Evaluation of pseudostatic active earth pressure coefficient of cantilever retaining walls

Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Evaluation of pseudostatic active earth pressure coefficient of cantilever retaining walls Aldo Evangelista a, Anna Scotto di Santolo a,n, Armando Lucio Simonelli b a b

Department of Hydraulic, Geotechnical and Environmental Engineering, University of Naples Federico II, 80125 Naples, Italy Department of Engineering, University of Sannio, 82100 Benevento, Italy

a r t i c l e in fo

abstract

Article history: Received 24 December 2009 Received in revised form 21 June 2010 Accepted 26 June 2010

A stress plasticity solution is proposed for evaluating the gravitational and dynamic active earth pressures on cantilever retaining walls with long heel. The solution takes into account the friction angle of the soil, wall roughness, backfill inclination and horizontal and vertical seismic accelerations. It is validated by means of the comparison with both traditional limit equilibrium methods (e.g. Mononobe– Okabe equations) and static and pseudostatic numerical FLAC analyses. For numerical analyses the soil is modelled as an elasto-plastic non-dilatant medium obeying the Mohr–Coulomb yield criterion, while the wall is elastic. The solutions for the horizontal and vertical seismic coefficients are proposed, which allow one to determine the intensity of the active thrust and its inclination d with respect to the horizontal. It is demonstrated that the latter also depends on the soil friction angle j. The inclination in seismic conditions dE is greater than the one in static conditions, dS, usually adopted in both cases. As a matter of fact, since wall stability conditions improve with the increase of inclination d, the present method gives solutions that are less onerous than traditional ones, producing less conservative wall designs. Finally pseudostatic results are compared with proper dynamic analyses (by FLAC code) performed utilising four Italian accelerometric time-histories as input ground motion. & 2010 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3. 4.

5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New stress pseudostatic plasticity solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between NSPPS and Mononobe–Okabe solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation of the proposed NSPPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Description of the wall–soil system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Definition of variables and numerical procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Evaluation of static thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Evaluation of pseudostatic thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Evaluation of dynamic thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of frequency content on dynamic thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Reinforced concrete cantilever retaining walls have been largely built starting from the second half of the last century. Walls of considerable height have been realised, thanks to the relevant contribution to equilibrium provided by the soil resting

n

Corresponding author. E-mail address: [email protected] (A. Scotto di Santolo).

0267-7261/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2010.06.018

1119 1121 1122 1123 1123 1124 1125 1125 1126 1127 1128 1128

on the internal foundation slab, which contributes to stability by means of its own weight. Cantilever walls can be considered as the precursors of sheetpile walls, since the role of structural weight is not predominant and the equilibrium conditions depend mainly on the backfill soil actions and the resistance of foundation soils. The dual function of the soil above the internal base makes it difficult to interpret the behaviour of the soil–wall system, especially in the presence of seismic loads. In fact many Codes, e.g. Eurocodes (Eurocode8, Part 5, 2003) [1] and the Italian Building Code (Norme Tecniche per le

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A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

Nomenclature

amax

inclination of the thrust with respect to the horizontal inclination of the thrust with respect to the horizontal in static conditions dE inclination of the thrust with respect to the horizontal in seismic conditions dMO inclination of the thrust with respect to the horizontal in pseudostatic conditions adopted by the Mononobe and Okabe approach dPS inclination of the thrust with respect to the horizontal evaluated by the FLAC pseudostatic approach j angle of internal friction g unit weight of soil c dilatancy angle e slope of backfill a and b inclinations of failure surfaces with respect to the vertical direction g gravity acceleration r (equal to g/g) mass density kh coefficient of horizontal acceleration kv coefficient of vertical acceleration sv normal stress acting on the horizontal plane tv shear stresses acting on the horizontal plane sa active stresses acting on the vertical plane ta active shear stresses acting on the vertical plane kah horizontal active earth pressure coefficient kav vertical active earth pressure coefficient se normal stress acting on the plane parallel to the slope at depth z te shear stress acting on plane parallel to the slope at depth z E elastic modulus of soil n Poisson coefficient of soil gc unit weight of concrete Ec elastic modulus of concrete nc Poisson coefficient of concrete Dl length of the elements of the square finite difference grid l wavelength associated with the highest frequency (fmax) of the input motion vs shear wave velocity of the backfill

bM

d dS

Costruzioni, 2008) [2], do not explicitly refer to cantilever walls and the role of the backfill soil lying over the internal slab. A traditional approach for verifying geotechnical and structural behaviour of cantilever walls, generally proposed in the literature, is based on the well-known limit equilibrium analysis in plane strain conditions [3], assuming that the wall movements are large enough to mobilise the shear strength in the backfill soil. Unlike the gravity wall, where the thrust on the inner wall face is calculated, in the case of cantilever walls the thrust is evaluated along an ideal vertical surface AV passing through the innermost point of the wall base (wall heel) (Fig. 1). It could be assumed that the thrust on the surface AV has an inclination equal to the soil friction angle j. In 1957 Huntington [4] showed that this assumption is both erroneous and unconservative if the backfill inclination e is lower than j. As a matter of fact, in the hypothesis normally adopted of backfill behaving like an infinite slope, the inclination of the thrust on AV surface is equal to the inclination of the backfill, at least in the static field. The failure surfaces are planes with inclinations a and b with respect to the vertical direction; the angle between the two surfaces is (90  j) (Fig. 2).

