Dynamic active earth pressure on cantilever retaining walls

Computers and Geotechnics 38 (2011) 1041–1051

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Dynamic active earth pressure on cantilever retaining walls

Anna Scotto di Santolo ⇑, Aldo EvangelistaDipartimento di Ingegneria Idraulica, Geotecnica e Ambientale, Università di Napoli Federico II, Via Claudio 21, 80125 Naples, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 March 2011Received in revised form 12 July 2011Accepted 29 July 2011Available online 27 August 2011

Keywords:Retaining wallsActive thrustPseudo-static conditionsDynamic analysisPermanent displacements

0266-352X/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compgeo.2011.07.015

⇑ Corresponding author. Tel.: +39 081 7685915; faxE-mail addresses: [email protected] (A. Scotto d

(A. Evangelista).

In normal practice, the active earth pressure on cantilever retaining wall is evaluated with differentprocedures relating to an ideal vertical plane passing through the heel of the wall. If the wall presentsa long heel, failure planes do not interfere with the vertical stem, so that the limit Rankine conditionscan develop freely in the backfill. The inclination of lateral actions along the ideal plane is assumed tobe constant and depends on the geometry of the ground level and on the friction angle u. The Authorsrecently proposed a new method to evaluate the active earth pressure coefficient due to seismic loadingwith a pseudo-static stress plasticity solution. The present paper describes the application of this methodto a retaining wall supporting a u soil backfill with an irregular surface. For two different configurationsof wall-soil system, the behaviour is also studied by continuum FDM dynamic analyses, utilising fourItalian accelerometric time-histories scaled at the same peak ground acceleration. The comparisonbetween different procedures is also analysed.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The cantilever wall was developed after the Second World War,following the introduction of reinforced-concrete constructiontechniques. As known, these retaining structures are usually builtto support backfill heights when gravity walls are uneconomical.

In these walls, the ground over the inner foundation slab plays avery important dual role:

� Its weight provides a significant stabilising effect on the retain-ing structure.� It is also responsible for the loads supported by the vertical

stem and induced on other structural wall elements.

The contrasting dual functions of the soil resting on the innerfoundation slab makes the interpretation of the structurebehaviour difficult, especially in the presence of seismic loads.

Generally, construction codes refer mainly to gravity wall,providing guidance and rules intended specifically for their design[1–3]. However, in the technical literature several recent contribu-tions can also be found on the behaviour of cantilever wall [4–6]. Innormal practice, the active earth pressure on cantilever retainingwalls is evaluated at the vertical plane through the heel of the wall(virtual back), Fig. 1. In cantilever walls with a long heel, failureplanes can develop freely in the backfill, because there is no inter-ference with the structure element of the wall, Fig. 1. Normally, the

ll rights reserved.

: +39 081 7683481.i Santolo), [email protected]

inclination of lateral actions along the virtual back is assumed con-stant and dependent on the geometry of the ground level and onthe friction angle, u.

In a previous work, the Authors considered methods to evaluateactive earth pressure on these structures and proposed a new one,New Stress Pseudostatic Plasticity Solution (NSPPS), which is valid forstatic and pseudo-static conditions [7]. The Authors analysed awall retaining cohesionless soil with a regular plane and a longheel (Fig. 2a). In the present paper, the Authors refer to the moregeneral case of a wall retaining an irregular backfill. The latter iscomposed by a horizontal surface up to the virtual vertical planeAV passing through the heel and by an inclined surface downhillup to the crest of the wall (with e < u), Fig. 2b. The proposed meth-od evaluates, like the remaining ones, the active thrust on the idealvertical surface AV and is based on the traditional assumptions ofrigid-plastic behaviour of soil and therefore disregards the magni-tude of the displacements necessary to the mobilisation of all shearstrength of soil. As known, this simple constitutive behaviour, isused in the calculation of permanent displacements of walls underdynamic actions adopting Newmark approaches [9].

The Authors felt the need to remove rigid assumptions and toevaluate the behaviour of the walls into a more general model,even not in a failure condition.

It appeared sufficient to adopt a FLAC 2D [8] numerical method,employing a perfectly-elastic–plastic constitutive law, removingthe approximations inherent in the traditional methods.

Finally some considerations between different results related todifferent procedures are reported.

