2.31 2.72

Norcia E-NCB090 Umbria, 1997 0.382 0.25 25.13

Table1. Seismic records used in this study.

3 THE FE MODEL

The problem under study has been analyzed using the commercial FE code PLAXIS

® v8.5 (Plaxis 2004) equipped with the Dynamic Module.

Two different soil models have been adopted for the granular soil: the classical Mohr-Coulomb elas-tic-perfectly plastic model and the isotropic harden-ing “Hardening-Soil Model” (HSM), both available in the material models library of PLAXIS

®. Figure 3 shows the finite element discretization

adopted, consisting of 1234 cubic strain triangles. In the horizontal direction, the FE model extends 75 m on both sides of the wall. Standard absorbent ele-ments (Lysmer & Kuhlemeyer 1969) have been placed at the lateral boundaries to simulate the radia-tion condition.

The geostatic stress state was computed assuming an earth pressure coefficient at rest K0 = 0.5. The in-stallation effects have not been taken into account. Before the application of the seismic loading, the excavation of the soil has been simulated by remov-ing the soil layers inside the excavation in a single (quasi-static) stage.

In the dynamic stage of each analysis, the viscous damping of the soil has been modelled via the Rayleigh approach, in which the discrete damping matrix C is computed as a linear combination of the mass (M) and stiffness (K) matrices:

R R

= α + βC M K (2)

The two Rayleigh coefficients αR and βR have been calculated according to the double frequency method as suggested by Lanzo et al. (2004), assuming that the soil damping ratio, D, is constant between the first natural frequency ω1 of the deposit and a fre-quency ωn = nω1, where n is the first odd integer larger than the ratio ωN1/ω1 between the fundamental frequency of the seismic signal (ωN1) and the first natural frequency of the deposit (ω1).

The fundamental frequencies of the seismic sig-nals are given in table 1, while the first natural fre-quency of the deposit was calculated, as a first ap-proximation, with the expression valid for a linear elastic layer of thickness H and shear wave velocity Vs:

1 2s

V

H

πω = (3)

For the shear velocity profile reported in Figure 2, ω1 varies between 7 and 16 rad/s.

To evaluate the influence of damping ratio on the numerical predictions, three different values of D (5%, 10% and 15%) have been considered. To take into account the change of soil stiffness (and shear wave velocity) with depth, the soil layer has been divided into 6 sublayers (see Figure 3), each charac-terized by the corresponding average values of Vs. The Rayleigh coefficients have then been computed for each sublayer and each seismic input adopted. The Rayleigh coefficients computed for the 3 earth-quakes assuming D = 5.0% are given in Table 2.

A-TMZ000 A-STU270 E-NCB090

layer #

z [m]

αR βR αR βR αR βR

1 0-1 0.516 0.0036 0.344 0.0073 0.619 0.0015 2 1-4 0.674 0.0028 0.449 0.0056 0.786 0.0014 3 4-6 0.750 0.0025 0.500 0.0050 0.875 0.0013 4 6-10 0.879 0.0021 0.586 0.0043 0.977 0.0014 5 10-20 0.711 0.0035 0.680 0.0037 1.133 0.0012 6 20-30 0.772 0.0032 0.772 0.0032 1.286 0.0011

Table 2. Rayleigh coefficients for 5% damping.

4 SOIL CONSTITUTIVE MODELS

4.1 Mohr-Coulomb elastoplastic model

As a prototype of the classical approach to constitu-tive modelling of soil behavior, the elastic-perfectly plastic Mohr-Coulomb model with non-associative flow rule – commonly found in any commercial FE code – has been used for the first series of simula-tions.

The complete set of parameters adopted in the seismic simulations is given in Table 3. It is worth noting that a non-associative flow rule, with a dila-tancy angle ψ = 2° has been used in all the simula-tions, and that a very small value of cohesion c’ has been imposed to avoid numerical instabilities close to the ground surface, where the mean effective stress is very small.

Figure 3. Finite element mesh

z (m)

γ (kN/m3)

ν (-)

E0,ref (kPa)

cref (kPa)

ϕ’ (°)

ψ (°)

E0,incr (kPa)

zref (m)

0-1 20 0.3 91379 0.5 35 2 0 0

1-4 20 0.3 91379 0.5 35 2 30460 1

4-6 20 0.3 182758 0.5 35 2 20537 4

6-10 20 0.3 223832 0.5 35 2 16283 6

10-20 20 0.3 288966 0.5 35 2 11969 10

20-30 20 0.3 408660 0.5 35 2 9184 20

Table 3. Parameters of the Mohr-Coulomb model.