ag PGA Ia Tp

Tm Swh Swv Mw Hw Sah Sav Ha FS SahPS SavPS Sahe Save Sahe

dE

max

max

Mw

max

maximum acceleration expected at the site ground surface reduction coefficient for retaining walls according to Italian code [2] horizontal acceleration expected on site A according to Italian code [2] peak ground acceleration of the recorded accelerogram Arias Intensity of the original signal predominant period corresponding to the maximum spectral acceleration (computed for 5% viscous damping) the mean period, as defined by Rathje et al. [16] on the basis of the Fourier spectrum of the signal horizontal component of the thrust on wall stem (A0 V0 ) vertical component (shear force) on wall stem (A0 V0 ) bending moment at the base of the stem (A0 A00 ) height of thrust on wall stem A0 V0 horizontal component of the thrust on the ideal surface AV vertical component of the thrust on the ideal surface AV (shear force) height of thrust on the ideal surface AV global sliding safety coefficient along the cantilever foundation interface horizontal thrusts components on AV evaluated by FLAC pseudostatic approach vertical thrusts components on AV evaluated by FLAC pseudostatic approach horizontal component of the thrust on AV evaluated by dynamic FLAC analyses vertical component of the thrust on AV evaluated by dynamic FLAC analyses maximum horizontal component of the thrust on AV evaluated by dynamic FLAC analyses maximum inclination of the thrust with respect to the horizontal evaluated by dynamic FLAC pseudostatic approach maximum bending moment at the base of the stem (A0 A00 ) during dynamic FLAC analyses

The load on the wall stem (A0 V0 ) is assumed to be equal to the thrust acting on the upper part h of the ideal surface AV. It is clear that this assumption is approximate for walls whose internal slab is short and for small inclinations of the backfill, as the failure

ε

V

ε β

α

Sa δ=ε

Sa δ=ε

A Fig. 1. Thrusts on: (a) cantilever walls and (b) gravity walls.

A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

ε V V’ α

β Sa

h

H

Sw Swh A’

D

ε

Swv

δS

Sav α+β = (90-ϕ)

Sah Ha

δS = ε

A B

Sa

Sae

δ=0

δ = δe

Fig. 2. Thrusts on cantilever walls: (a) inclined backfill: static conditions and (b) horizontal backfill: static and seismic conditions.

1121

unsafe because they may underestimate the active earth pressures and overestimate the passive one. The error in the active case is quite small. According to Collins [9] these analyses can be considered as kinematic solutions of limit analysis. Another group of limit analysis methods is based on stress solutions; among them the best known is the Rankine [10] procedure, usually applied to cantilever wall assuming for the backfill the infinite slope configuration. The limit analysis solutions proposed in geotechnical literature are generally of the first type. In the present paper a new stress pseudostatic plasticity solution (NSPPS) is proposed for evaluating the gravitational and dynamic active earth pressures, adopting pseudostatic seismic loading. This method allows one to evaluate the inclination dE of the thrust along the vertical surface AV. In the following it will be demonstrated that the value of dE depends on the seismic coefficients kh and kv, the soil friction angle j and the backfill inclination e. It will also be shown that dE is greater than the corresponding inclination in static conditions, dS, usually adopted for both static and seismic conditions. It is worthwhile pointing out that if in Mononobe and Okabe formula dMO ¼ dE is assumed, then the new method leads to equal values of thrust coefficients. As is known, wall stability conditions improve with the increase of d; hence the present method gives solutions that are less onerous than traditional ones. Further, if wall displacements are computed by methods based on the Newmark traditional approach [11], higher threshold acceleration values are determined and lower displacements are obviously obtained. The final aim of the study is the comparison of the results of the geotechnical and structural design of a cantilever wall obtained by

 the pseudostatic approach with different inclinations of the thrust (d ¼ dS or d ¼ dE);

 numerical pseudostatic analyses;  proper dynamic analyses (by FLACs code) performed utilising, as input ground motion, four Italian accelerometric timehistories scaled at the same peak ground acceleration. 2. New stress pseudostatic plasticity solution