The studies conducted with the different approaches (static,pseudo-static and seismic) have quantified the following variables:

Nomenclature

u internal friction anglec unit weight of soilw dilatancy anglee slope of the backfilla, b slope of the failure surface with respect to the vertical

directiong gravitational accelerationq (equal to c/g) mass densityd inclination of the thrust with respect to the horizontalW weight of the wall-soil systemkh horizontal seismic coefficientkv vertical seismic coefficienth pseudo-static anglere normal stress acting on plane parallel to the slope at

depth zse shear stress acting on plane parallel to the slope at

depth zra active stress acting on the vertical planesa active shear stress acting on the vertical planeka active earth-pressure coefficientkah horizontal component of active earth-pressure coeffi-

cientkav vertical component of active earth-pressure coefficientE elastic modulus of soilm Poisson coefficient of soilcc unit weight of concreteEc elastic modulus of concretemc Poisson coefficient of concreteamax maximum acceleration expected at the site ground sur-

facebM reduction coefficient for retaining walls according to

Italian code [3]ag horizontal acceleration expected on site A according to

Italian code [3]PGA peak ground acceleration of the recorded accelerogramIa Arias Intensity of the original signal

Tp predominant period corresponding to the maximumspectral acceleration (computed for 5% viscous damp-ing)

Tm the mean period [12] on the basis of the Fourier spec-trum of the signal

ac critical acceleration for slidingkc critical acceleration coefficient for slidingd permanent horizontal displacement of the foundation

on the subsoil induced by seismic loadsSwh horizontal component of the thrust on wall stem (A0V0)Swv vertical component (shear force) on wall stem (A0V0)Mw bending moment at the base of the stemzw height of thrust on wall stem A0V0

Sah horizontal component of the thrust on the ideal surfaceAV

Sav vertical component of the thrust on the ideal surface AV(shear force)

za height of thrust on the ideal surface AVPd pressure downhill of the surface AV for irregular surfacePu pressure uphill of the surface AV for irregular surfaceFS global sliding safety coefficient along the cantilever

foundation interfaceSahPS horizontal thrust component on AV evaluated by FLAC

pseudostatic analysesSavPS vertical thrust component on AV evaluated by FLAC

pseudostatic analysesMPS maximum bending moment at the base of the stem dur-

ing pseudo static FLAC analysesSah max maximum horizontal thrust component on AV evalu-

ated by dynamic FLAC analysisdmax maximum inclination of the thrust with respect to the

horizontal evaluated by dynamic FLAC approachSwh max maximum horizontal component of the thrust on wall

stem (A0V0)Mw max maximum bending moment at the base of the stem dur-

ing dynamic FLAC analyses

B

hSav

Sah

Sa

Swh

SwSwv

α β

δ

(α+β) = (90-ϕ)

z za

V

A’’

A

H

ε

D

V’

A’

Fig. 1. Active pressure against a cantilever wall.

1042 A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051

� Sah = horizontal component of the thrust on the ideal surfaceAV;� Sav = vertical component of the thrust on the ideal surface AV

(shear force);� d = tan�1(Sav/Sah) = thrust inclination as compared with the hor-

izontal acting on AV;� kah = horizontal component of active earth-pressure coefficient;

� kav = vertical component of active earth-pressure coefficient;� FS = global sliding safety coefficient along the cantilever foun-

dation interface;� Ma = bending moment at the base of the stem (A0A00) evaluated� za = height of thrust on the ideal surface AV;� Swh = horizontal thrust component on the wall stem (A0V0);� Swv = vertical component (shear force) on wall stem (A0V0);� Mw = bending moment at the base of the stem (A0A00);� zw = height of thrust on the wall stem A0V0;� d = permanent horizontal displacement of the foundation on the

subsoil induced by seismic loads;

The suffixes PS are added only in the numerical pseudo-staticcases.

2. New pseudostatic method for evaluating active earthpressure

2.1. Description for regular plane backfill

The method to evaluate the active earth pressure on cantileverwall retaining proposed by Evangelista and co-workers [7] is basedon the stress plasticity theory. It refers to the active stress statepresent in an undefined slope consisting of a cohesionless soil withunit weight, c, subjected to the action of gravitational (g) and seis-

(a)

(b) V’

A’

8.2m

V

A’’

A

7.5m

0.8m 10

.8 m

γ = 15 kN/m3

c’= 0 ϕ’= 30°

SaeδSw

kh gρ

γ (1±kv)θ

0.8m

V

A’’

A

10.8

m

V’

A’

8.2m

10m

Sa

za

0.8m

Sw Sae

δ

γ = 15 kN/m3

c’= 0 ϕ’= 30°

kh g ρ

γ (1±kv)θ

0.8m

zwA’

A’

Fig. 2. Thrusts on cantilever walls: (a) horizontal backfill: Wall A; (b) inclinedbackfill: Wall B.

A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051 1043

mic forces of magnitudes (kh � g) and (kv � g) in the horizontal andvertical directions, respectively; see Fig. 2. The pseudo static angleresulting from the gravitational and the inertial body forces ish = tan�1(kh/1 ± kv). In the present study, for simplicity’s sake, thevertical seismic coefficient kv was assumed to be equal to zero.Positive values of kh indicate inertial actions directed towards thestem.