For each sublayer a linear variation of the Young modulus has been assumed, according to:

( )0 0, 0,ref incr refE E E z z= + − (4)

As the model do not allow to change the soil stiff-ness with the strain level, a reduced “static” stiffness has been adopted during the preliminary construc-tion stage. Three different scenarios have been con-sidered for the static moduli, characterized by a modulus reduction ratio E/E0 equal to 1/3, 1/5 and 1/10, respectively.

4.2 Hardening Soil model

As a second, more advanced constitutive formula-tion for the sand soil, the Hardening Soil model (HSM, Schanz 1998) available in the PLAXIS

® mate-rials library has been considered.

The HSM is an isotropic hardening plasticity model intended to describe the mechanical behav-iour of sand, gravel and stiff, heavily overconsoli-dated cohesive soils. Its yield function is given by:

50

1 2

(1 / )p

a ur

q qf

E q q E= − − γ

− (5)

where E50 is the secant modulus at 50% failure load in drained triaxial compression, Eur is the Young's modulus describing the elastic response of the mate-rial, and qa is given by:

( )( )

32sin ' cot '

1 sin '

′ ′ϕ ϕ + σ= =

− ϕ

f

a

f f

q cq

R R (6)

where qf is the stress deviator at failure, provided by the Mohr-Coulomb criterion, σ’3 the minor principal effective stress, and Rf a material parameter.

In the following, the small strain stiffness, Eur, and the secant modulus at 50% failure load, E50, will be referred to as E0 and E, respectively for consistency with the Mohr-Coulomb model. The only (scalar) hardening parameter of the model is the plastic shear strain γp = 2ε1

p−εvp, where ε1

p and εvp are the major

principal component and the volumetric component of plastic strain, respectively. The meaning of all pa-rameters appearing in Eq. (5) and (6) is illustrated in Figure 5.

In the current version of the model, both moduli depend on current stress state via the expression:

3cos ' sin '

cos ' sin 'α α

′ ′ϕ − σ ϕ= ′ ϕ + ϕ

m

ref

ref

cE E

c p (8)

where α stands for “50” or “ur”, pref is a reference pressure, and m a material parameter, typically in the range 0.5 ≤ m ≤ 1.0. In order to assess the effect of the stiffness ratio Eur/E50 on the computed results, three different values have been considered in the simulations, namely 3.0, 5.0 and 10.0.

Figure 4. Hard Soil Model: stress-strain relationship corre-sponding to the adopted hardening rule (after PLAXIS 2004).

The flow rule adopted in HSM is characterized by a classical linear relation, with the mobilized dilatancy angle given by:

sin ' sin '

sin1 sin ' sin '

ϕ − ϕψ =

− ϕ ϕm cv

m

m cv

(9)

where ϕ’cv is a material constant (the friction angle at critical state) and:

1 3

1 3

' 'sin '

' ' 2 'cot '

σ − σϕ =

σ + σ − ϕm

c (10)

According to Eq. (9), ψm depends on the values of friction and dilatancy angles at failure, ϕ’ and ψ, which control the quantity ϕ’cv. Note that Equation (9) is comparable to the Rowe’s stress-dilatancy the-ory, as discussed by Schanz & Vermeer (1996). The complete list of material parameters adopted in the HSM simulations is given in Table 4.

Parameter Set 1 Set 2 Set 3

pref (kPa) 100 100 100

Eur,ref (kPa) 288835 288835 288835

E50,ref (kPa) 96278 57767 28883

m (-) 0.50 0.50 0.50

cref (kPa) 0.50 0.50 0.50

ϕ’ (°) 35 35 35

ψ (°) 2 2 2

νur (-) 0.20 0.20 0.20

Table 4. Parameters of the Hardening Soil Model.

5 SELECTED RESULTS

In the present work, an extensive parametric study based on 36 dynamic FE simulations has been car-ried out. Due to space limitations, only a selection of the results obtained for the Tolmezzo earthquake is presented in the following. The discussion of the re-sults is therefore focused on the effects of the consti-tutive model adopted and the amount of viscous damping considered.

5.1 Mohr-Coulomb model

Some selected results from the FE simulations per-formed with the Mohr-Coulomb model are presented in Figures 5 to 7. The distribution of horizontal stresses computed for the two extreme values of the damping ratio D at E0/E = 3 are show in Figure 5.

Figure 5. Mohr-Coulomb model: horizontal stresses. ST: static; PS: post-seismic; LE: limit equilibrium.