Fig. 3. Stress state in a horizontal semi-space in active state and in the presence of pseudostatic actions with kh a 0 and kv ¼ 0.

plane cannot freely develop to the surface but intercepts the body of the wall [5]. The error, however, is generally small and does not exceed 2% [4]. The thrusts are calculated with the traditional limit equilibrium methods and pertain to the limit active state induced by an infinite slope. Under seismic loads, the thrust is usually calculated by means of the pseudostatic Mononobe–Okabe (MO) method [6,7]. The analysis is a direct modification of Coulomb wedge analysis [3] where the earthquake effects are replaced by a pseudostatic inertia force whose magnitude is computed on the base of the seismic coefficient concept; as for Coulomb analysis, also in the MO method the failure surface is assumed planar, regardless of the fact that the critical surface may be curved [8]. These solutions are

In this section the new method (NSPPS) is explained. The problem under investigation is a slope dry cohesionless soil, with unit weight g, retained by a cantilever wall under combined action of gravity (g) and seismic body forces (khg) and (kvg) in the horizontal and vertical direction respectively. In this analysis for the sake of simplicity the vertical seismic coefficient kv is assumed equal to zero. Positive kh denotes inertial action towards the wall (Fig. 2). If it is assumed that the backfill is horizontal (slope e ¼0) then at the generic depth z a possible statically admissible stress field is

sv ¼ gz

ð1Þ

tv ¼ kh gz

ð2Þ

where sv is the normal (compression are assumed positive) and tv is the shear stresses (positive according to the Mohr convention), acting on the horizontal plane. According to the theorem of conjugate shear stresses, the same shear stress tv acts on the vertical plane also. Therefore if in the point the failure conditions are achieved, the stress state is represented by the Mohr circle reported in Fig. 3. The active stresses sa and ta on the vertical plane AV can be derived according to the following relations:

sa ¼ 2 OC sv

ð3Þ

ta ¼ tv

ð4Þ

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A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

where

10 1

y ¼ tan ðkh Þ

ð5Þ

  qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 OC ¼ sv 1 þ k2h

ð6Þ

z

ϕ 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ cosy þ sin2 fsin2 y

ð7Þ

se ¼ g cos e z kh g sin e z

ð8Þ

te ¼ g sin e z þ kh g cos e z

ð9Þ

where se and te are the normal and shear stress acting on plane parallel to the slope at depth z. Being z ¼ z cos e, then

se ¼ gzðcos2 ekh sin e cos eÞ

ð10Þ

te ¼ gzðsin e cos e þ kh cos2 eÞ

ð11Þ

Considering in Fig. 5 the Mohr circle passing through the stresses (se, te) and tangent to the failure surface, the distance of the centre from the origin (OC ) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   OC ¼ 2 se  4s2e 4ð1sin2 fÞ t2e þ s2e ð12Þ 2ð1sin2 fÞ

0 O 0

ð13Þ

Table 1 Values of horizontal earth pressure coefficient kah in function of j and kh for e ¼0. kah

u (deg.)

20 25 30 35 40 45

kh 0.05

0.1

0.15

0.2

0.25

0.30

0.35

0.4976 0.4118 0.3383 0.2754 0.2213 0.1751

0.5202 0.4298 0.3535 0.2885 0.2331 0.1858

0.5591 0.4606 0.3791 0.3107 0.2527 0.2036

0.6177 0.5053 0.4159 0.3422 0.2806 0.2287

0.702 0.5662 0.4648 0.3837 0.3169 0.2614

0.8263 0.6471 0.5275 0.4359 0.3622 0.3019

1.0524 0.7549 0.6063 0.4999 0.4171 0.3505

ε

ε z ε

ξ

τa

ε ζ

τε

R C 10

σε 15

20

σ

-5

-10 Fig. 5. Stress state in an infinite slope in active state in the presence of pseudostatic loading with kh a0 and kv ¼0.

The following relations among the angles indicated in Fig. 5 can be derived: ! s OC z ¼ tan1 e ð14Þ

t

x¼

o¼

p 2

ez

p 2

x þ e

ð15Þ

ð16Þ

Finally the active stress components can be expressed as follows:

sa ¼ OC R sin o

ð17Þ

ta ¼ Rcos o

ð18Þ

The solution is possible when ½4s2e 4ð1sin2 jÞðt2e þ s2e Þ is greater than 0. The active stress (sa, ta) has been computed for different values of j, kh, and e. The relative earth pressure coefficients kah and kav, equal to sa/gz and ta/gz, respectively, are plotted in Figs. 6 and 7 for j equal to 301 and 401, respectively.