Assuming that the ground surface is a regular plane sloping e(0 < e < u � h), a possible distribution of stresses at the plane strainconditions and in equilibrium with the external loads at the gener-ic depth z is:

re ¼ c � z � ðcos2 e� kh sin e cos eÞ ð1Þ

se ¼ c � z � ðsin e cos eþ kh cos2 eÞ ð2Þ

where re and se are the normal and shear stresses acting on a planeparallel to the surface (Fig. 3).

The limit active stress state is obtained by considering that atfailure the Mohr circle must undergo the point S(re, se), in Fig. 3.The distance between the centre of the circle from the origin is:

ε

γε

ε

kh γ

z

z*

ε

z* = z cosε

Fig. 3. Stress tensor for undefinite slope with inclination (e + h) 6 u

OC ¼2 � re �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4r2

e � 4 � ð1� sin2 uÞ � s2e þ r2

e� �q

2 � ð1� sin2 uÞð3Þ

and the radius of the Mohr circle is:

R ¼ OC � sin u ð4Þ

Moreover, the following relationships can be used to evaluatethe angles indicated in Fig. 3:

f ¼ tan�1 re � OCse

!ð5Þ

n ¼ p2� e� f ð6Þ

x ¼ p2� nþ e ð7Þ

The limit active state of stress can be determined only if the dis-criminant in Eq. (3) is not negative (necessary condition). The stateof stress at the failure condition, in equilibrium with the externalloads, on vertical direction is then:

ra ¼ OC � R � sinx ð8Þ

sa ¼ R � cos x ð9Þ

The limit active state of stress (ra, sa) was estimated for differ-ent values of e and u with the seismic coefficient kh ranging from0.05 to 0.35. The earth active pressure coefficients kah = ra/cz andkav = sa/cz is then derived. The results obtained are plotted in Figs. 4and 5 for friction angles of 30� and 40�, respectively.

In the simple case of horizontal backfill (e = 0) (called hereafterWall A) and in static conditions (h = 0) the solutions (8) and (9) areequal to the solutions of Rankine with d = 0. For horizontal backfillsubjected to seismic loads (h – 0), d is greater than zero. In thissimple case, the evaluation of earth pressure coefficients of theadopted procedure obtains the same values as the Mononobe–Oka-be approach [10,11] imposing the same d.

In Fig. 6 the trend of inclination d of active lateral loads on a ver-tical plane is reported as function of horizontal seismic accelera-tion kh and backfill inclination e for u equal to 30� and 40�. Itcan be observed that in the presence of pseudo-static seismic ac-tions, d increases significantly approaching the limit condition ofd equal to u.

2.2. Description for irregular plane backfill

When the backfill surface is irregular, the inclination of theforce along the plane AV must be evaluated. In the analysed case(Fig. 2b) (called hereafter Wall B), the Authors cannot use the solu-

α β

S

in the presence of pseudostatic loading with kh – 0 and kv = 0.

0

0,2

0,4

0,6

0,8

1

kh

k ah

ϕ=30°

ε =0°5°10°15°

20°

0,0

0,2

0,4

0,6

0 0,1 0,2 0,3 0,4

0 0,1 0,2 0,3 0,4

kh

k av

ϕ=30°

ε =0°5°10°

15°

20°

Fig. 4. Horizontal and vertical components of earth pressure coefficient forcohesionless backfill with u = 30� for different values of e and kh evaluated byNSPPS.

0

0,2

0,4

0,6

0,8

1

kh

kh

k ah

ϕ=40°

ε =0°5°10°15°

20°25°

0

0,2

0,4

0,6

0 0,1 0,2 0,3 0,4

0 0,1 0,2 0,3 0,4

k av

ϕ=40°

ε =0°5°

10°15°

20°

25°

Fig. 5. Horizontal and vertical components of earth pressure coefficient forcohesionless backfill with u = 40� for different values of e and kh evaluated byNSPPS.

Fig. 6. Variation of inclination of active thrust d on vertical plane as function ofhorizontal seismic acceleration kh and backfill inclination e.

1044 A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051

tions for the infinite slope, which would lead to imposing the forceparallel to the slope (d = e + h with d < u). As a first approximation,we could use the mean value of the slope between the up anddownhill directions. The solution to determine the active pressurefor an irregular ground surface, when Rankine’s conditions prevail,is reported by Huntington [14] and is shown in Fig. 7. Huntingtonsuggested assuming two opposite embankments arranged uphilland downhill loading the same surface AV. The forces due to thedownhill and uphill fills, Pd and Pu, were calculated by a trialmethod looking for those wedges for which the Pd equals Pu inthe opposite direction on the same line of action (Fig. 8).

This principle has been extended in the present work to situa-tions where there are seismic actions.