In the figures, the post-seismic distributions of σh (PS) are compared with the static, pre-seismic condi-tions (ST) and with the limit equilibrium earth pres-sure distributions (LE) computed using Blum’s method and Chang (1981) earth pressure theory.

Under static conditions, active and passive limit states are fully mobilized down to a depth of about

5 m from the original ground surface. Below this depth, in front of the wall σh remains approximately constant, while behind the wall it increases up to the full geostatic value (K0 = 0.5).

Figure 6. Mohr-Coulomb model: bending moments. ST: static; PS: post-seismic; EN: env. of max. values;

LE: limit equilibrium.

The seismic shaking produces significant permanent changes in earth pressure distributions on the two sides of the wall. The largest variations occur in the zone located between 4 m (the final excavation level) and 8 m below the original ground surface, where the normal stresses on both sides of the wall are more that doubled.

Also, in the uppermost 1 m of the wall, the post-seismic stress distribution on the active side shows a significant pressure bulge, which has a strong impact on the wall bending moments.

Such differences between the two stress distribu-tions are associated to the development of irreversi-ble, plastic deformations in the soil mass during the seismic stage.

The comparison between Figure 5a and 5b indi-cates that such phenomenon is more pronounced for low damping values, as in this case the magnitude of deviatoric stress increments induced in the soil by the seismic shaking decreases with increasing D.

Figure 7. Mohr-Coulomb model: max. bending moments under seismic conditions.

The observations made on the horizontal stress dis-tributions are consistent with the computed distribu-tions of wall bending moments, shown in Figures 6a and 6b. In the figures, the static, pre-seismic bending moment distribution (ST) is compared with the final, post-seismic one (PS) and with the envelope of maximum bending moment experienced during the seismic excitation (EN), calculated using a dynamic to static stiffness ratio of 3 and the two extreme val-ues of the damping ratio.

The results clearly show that, for both values of the damping ratio, the maximum moments and the post-seismic moment are largely greater than the ini-tial, static ones. As expected, the reduction in stress levels experienced by the wall as the damping ratio increases reflects into a decrease of the peak bending moments with D, as it is clearly apparent from the data in Figure 6. On the contrary, the maximum bending moment experienced by the wall during the seismic stage is practically independent of the stiff-ness ratio E0/E. In Figures 6a and 6b, the results of FE simulations are also compared with the moment distributions computed with the Limit Equilibrium approach. The curve labeled LE(stat) refers to the initial static conditions, while the curve labeled LE(dyn) provides the bending moments distribution computed with the pseudo-static approach according to the 1996 Italian seismic code (kh = 0.1), using Chang (1981) earth pressure theory. It is worth not-ing that the maximum and post-seismic bending moments computed for both D = 5% and D = 15% are much larger than the pseudo-static stress distri-butions provided by the Limit Equilibrium analysis, although the LE solutions are typically considered to be overly conservative.

5.2 Hardening Soil model

Figures 8 to 10 illustrate some selected results of the FE simulations performed with the Hardening Soil model for the same values of damping ratio D and

stiffness ratio E0/E. A direct comparison with the data reported on Figures 5 to 7 allow to assess the influence of the constitutive model on the dynamic response of the soil-wall system.

From a qualitative standpoint, the stress distribu-tions at the soil-wall interfaces obtained with the HS model appear similar to those given by the MC model. In particular, large permanent increases of σh are observed on both sides of the wall after the seis-mic event, although in this case the largest differ-ences on the “active” side of the wall are more con-centrated towards the wall ends. Again, the permanent increase in soil pressures is to be attrib-uted to the onset of plastic yielding in the soil mass, which is strongly affected by the amount of viscous damping included in the simulation.

Figure 8. Hardening soil model: horizontal stresses. ST: static; PS: post-seismic; LE: limit equilibrium.

The pattern of the bending moment distributions along the wall – static, initial (ST); post-seismic (PS), and envelope of maxima during the seismic stage – is given in Figure 9, together with the static and the (pseudo-static) seismic LE solutions.

The three distributions of bending moments ob-tained in the simulations with the HS model qualita-tively resembles the ones given by the MC model. It is worth noting that, for D = 5%, the post-seismic

moments provided by the HS model are slightly lar-ger than those obtained with the MC model. On the contrary, the adopted constitutive model has no sub-stantial influence for D = 15%.

Very large differences are observed between the max. bending moments experienced during the earthquake and the initial, static ones.