3. Comparison between NSPPS and Mononobe–Okabe solution

kh γ ε

z*

5 σa

and the radius of the Mohr circle is R ¼ OC sin f

ξ ω

τ

The earth active pressure coefficients kah and kav are equal to sa/gz and ta/gz, respectively. Table 1 reports the kah values obtained by the proposed method. In this case kav a0 and it is equal to kh, according to Eq. (2). In the case of inclined backfill of slope e (Fig. 4) at the generic depth z a possible statically admissible stress field, which is such that it satisfied all stress boundary conditions, is

P

γ

z* = z cosε Fig. 4. Infinite slope with inclination (e + tan  1 kh) r j in the presence of pseudostatic loading with kh a 0 and kv ¼ 0.

The seismic lateral earth pressures obtained from the proposed method are compared with those obtained from the well-known Mononobe–Okabe analysis, generally adopted in current practice for seismic design of retaining walls. For instance, for the case j ¼40 and e ¼ 51, and for horizontal seismic coefficient kh varying between 0.1 and 0.3, the proposed method provides the values of kah and kav reported in Table 2. The inclination d of the thrust with respect to the horizontal can be easily evaluated. If these d values are assumed in the MO formula, the same values of thrust coefficient reported in Table 2 are obtained.

A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

1

1123

1

ϕ = 30°

ϕ = 40°

0.8

0.8 15°

0.6

10°

5°

ε = 0°

0.6

15°

kah

kah

20°

20°

0.4

0.4

0.2

0.2

10° 5° ε = 0°

25°

0

0 0

0.1

0.2 kh

0.3

0.1

0

0.4

0.6

0.2 kh

0.3

0.4

0.6 ϕ = 30°

ϕ = 40° 15° 15°

kav

20°

0.4 10°

10°

20°

5° ε = 0°

kav

0.4

5° ε = 0°

25°

0.2

0.2

0

0.0 0.2 kh

0.3

0

0.4

Fig. 6. Horizontal and vertical earth pressure coefficients for different values of e and kh for j ¼ 301.

4. Validation of the proposed NSPPS In order to validate the proposed method, the behaviour of a cantilever wall has been studied by means of numerical analyses with FLAC 2D code [12]. An extreme high and thin wall has been chosen for the analyses, with the aim of emphasizing the role of the deformability of the wall–soil system. The wall is 10.80 m high and retains a backfill made up of dry cohesionless pyroclastic material (pozzolana) with a unit weight g equal to 15 kN/m3 and friction angle j ¼301 (Fig. 8). The foundation subsoil is made up of neapolitan yellow tuff (NYT) with g ¼15 kN/m3 too. The geometry of the wall allows the development of Rankine failure surfaces that do not intersect the stem of the wall (A0 V0 ). By applying the method to the wall–soil system under examination (Fig. 8), characterised by e ¼0 and j ¼301, and assuming the seismic coefficient values kh ¼0.1 and kv ¼0, the earth pressure coefficients kah and kav equal to 0.3535 and 0.1, respectively, and therefore d equal to 161, are obtained.

0.2

0.3

0.4

kh Fig. 7. Horizontal and vertical earth pressure coefficients for different values of e and kh for j ¼401.

Table 2 Comparison between thrust seismic coefficients for j ¼ 401, e ¼51 and different values of kh. kh

NSPPS

0.1 0.2 0.3

Mononobe–Okabe

kah

kav

dE

kah

kav

0.2425 0.3013 0.4004

0.1212 0.2264 0.3350

26.56 36.92 39.92

0.2425 0.3013 0.4003

0.1212 0.2264 0.3350

0.8m V’

4.1. Description of the wall–soil system The backfill soil is modelled as an elasto-plastic medium (elastic modulus E¼50 MPa and Poisson coefficient n ¼0.3) obeying the Mohr–Coulomb yield criterion (cohesion c0 ¼0; friction angle j ¼301) and non-associated flow (c ¼0); the subsoil material (volcanic tuff) is considered elastic (E¼5 GPa and n ¼ 0.1), but is covered by a thin layer of pyroclastic material in order to allow sliding kinematics. The wall has been schematized with elastic elements characterised by unit weight gc equal to 24 kN/m3, Ec ¼20 GPa and nc ¼0.1.

0.1

z

V

Sa

10m

0.1

10.8 m

0

A’

x

Ha A

α=β γt = 15 kN/m3 c’= 0 ϕ’ = 30° ; ψ = 0° G = 19 MPa ; B = 41 MPa E = 50 MPa ν = 0.3 γcls = 24 kN/m3 E = 20000 MPa ν = 0.1

8.2m Fig. 8. Model of the wall–soil system for static analysis.