3. Pseudostatic analysis with NSPPS methods for walls withregular and irregular backfill

Two cantilever walls were considered, as shown in Fig. 2. Thefirst, Wall A, consists of a 10.80-m-high from foundation plain, sup-porting a horizontal backfill. The second, Wall B, is 8.3 m high fromthe foundation level and retains an irregular backfill with an angleof 25� starting by the crest of the wall until the virtual AV surfaceand flat uphill. The height of the virtual back AV is equal for thetwo walls. In both cases, the backfill consists of incoherent pyro-clastic soil with a friction angle u of 30� and a unit weight c equalto 15 kN/m3. The subsoil consists of a rocky formation with a baseinterface characterised by a roughness angle equal to 30�. Thelength of the internal base of the wall allows the development ofRankine failure surfaces in the backfill (not intersecting the verticalstem of the wall).

Seismic actions are referred to different Italian areas. In thepseudo-static condition the equivalent coefficient kh was obtained

Fig. 7. Active pressure by trial method for Rankine’s conditions (from Huntington[14]).

-δδ

-25°

Fig. 8. Results of the proposed method for irregular backfill with NSPPS comparedwith Mononobe and Okabe for u = 30� and kh = 0.1 and kv = 0.

A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051 1045

according to the NTC (National Technical Building Code) [3] by thefollowing approach:

kh ¼ bM �amax

gð10Þ

where amax is the maximum acceleration expected at the site, whichis assumed equal to 0.35g, bM is a reduction coefficient for theretaining wall, equal to 0.31 for soil classes A–D, in a range of value

of acceleration on bedrock 0.2–0.4g. A value of kh equal to 0.1 wasobtained, while kv = ±0.5kh was assumed equal to zero.

A comparison between the results for static and pseudo-staticconditions is presented in Table 1.

Applying the proposed method, NSPPS, to Wall A under a horizon-tal pseudo-static coefficient kh of 0.1 and kv equal to 0, the Authorsobtained the following active coefficient ka = 0.305, with a thrustinclination d = 19.2�. The horizontal thrust component is equal to309 kN/m on AV and is equal to 255 kN/m on A00V (Fig. 2). The bend-ing moment on the stem is 1115 kN m/m. The moments were calcu-lated neglecting the inertial forces, due to the weight of the wall, andthe tangential force acting on the inner face of the stem.

For Wall B, with the irregular backfill surface, as described inSection 2.2, the thrusts Pd and Pu are plotted in Fig. 8, as a contin-uous line.

In the same figure are plotted the results obtained by theMononobe–Okabe method shown as symbols for comparison, witha kh = 0.1 and a kv = 0. In these conditions the active earth pressurecoefficient ka is equal to 0.367 with an inclination d equal to f20.60�, while in static conditions, ka is equal to 0.310 with an incli-nation of 9.5�. The horizontal thrust component is 300 kN/m on AVand is 145 kN/m on A00V. The bending moment is 362 kN m/m. Rel-ative values are collected in Table 1. It can observed that Wall B isloaded more than A owing to seismic actions.

4. Simplified dynamic analysis with NSPPS for evaluation ofsliding displacements

Recently, many contributions have been made in Italian andInternational research on the application of design-performanceanalysis methods to geotechnical structures and in particular toretaining walls. these methods have generally been based on anextension of the model of a rigid block sliding on an inclined planefirst proposed by Newmark in 1965 [9] to evaluate the effects ofearthquakes on dams and embankments.

The new Technical Italian Code [3] specifies the applicability ofthe method of displacement in reviews of retaining walls in theultimate limit and serviceability limit states. In all cases, verifica-tion of the applicability of studies of movement induced by seismicaction must be properly evaluated according to the relevant limitstate (ULS or SLE) and life-reference work. It is obvious that themethod allows for transient stability at critical conditions, at whichplastic displacements occur. For retaining walls, the aforemen-tioned movements are normally sliding along the foundation.

The assessment of permanent sliding displacement of the wall,induced by an earthquake, is generally based on the determinationof the parameter ac, the critical acceleration at which the resis-tance to motion is completely overcome, as long as the speed is po-sitive. Normally ac is evaluated with the inclination of the activethrust constant and equal to static one. It is possible to integratethe equation of relative motion, between the wall and base, forthe point at which ac is exceeded by current acceleration, causingthe generation of the relative velocity between the wall and thesubsoil. In these analyses, a rigid, perfectly plastic soil model isusually adopted, and a critical acceleration essentially independentof the history of movements of the wall, for a constant inclinationof the active thrust.

The NTC [3] parametric analysis of this type allowed the deter-mination of the calibration coefficients bM, according to Eq. (10),reducing seismic action in field applications.

As previously shown for cantilever walls the thrust during aseismic event is usually calculated, with pseudo-static approach.However during an earthquake, there is continuous variation ofthe shear force along the virtual back of the wall and, therefore,a continuous change of inclination d of the active thrust.