The computed dynamic and post-seismic loads on the structure tend to increase substantially as the damping ratio decreases. A more complete picture on the effects of damping ratio and stiffness ratio on the max. bending moments during the seismic stage is given in Figure 10, where the max bending mo-ments obtained with the MC model for E0/E = 3 are also plotted for comparison.

Figure 9. Hardening soil model: bending moments. ST: static; PS: post-seismic; EN: env. of max. values;

LE: limit equilibrium.

The max. bending moment decreases by about 30% when the damping ratio increases from 5% to 15% (the corresponding reduction for the MC model is about 40%). In this case, it is interesting to note that the response of the wall is also significantly affected by the stiffness ratio E0/E. In particular, the larger is the stiffness ratio, the lower is the maximum bend-ing moment.

Figure 10. Hardening soil model: max. bending moments under seismic conditions.

Figure 11. Max. bending moments at the end of the construc-tion stage vs stiffness ratio for the two models considered.

The reason for this observation is to be found in the effects of the constitutive model formulation on the dynamic response of the system, as the soil model adopted has only a limited influence on the initial, pre-seismic loading conditions of the structure, as shown in Figure 11, where the max. bending mo-ments at the end of construction are plotted as a function of the stiffness ratio.

6 CONCLUSIONS

The current availability of robust and accurate im-plementations of the finite element method in com-mercial codes capable of analyzing soil dynamics problems, together with the ever increasing compu-tational power of MS-Windows or UNIX/Linux workstations represents a significant progress in the analysis and design of such complex geotechnical structures as flexible retaining walls subject to earthquake loading. However, the impact on the re-sults of the different choices made in the analysis, in

terms of both the details of the computational strat-egy and the description of soil behavior under cy-clic/dynamic conditions still remains to be investi-gated.

In the present work, some selected results from an extensive parametric study have been presented to provide some insight on the effects of the amount of viscous damping introduced in the model and of the constitutive framework adopted to describe the me-chanical response of the soil (perfect plasticity vs hardening plasticity). The choice has deliberately been made to confine the investigation to a specific FE platform and constitutive models widely avail-able to practicing engineers, to assess the possibili-ties offered by commercial design tools in the cur-rent geotechnical practice.

The results of the simulations have shown that large differences (on the unsafe side) can be ex-pected in wall bending moments between conven-tional limit equilibrium solutions and a full soil-structure interaction analysis, even for the simplest case of a cantilevered retaining wall in a dry soil.

The FE results show that the computed response of the wall/soil system is largely affected by the choice of the Rayleigh damping coefficients, inde-pendently of the specific constitutive model adopted for the soil. A word of caution is therefore necessary when FE codes implementing the Rayleigh approach are used in routine design. The approach suggested by Lanzo et al. (2004) can be used for the rational selection of appropriate values of αR and βR for any given value of the damping ratio D.

A significant feature of all the FE results is the large permanent increase of wall bending moments after the earthquake. The residual load are often a significant fraction of the max. loads experienced during the shaking. This phenomenon, which is a di-rect consequence of the inelastic nature of the soil, cannot be modelled if the soil is treated as an elastic or hypoelastic material. Even if the constitutive models considered in this study are quite simple and not capable of reproducing the hysteretic behavior of cyclically loaded soils, yet the influence of soil yielding on the computed wall stresses is quite large.

In this respect, it is also worth noting that, for complex models such as the HS model, the dynamic response of the system may be significantly affected by the amount of stiffness degradation with the strain level (i.e., the stiffness ratio E0/E for the HS model) assumed for the soil. A proper quantitative modelling of the gradual transition of the material stiffness from high values at small to smaller values at large strain levels seems therefore to be of impor-tance for the accurate prediction of the seismic re-sponse of the structure even when using advanced constitutive equations to model the cyclic/dynamic behavior of the soil.

Finally, it is worth mentioning that more sophisti-cated analyses with advanced constitutive models

and physical testing on small scale models in the centrifuge are currently under way in order to quan-tify the reliability of the predictions obtained with the FEM code adopted in this work, as well as the predictive capabilities of other “conventional” de-sign tools.

ACKNOWLEDGEMENTS

The work presented in this paper is part of ReLUIS research project, founded by the Italian Department of Civil Protection. The strong motion data utilized in this study was developed as part of an ongoing joint project involving researchers from the University of Rome La Sapienza and the University of Califor-nia, Los Angeles, with support from the Pacific Earthquake Engineering Research center. Preliminary results from this group were published by Scasserra et al. (2006), but the data utilized here have not been published.

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