The dimensions of the finite difference zones have to follow the recommendation proposed by Itasca [12] in order to obtain accurate computation of model response and numerical distortion

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A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

of propagating ground motion. The length of the elements (Dl) has to be smaller than one-tenth to one-eighth of the wavelength (l) associated with the highest frequency (fmax) of the input motion. A square finite difference grid has been used with Dl equal to 0.2 m (Fig. 9). The backfill is characterised by a shear wave velocity vs ¼226.5 m/s. This grid is adequate to propagate shear wave having frequency up to about 113 Hz. This value is well above the natural frequency of the modelled system (about 6 Hz). It has been preferred to avoid interface elements among the various materials, by inserting thin layers with suitable properties. Along the backfill–wall interface a layer of elements having the same stiffness of pozzolana, but reduced strength (j ¼101), has been placed. In creating the model first the wall elements have been activated, then the backfill ones in subsequent layers half meter in thickness. Using an elasto-plastic constitutive model in conjunction with the Mohr–Coulomb yield criterion, a hysteretic damping D is described by Green et al. [13] according to the following expression:   2 G D¼ 1 ð19Þ Gmax p where G is the secant shear modulus and Gmax is the maximum value, at small strains. 4.2. Definition of variables and numerical procedures

 Swh ¼horizontal component of the thrust on wall stem (A0 V0 );  Swv ¼vertical component (shear force) on wall stem (A0 V0 );

1.500

Lx = Ly = 20cm

1.000

0.500

0.000

0.750

1.250

1.750

  

Mw ¼ bending moment at the base of the stem (A0 A00 ); Hw ¼height of thrust on wall stem A0 V0 ; Sah ¼horizontal component of the thrust on the ideal surface AV; Sav ¼ vertical component of the thrust on the ideal surface AV (shear force); d ¼tan  1(Sav/Sah)¼thrust inclination with respect to the horizontal, acting on AV; Ha ¼height of thrust on the ideal surface AV; FS ¼global sliding safety coefficient along the cantilever foundation interface.

These variables are reported as a function of the wall sliding amount dx in the static case, and as a function of time in the dynamic case. For static analysis, after the initial phase, alternating phases of slow sliding of the wall foundation (velocity of 1  10  8 m/s) and of stabilization of the unbalanced forces have been imposed, up to the generation of the failure conditions. In the pseudostatic case a constant acceleration equal to 0.1g has been applied at any point of the model. This value has been derived from the hypothesis that the maximum acceleration amax expected at the site ground surface is 0.35g; in fact, in this case, the seismic coefficient of the pseudostatic acceleration has been computed, according to the Italian Building Code [2], by the following formula: kh ¼ bM

The following variables have been then computed by means of a ‘‘fish’’ written by the authors:

0.250

   

2.250

Fig. 9. FLAC model grid of the wall–soil system.

amax g

ð20Þ

where bM is the reduction coefficient for retaining walls that can tolerate displacements, and is equal to 0.31 for any soil category (A–D) and for values of ag ranging between 0.2 and 0.4g (where ag is the horizontal acceleration expected on site A). The analyses have been carried out in the dynamic field (dyn on) in order to be able to impose quiet boundary conditions (as proposed by Kuhlemeyer and Lysmer [14]); a 3% Rayleigh damping factor has been introduced [15]. The dynamic response of the retaining wall to four different accelerometric time-histories has been studied. The input motions have been derived by four accelerograms recorded in Italy at Tolmezzo and San Rocco (Friuli, 1976), Sturno (Irpinia, 1980) and Norcia (Umbria-Marche, 1997). The main features of the recorded accelerograms are listed in Table 3. The peak ground acceleration, PGA, is the maximum value of the recorded accelerogram. The frequency content of the waveform is quantified through the predominant period Tp, corresponding to the maximum spectral acceleration in an acceleration response spectrum (computed for 5% viscous damping) and through the mean period, Tm, as defined by Rathje et al. [16] on the basis of the Fourier spectrum of the signal. Actually, Tm should provide a better indication of the frequency content of the recordings because it averages the spectrum over the whole period range of amplification. The energy content of the waveform is quantified through the Arias Intensity value. The accelerograms have been then scaled at a peak value amax ¼0.35g, in order to simulate the earthquake intensity that could be expected on a site classified as B or C category in high seismicity Italian regions for an event having a return period of 475 years (Fig. 10).

Table 3 Strong motion features. Site (Earthquake)

PGA (g)

Mean period Tm (s)

Predominant period TP (s)

Arias Intensity Ia (cm/s)

San Rocco (Friuli 1976) Sturno (Irpinia 1980) Norcia (Umbria-Marche 1997) Tolmezzo (Friuli 1976)

0.09 0.32 0.38 0.35

0.29 0.86 0.17 0.39

0.10 0.20 0.12 0.26

3.22 139.35 34.25 78.65

A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

0.3

300

Swh

0.1 15

20

-0.3

20

200

15 10

100 δs

-0.4

5

0

0.4 0.3

0

W-SR0000

0.2

δ (°)

10

S (kN/m)

5

-0.2

a (g)

30 25

0

0.002

0.004 Sav

0.006

-100

0 0.008 -5 -10

0.1 0 -0.1 0

35

Sah

A-STU 270

0.2 a (g)

40

400

0.4

-0.1 0

1125

dx (m) 5

10

15

20

Fig. 11. Static FLAC model: thrust components and inclination d versus wall sliding displacement dx.