Table 1Results obtained for Walls A and B in static and Pseudo-static conditions.

u = 30� Static Pseudo-static (kh = 0.1 and kv = 0) NSPPS Pseudo-static (kh = 0.1 and kv = 0) FLAC

Wall A B A B A B

kah (1) 0.333 0.306 0.354 0.343 0.469 0.51kav (1) 0 0.051 0.100 0.129 0.120 0.17d (�) 0 9.5 15.78 20.62 15.0 18.8Sah (kN/m) 291 267 309 300 410 389Sav (kN/m) 0 45 87.50 112.80 110 114za (m) 3.60 3.60 3.60 3.60 3.60 3.60Swh (kN/m) 250 129 255 145 330 293Mw (kNm/m) 832.5 322.7 885 361.7 1100 821zw (m) 3.33 2.5 3.33 2.5 3.60 2.46FS (1) 2.53 2.50 1.80 1.72 1.51 1.49

1046 A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051

In the classic approach of Newmark the threshold accelerationis assumed constant during the motion. Improvements to themethod for gravity walls were made by Zarrabi [16], in whichthe value of the acceleration threshold changes over time depend-ing on the level of acceleration. This method was validated by test-ing prototypes of a gravity wall with a smooth inner facing on ashaking table [17,18].

In the current work, the Authors modified the Newmark ap-proach for cantilever walls based on the new calculation methodof the earth pressure proposed in Section 2. In Pseudo-static condi-tion, for instance in Fig. 9 this coefficient kc versus the inclination ofthrust d was reported for a horizontal backfill (e = 0), for u equal to30� and 40� and kh ranging from 0.05 to 0.35. It is observed that thekc increases as d increasing and for higher friction angles. In simpli-fied dynamic conditions the critical threshold acceleration coeffi-cient kc was iteratively evaluated by the NSPPS to determine theseismic actions that produce a sliding foundation. For every pointof the accelerogram we evaluate: active (Sah) and inertial (W � kh)forces and limit shear strength [(Sav + W) tan u] along the founda-tion plane. According to Newton’s law, the total force F applied toa body of mass M which undergoes an acceleration a is equal to:

F ¼ Ma ¼ Sah þW � kh � ðSav þWÞ tan u ð11Þ

As previously stated, in this study, for simplicity’s sake, the ver-tical seismic coefficient kv was assumed to be equal to zero. Posi-tive values of kh indicate inertial actions directed towards the stem.

4.1. Application to cantilever Walls A and B

The displacements induced on the walls were calculated usingthe classic Newmark [9] model but with the thrusts evaluated by

Fig. 9. Horizontal critical acceleration coefficient kc versus inclination of the thrustd for e = 0 and u = 30� and 40�.

the Mononobe–Okabe approach (MO) and through the NSPPS pro-cedure proposed by the Authors. The accelerations threshold ac ofthe A and B soil-wall systems, beyond which sliding motion begins,are: 0.233g and 0.281g for Wall A and 0.234g and 0.290g for Wall B,according to MO and NSPPS procedures. Four accelerometric timehistories of Italian earthquakes scaled at the same PGA equal to0.35g were utilised, Fig. 10. In particular the four input motionsare Tolmezzo and San Rocco (from the earthquake of Friuli,1976), Sturno (Irpinia, 1980) and Norcia (Umbria-Marche, 1997).The main features of the recorded accelerograms are listed inTable 2; the frequency content of the waveform is quantifiedthrough the predominant period, Tp, corresponding to the maxi-mum spectral acceleration in an acceleration-response spectrum(computed for 5% viscous damping) and through the mean period,Tm, as defined by Rathje et al. [12] on the basis of the Fourier spec-trum of the signal. Actually, Tm should provide a better indicationof the frequency content of the recordings because it averagesthe spectrum over the whole periodic range of amplification.

In simplified dynamic conditions the critical threshold acceler-ation coefficient kc was iteratively evaluated by the NSPPS to deter-mine the seismic actions that produce a sliding foundation. In thiscase the critical acceleration is not constant but varies with timeaccording to the value of kah, d and seismic input. For instance, inFig. 11a the horizontal seismic earth pressure coefficient kah eval-uated with the NSPPS method for the Tolmezzo earthquake, e = 0and u = 30�, is reported. In Fig. 11b the cumulative frequency ofthe number of observations, in which this coefficient kah is belowa predetermined value more or equal to one, was shown for allground motions. It is noted that the areas limited by the curvesand the y-axis represent, to a first approximation, the intensity ofthe earthquake. In the analysed case, it is reduced by passing fromSturno to Tolmezzo, S. Rocco and Norcia (following the classifica-tion of the Arias Intensity given in Table 2). This allows us to saythat amax is not the only significant parameter.

The cumulative displacements calculated, for example, forWalls A and B and the San Rocco accelerogram are reported inFig. 12 for the different procedures. The comparison between thecumulative displacements of Walls A and B evaluated with theNSPPS proposed method and those evaluated with the traditionalmethod of Newmark for different earthquakes are reported inTable 3. As was expected, the displacements decrease with increas-ing thrust inclination and are very small due to a high kc almostequal to amax/g for the wall soil system analysed.

5. Dynamic numerical analysis of wall-soil systems A and B

It is well known that the validation of a new method requires anexpensive physical model such as the use of the shaking table orthe dynamic centrifuge, and observations on real walls. The use

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20

a (g

)

A-STU 270

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20

a (g

)

ATMZ 000

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20a (g

)

E-NCB090

(a) (c)

(b) (d)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20

a (g

)

W-SR0000

Fig. 10. Accelerograms scaled at 0.35g: (a) Sturno; (b) San Rocco; (c) Norcia; (d) Tolmezzo.

Table 2Strong motion features.

Site (earthquake) PGA (g) Tm (s) TP (s) Ia (cm/s)

Norcia (Umbria-Marche, 1997) 0.38 0.17 0.12 34.25San Rocco (Friuli, 1976) 0.09 0.29 0.10 3.22Sturno (Irpinia, 1980) 0.32 0.86 0.20 139.35Tolmezzo (Friuli, 1976) 0.35 0.39 0.26 78.65

PGA – Peak Ground Acceleration; Tm – mean period; TP – predominant period;Ia – Arias Intensity.

(a)

0,2

0,3

0,4

0,5

0,6

0,7

0 5 10 15t (s)

k ah

(b)

0

20

40

60

80

100

21,51

k ah /k ah Static

Cum

ulat

ive

Freq

uenc

y

SturnoTolmezzoNorciaSanRoccoP

R

Fig. 11. (a) Variation of horizontal active earth pressure coefficient with NSPPS dueto Tolmezzo earthquake for Wall A; (b) Cumulative frequency diagram of the ratiobetween kah evaluated with NSPPS and kah in static condition; P = % of cases onwhich kah is below a R value for the four earthquakes.

A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051 1047

of a numerical model is however a cheap and useful tool, especiallysignificant if the tester method has a superior degree of valuation.

The two wall-soil systems (A and B) were analysed using theFLAC code, which utilises an explicit finite-difference method forsolving differential motion equation. For both models the Authorsused a square grid 0.2 m on each side, Fig. 13, in order to obtainaccurate and stable solutions and avoid a numerical distortion inthe propagation of seismic input motion. The backfill was modelledwith an elasto-plastic constitutive law (elastic modulus E =50 MPa, and Poisson coefficient m = 0.3) in conjunction with theMohr–Coulomb yield criterion (cohesion c = 0; friction angleu = 30�, and dilatancy w = 0). The wall was modelled as elasticwith cc = 24 kN/m3, Ec = 20 GPa, and mc = 0.1. The subsoil consistsof an elastic rocky rigid and scabrous formation (c = 0 u = 30�). Itwas preferred not to use interface elements between the differentmaterials but rather small layers of appropriate properties. Themodel was created by the activation of the elements constitutingthe wall and subsequently those of the backfill for successive layersof one metre thickness [13]. The results in the static field are re-ported in a previous work only for Wall A [7].

The model was subjected to a constant horizontal acceleration0.1g at all points of the system, according to the Italian BuildingCode Eq. (10), to conduct a pseudo-static analysis. Also for the dy-namic analysis, the four accelerometric time histories of Italian

earthquakes scaled at the same PGA, assumed equal to 0.35g, wereutilised (Fig. 10 and Table 2).

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 2 4 6 8 10t (s)

d (m

m)

MO δ=16°

NSPPS

MO δ=0°

Fig. 12. Comparison between permanent displacements of Wall A due to the SanRocco accelerogram evaluated by different approaches.

Table 3Permanent sliding displacements (mm) of Walls A and B obtained by differentmodels.

Input d (mm)

Wall A Wall B

MO NSPPS MO NSPPS

(a) Sturno 0.973 0.119 1.784 0.335(b) San Rocco 0.500 0.118 0.704 0.300(c) Norcia 0.0083 0.0024 0.112 0.051(d) Tolmezzo 0.468 0.0942 0.725 0.265

1048 A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051

Both analyses were carried out in a dynamic-field mode in orderto use the viscous boundary conditions of Kuhlemeyer and Lysmer[15], with a Rayleigh damping of 5%.

5.1. Results of numerical analysis

For the pseudo-static analysis, the results obtained are summa-rised in Table 1 together with that of NSPPS. For Wall A, the vertical

(a)

(b)

10.8

m

8.2m

8.2m

V

A

V’

A’

V

A

10.8

m

V’

A’

Fig. 13. Cantilever walls: (

SavPS and horizontal component SahPS of the thrust on the AV sur-face were calculated as 410 and 110 kN/m, respectively. The rela-tive values of the earth coefficients kav and kah were 0.469 and0.120. The inclination of the thrust dPS assumed a mean value equalto 15�. The maximum bending moment of the steam wall Mw PSwas 1100 kN m/m. In case B, we obtained horizontal thrust compo-nents SahPS of 389 kN/m on AV and 293 kN/m on A0V0 with dPS of18.8�. The maximum bending moment, MwPS, was 537 kN m/m.In two cases the FLAC values are very close to those obtained withthe NSPPS method. The numerical results slightly overestimatedthe theoretical values.