-0.2 -0.3

10

1000

-0.4

0.3

Mw (kNm/m)

E-NCB090

a (g)

0.2 0.1 0 -0.1 0

5

10

15

8

600

6

400

Hw

4

200

Ha

2

20

-0.2 -0.3

0

-0.4

0

0.002

0.004

0.006

0 0.008

dx (m)

0.4 ATMZ 000

0.3

Fig. 12. Static FLAC model: bending moments of the stem wall and point of application of thrusts vs. wall sliding displacement dx.

0.2 a (g)

800

H (m)

Mw

0.4

0.1 0 -0.1 0

5

10

15

20

-0.2 -0.3 -0.4 Fig. 10. Analysis input motions (scaled at 0.35g): (a) modified Sturno accelerogram; (b) modified San Rocco accelerogram; (c) modified Norcia accelerogram and (d) modified Tolmezzo accelerogram.

4.3. Evaluation of static thrust Numerical analyses by FLAC 2D have been properly carried out with the aim of studying the development of the failure conditions in the backfill behind the wall. The stress state varies in function of the imposed sliding displacement (dx). In Fig. 11 the horizontal and vertical thrust components along both real A0 V0 and ideal AV surfaces are reported (see Fig. 8). The failure conditions are reached for a sliding of about 4 mm. In the same figure (Fig. 11) the thrust inclination (d) versus sliding displacement dx is also reported: d values are computed as average values along the ideal AV surface. It is clear that the inclination d of the thrust does not represent a real property of the soil, but it varies as a function of the stress and strain distribution behind the wall.

In the present case (horizontal backfill) d is expected to be equal to 0; actually very low d values have been computed. The horizontal thrust component, evaluated by the diagram in Fig. 11, is Sah ¼300 kN/m, while the vertical component is practically negligible. The integral of the vertical stresses is V¼870 kN/m; thus a horizontal thrust coefficient kah ¼0.343 is computed. The theoretical coefficients calculated by means of the Rankine theory are kah ¼0.333 and kav ¼ 0. The obtained failure surfaces, identifying an angle a + b equal to  601, are coherent with those by Rankine. In conclusion, comparing FLAC with the Rankine theory results, a difference of 3% in the thrust is observed. The stem bending moments are plotted in Fig. 12 in function of the sliding displacement dx; in the same Fig. 12 both the thrust application point Ha along the ideal surface AV and the application point Hw acting on the stem are reported. The maximum bending moment is equal to 780 kN m/m, which is practically coincident with that obtained with the traditional method (830 kN m/m).

4.4. Evaluation of pseudostatic thrust The same model adopted for the static case has been utilised for pseudostatic analysis by FLAC 2D (Fig. 8). As previously mentioned, a constant horizontal acceleration equal to 0.1g has been applied.

A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

On the AV plane the following thrusts components have been calculated: SahPS ¼410 and SavPS ¼110 kN/m. From these values the following coefficients are evaluated: kah ¼0.469 and kav ¼ 0.120. The vertical coefficient is very close to the coefficient of the imposed horizontal acceleration, equal to 0.1. Regarding the thrust inclination on the ideal surface AV, the pseudostatic FLAC application gives a value of dPS equal to 151, for seismic coefficients kh ¼ 0.1 and kv ¼0; the same value has been obtained by the proposed solution. This evidence validates the method NSPPS proposed. The maximum bending moment MPS is equal to 1100 kN m/m.

30 20 10

δ (°)

1126

0

δ2δ

δ1

0 δ3

0.5

1

1.5

2

1.5

2

-10 -20

4.5. Evaluation of dynamic thrust

-30

t (s)

30 20 10

δ (°)

First the dynamic analyses have been performed applying at the base of the wall the velocity time-history deduced by the San Rocco accelerogram, previously scaled at a peak value PGA ¼0.35g (Fig. 10). The results have been reported in Fig. 13a in terms of thrust components on the ideal surface AV (Sahe and Save) and on the stem A0 V0 (horizontal Swhe component only); in Fig. 13b the average inclination dPS of the thrust acting on AV surface is plotted vs. time. Moreover the inclination d of the thrust has also been evaluated as average value on three vertical segments of AV, high H/3 each; in Fig. 14 all computed inclination values are reported, for the comparison with the inclination obtained by the application of the proposed solution (Fig. 14b), as a function of the instantaneous acceleration value. In Fig. 15 the time-history of the bending moment (Mw) at the base of the wall stem (A0 A00 ) and the elevation of the thrust acting on the AV surface (Ht) and on the stem (Hw) are reported.

0 0

0.5

1

-10 -20 -30 t (s) Fig. 14. Comparison between d evaluated by Flac (a) and d obtained by the new proposed method (b).

600

400

1400

200

1200

SahPS

Swhe

300 Save

SavPS

100 0 0

5

10

15

20

-100

8

MPS

1000

6

800

Ha

4

600 400

2

Hw

200 0

-200

0 0

t (s) 30

5

10 t (s)

15

20

Fig. 15. Dynamic FLAC model: time-histories of bending moments on wall stem and elevations of resultant thrust on AV surface (Ha) and wall stem (Hw).

20

δPS

10 δ (°)

10

H (m)

1600

Mw (kNm/m)

S (kN/m)

Sahe 500

0 0

5

10

15

20

-10 -20 -30 t (s) Fig. 13. Dynamic FLAC model: time-histories of thrust components on surfaces AV and A0 V0 (a) and d inclination on AV (b).

Finally in Fig. 16 the global safety factor (FS) of the wall against sliding is plotted vs. time; for comparison, the result of the pseudostatic analysis is also reported. The comparison among the thrusts obtained with the different approaches is now examined (Fig. 13a). The thrust SahPS on the ideal surface AV calculated by the pseudostatic FLAC (kh ¼0.1; kv ¼0) was equal to 410 kN/m; this value is quite lower than the one obtained by the dynamic FLAC on the same surface, but is very close to the thrust acting on the wall stem. The difference between the two thrust values can be justified considering the different height of the AV surface (10.8 m) and wall stem A0 V0 (10.0 m). The thrust varies with the exciting acceleration: the

A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

In Figs. 17–19 the time-histories of thrusts, their inclinations and bending moments at the base of the wall stem are reported for each of the four input motions. In Table 4 the maximum values

4.5 4 3.5 3 FS

1127

Table 4 Comparison between Flac results for different earthquakes.

2.5 2 1.5 1

FSPS

0.5

Site

Sahe

Sturno Tolmezzo San Rocco Norcia

700 540 510 440

(kN/m)

max

dE

max

Mw

(deg.)

25.6 26.3 21.6 19.6

max

(kN m)

1900 1600 1500 1200

0 0

5

10

15

20

t (s)

Sahe

700

Fig. 16. Dynamic FLAC model: time-history of sliding global safety factor.

1. the inclination d of the thrust on AV surface continuously changes during the earthquake (Fig. 13b); 2. the values of d calculated on the three distinct elements of AV are slightly different and depend on the soil accelerations, and therefore on the input motion (Fig. 14a). 3. there is a good agreement between d time-histories obtained by FLAC and the proposed solution (Fig. 14).

S (kN/m)

San Rocco

Tolmezzo

500 300

SahPS

Norcia Save

SavPS

100 5

-100 0

10

15

20

-300 t (s) Fig. 17. Time-histories of thrust components on AV surface, for the different seismic input motions.

30 Tolmezzo

Sturno

20

δPS

10 δ (°)

maximum value on AV, equal to 510 kN/m, is recorded around 2 s. At the same instant the maximum moment equal to 1500 kN m/m occurs. The value of the thrust vertical component Save is quite close to the one (SavPS) computed by pseudostatic FLAC (Fig. 13a). Regarding the bending moments (Fig. 15), they have been computed considering the thrust acting on the wall stem A0 V0 and ignoring the inertia forces acting on the structure. The maximum bending moment in static conditions is equal to 840 kN m/m; hence the dynamic moments are significantly higher than the static one. On the other hand the values of the normal force, induced by dynamic actions at the base of the stem, significantly vary from 0 to 130 and to 260 kN/m passing, respectively, from d ¼01, 151 and 21.61. Nevertheless these differences are negligible if compared to the weight of the stem. The time-history of the sliding global safety factor is plotted in Fig. 16. It starts from the static value 2.2, and never goes below a value of 1.2. The final value, at the end of the motion, is about 1.5; the same value comes out from the pseudostatic analysis. Therefore no relative displacements occur between the wall and any point under the base, according to FLAC dynamic analysis. The results suggest three important considerations:

Sturno

0 0

5

10

15 Norcia

-10 -20

20

San Rocco

-30 t (s) Fig. 18. Time-histories of inclination d for the different seismic input motions.

2000

In order to evaluate the effect of frequency content of seismic signals on dynamic response of retaining walls, the four different accelerograms described in Section 4.2 have been chosen as input motions; three of them (Tolmezzo, Sturno and Norcia) were recorded during main Italian strong motion events; the fourth (San Rocco) was recorded during an aftershock of the main earthquake. For the analyses, as previously mentioned, all the accelerograms have been scaled to the same peak value, PGA¼ 0.35g.