With respect to the previous analysis, the dynamic one not onlytakes into account the deformability of the system, but also as-sesses the seismic responses of two earth-wall systems at differentinput motion in term of frequency and energy content and investi-gates the role of the ground over the inner foundation slab on theseismic wave propagation. Dynamic analyses were performed withfour seismic time histories of different Italian earthquakes, re-ported in Table 2, scaled at the same maximum acceleration of0.35g.

Time histories of the horizontal and vertical active components(Sah and Sav) on the surface AV in cases A are reported in Fig. 14aand b. The active force obviously varies with the input motion:the maximum values of Sah were observed for the Sturno recordat about 7 s. The values are equal to 700 kN/m and 695 kN/m forWalls A and B, respectively. This amplification would seem to showthat the ground over the inner foundation slab is in phase with thewedge. In Fig. 15 the time histories of the bending moment at thebase of the wall stem, Mw, evaluated neglecting inertial actions, arereported. The maximum Mw occurred for Sturno record, equal to1850 and 1981 kN m/m for Walls A and B, respectively. Both valuesare greater than those assessed by the pseudo-static analysis, re-ported in the same figure as a dotted line.

As for a vertical load, induced by dynamic actions, they in-creased significantly until 300 kN/m) with the variation of d. How-ever, these differences are negligible when compared to the weightof the wall. In Fig. 16 are reported the time histories of the averageinclination d along the ideal surface AV for Walls A and B. The incli-nation of the thrust on AV varies continuously during the earth-

γ = 15 kN/m3

c’= 0 ϕ’ = 30°; ψ = 0° G = 19 MPa; B = 41 MPa E = 50 MPa ν = 0.3

γc = 24 kN/m3

Ec = 20,000 MPa νc = 0.1

γ = 15 kN/m3

c’= 0 ϕ’ = 30°; ψ = 0° G = 19 MPa; B = 41 MPa E = 50 MPa ν = 0.3

γc = 24 kN/m3

Ec = 20,000 MPa νc = 0.1

a) Wall A; (b) Wall B.

0

200

400

600

800

0 5 10 15 20t (s)

S ah (

kN/m

) S ahPS

Sturno

Norcia

TolmezzoSan Rocco

-300

-200

-100

0

100

200

300

0 5 10 15 20

t (s)

S av (

kN/m

)

SavPS

Sav

Sturno

Norcia

TolmezzoSan Rocco

Fig. 14. Dynamic FLAC results: time histories of thrust components on AV surfacesfor Wall A.

A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051 1049

quake. This inclination was also evaluated for Wall A by thenumerical code on individual sectors of the AV surface, equal toone-third, two-thirds and unity. The d time histories were reportedfor the first 2 s in Fig. 17a relating to the San Rocco earthquake.These values were slightly different and depended on the acceler-

(a)

0

500

1000

1500

2000

2500

t (s)

Mw (k

Nm

/m)

Mw (k

Nm

/m)

MPS

Sturno

Norcia

TolmezzoSan Rocco

(b)

0

500

1000

1500

2000

2500

0 5 10 15 20

0 5 10 15 20

t (s)

Sturno

San Rocco

Norcia

Tolmezzo

MPS

Fig. 15. Dynamic FLAC results: time histories of bending moment on wall stem; (a)Wall A; (b) Wall B.

ation of the ground motions and therefore on the seismic input. InFig. 17b the time history of d evaluated by NSPPS for the same seis-mic input was also reported. This curve falls close to those previ-ously evaluated.

Fig. 18 shows the time history of the global safety factor (SF) forthe slip of the foundation of the wall with the variation of seismicinputs in cases A and B. The initial static value was equal to 2.2 forboth the walls and then varied with time in the time-history sets.The final value, at the end of the seismic motion, varied dependingon the accelerometer history. Instantaneous values less than unityoccur only for case B. The final values are greater than or equal tothose obtained from the pseudo-static FLAC analysis (FSPS). There-fore, for Wall A for which the FLAC dynamic analysis alwaysyielded a safety factor higher than unity, there was no relativemovement between the base of the wall and the ground below.For Wall B, instead, very little permanent shifts occurred.

Overall, the Authors noted that the vertical wall stem was morestressed in the case of an irregular backfill (Wall B). A comparisonbetween the maximum instantaneous values assessed in both soil-wall systems with the dynamic FLAC method is given in Table 4.

It is noted that all variables are sometimes much higher thanthose obtained with the theoretical and numerical pseudo-staticmethods. The greatest differences between these values and thoseof the pseudo-static analysis, reported in Table 1, are determinedfor the Sturno and Tolmezzo earthquakes.