Tolmezz

1600 Mw (kNm/m)

5. Effects of frequency content on dynamic thrust

Sturno

1800

The numerical results seem to validate the analytical solution here proposed.

1400

San Rocco

1200 1000

MPS

Norcia

800 600 400 200 0 0

5

10 t (s)

15

20

Fig. 19. Time-histories of the bending moment at the stem base, for the different seismic input motions.

1128

A. Evangelista et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1119–1128

of the thrusts, inclination d and bending moment evaluated by FLAC dynamic analyses are reported. The results illustrated in Figs. 17–19 and Table 4 clearly indicate that the active thrust and the relative bending moment depend not only on the PGA value but also on the frequency content of the seismic input. As expected, the greater thrust and moment values have been produced by ‘‘Sturno’’ time-history; in fact it is characterised by the higher energy content (Arias Intensity of the original signal is Ia ¼139.35 cm/s) and by a main period TP (0.20 s) very close to the main period of the backfill (0.18 s).

6. Conclusions A new stress plasticity solution has been proposed, which allows one to determine the value and inclination d of seismic active thrust on earth retaining cantilever walls. This solution is validated by means of the comparison with both the results of traditional limit equilibrium approach (MO) and numerical dynamic analysis (FLAC). It is demonstrated that d angle is not a soil properties but depends also on the seismic acceleration coefficients in the pseudostatic methods and on the instantaneous value of timehistory acceleration in dynamic approaches. The inclination in seismic conditions, dE, is greater than the one in static conditions, dS, usually adopted for both static and seismic analyses. As is known, wall stability conditions improve with the increase of d; hence the present method gives solutions that are less onerous than traditional ones. Regarding the structural response, this solution produces higher normal stresses in the wall stem, which could have beneficial effects on structural design.

References [1] Eurocode8-Part 5. Foundations, retaining structures and geotechnical aspects, CEN European Committee for Standardization, Bruxelles, Belgium, December 2003 [Final Draft]. [2] Norme Tecniche per le Costruzioni. DM 14 gennaio 2008, G.U. n. 29, 4 febbraio, n. 30, 2008. [3] Coulomb CA. Essai sur une application des regles de maximis et minimis a quelques problemes de statique relatifs a l’architecture. Me´moires de Mathe´matique et de Physique, pre´sente´s a l’Acade´mie Royale des Sciences par divers Savans et lus dans ses Assemble´es, vol. 7, Paris, 1773. p. 343–82. [4] Huntington W. Earth pressures and retaining walls. New York: John Wiley; 1957. [5] Greco VR. Active earth thrust on cantilever walls with short hell. Canadian Geotechnical Journal 2001;38:401–9. [6] Okabe S. General theory on earth pressure and seismic stability of retaining wall and dam. Journal of Japanese Civil Engineering Society 1924;10(5):1277–323. [7] Mononobe N., Matsuo H. On the determination of earth pressure during earthquakes. In: Proceedings of the world engineering congress, Tokyo, vol. IX, 1929. p. 177–85. [8] Chen WF. Limit analysis and soil plasticity, 638. Amsterdam: Elsevier; 1975. [9] Collins. IL. A note on the interpretation of Coulomb analysis of the thrust on a rough retaining wall in terms of the limit theorems of plasticity theory. Ge´otechnique, Technical Note 1973;23(3):442–7. [10] Rankine WJM. On the stability of loose earth. Philosophical Transactions of the Royal Society of London 1857;147:928. [11] Newmark NM. Effect of earthquakes on dams and embankments. Ge´otechnique 1965;15(2):139–59. [12] Itasca FLAC (Fast Lagrangian Analysis of Continua). Minneapolis: Itasca Consulting Group, Inc., 2000. [13] Green RA, Olgun CG, Cameron WI. Response and modelling of cantilever retaining walls. Computed-Aided Civil and Infrastructure Engineering 2008;23:309–22. [14] Kuhlemeyer RL, Lysmer J. Finite elements method accuracy for wave propagation problems. Journal of Soil Mechanics and Foundations Division 1973;99(SM5):421–7. [15] Finn WDL. Dynamic analyses in geotechnical engineering. In: Von Thun JL, editor. Earthquake engineering and soil dynamics. II – Recent advances in ground-motion evaluation. Geotechnical Special Publication 20. ASCE; 1988. p. 523–91. [16] Rathje EM, Abrahamson NA, Bray JD. Simplified frequency content estimates of earthquake ground motions. Journal of Geotechnical and Geoenvironmental Engineering, ASCE 1998;124(2):150–9.