The numerical analysis FLAC provides a maximum value of ka

more than double the design one. Consequently it happens alsofor thrusts and bending moments. The same result is obtainedwhen applying the new procedure NSPPS if, instead of a single va-lue of amax, we consider an accelerometric history (simplified dy-namic approaches, see Fig. 11).

These aspects seem to suggest that the new Italian rules [3] arenot always on the safe side because the seismic parameters theyassume are too low. In this respect the Authors show that it must

(a)

-30

-20

-10

0

10

20

30

t (s)

δδ (°

)δδ

(°)

δPS

Norcia

SturnoTolmezzo

San Rocco

(b)

-30

-20

-10

0

10

20

30

0 5 10 15 20

10 15 200 5

t (s)

δPS

Norcia

Sturno

Tolmezzo

San Rocco

Fig. 16. Dynamic FLAC results: time histories of thrust inclination d on AV surfacesfor Wall B.

(a)

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2

t (s)

δ (°

)δ

(°)

δ1

δ2

δ3

δ

(b)

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2

t (s)

Fig. 17. Time histories of d evaluated by FLAC (a) and by NSPPS (b) for Wall A due toSan Rocco earthquake.

(a)

00.5

11.5

22.5

33.5

44.5

5

t (s)

FS

FSPS

Norcia

Sturno

Tolmezzo

San Rocco

(b)

00.5

11.5

22.5

33.5

44.5

5

0 5 10 15 20

0 5 10 15 20t (s)

FS

FSPS

Norcia

Sturno

Tolmezzo

San Rocco

Fig. 18. Dynamic FLAC model: time histories of sliding safety factor; (a) Wall A; (b)Wall B.

Table 4Results obtained for Walls A and B in dynamic conditions with FLAC analyses.

Earthquake Tolmezzo Sturno Norcia San Rocco

Wall A B A B A B A B

Sah max (kN/m) 540 550 700 695 440 339 510 407dmax (1) 26.3 27.8 25.6 27.5 19.6 24.0 21.6 13.4Swh max (kN/m) 468 407 550 631 375 273 456 318Mw max (kN m) 1600 1200 1850 1981 1200 818 1500 928

1050 A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051

distinguish between the phenomena with ductile behaviour andlow hazard (sliding on the floor of the foundation) and those withbrittle behaviour and high hazard (achievement of strength in awall section of reinforced concrete).

6. Conclusions

The Authors reported the results of the analyses performed toevaluate active pressure in different soil behaviour and displace-ments under seismic loads of two cantilever walls. The structuresare characterised by the same height of the backfill at the heel ofthe wall; but while the first (Wall A) supports a backfill with a pla-nar surface, the second (Wall B) presents a backfill with slope up-wards from the crest of the wall up to the virtual back of the wall.The length of the internal base allows the development of Rankinefailure surfaces inside the backfill.

The response of the two walls was investigated through the dif-ferent adopted approaches:

1. the first one concerns a theoretical pseudo-static analysis (thenew method, NSPPS, proposed by the Authors);

2. the second one is a simplified numerical pseudo-static analysisimplemented with the finite difference code, FLAC;

3. a simplified dynamic analysis to assess the sliding of the wallsalong their foundation, according to the traditional Newmarkapproach and with its modification in accordance with themethod at point 1); and

4. the last is a complex dynamic analysis conducted with FLACcode.

From the comparison of the results of different methods themost important aspects come out. In the theoretical pseudo-staticanalysis, the active thrust and its inclination d were evaluated bythe new method, NSPPS, proposed by the authors in a previouswork for regular backfill [7] and now extended to an irregularone (see Section 2).

The response of the soil-wall systems considered confirms thatthe theoretical and numerical simplified pseudo-static approachesgive proximal results regardless of the different hypotheses on soildeformability. The complex dynamic analysis, which considers thetime history of acceleration and its characteristics are moreaccurate.

The comparison between the NSPPS method and traditionalpseudo-static design approaches and numerical results in termsof permanent displacements evaluated by the Newmark modelsuggests that the former are more realistic than the current pseu-do-static analysis.

Regarding the structural responses of the retaining walls, itmust be noted that resistance of the stem-base section is the ele-ment that strongly affects the stability of the structure because itis the element most exposed to stresses induced by both staticand seismic actions.

The numerical FLAC analysis provides values of maximumbending moment more than double the design one, if the pseu-do-static coefficient is evaluated according to Eq. (10). New meth-od and numerical pseudo-static analyses give lower values thanthe traditional ones.

Nevertheless, the lack of well-documented case histories limitsthe conclusion of the proposed approach. For this reason an exper-imental activity is currently being carried out on prototypes on ashaking table.

A. Scotto di Santolo, A. Evangelista / Computers and Geotechnics 38 (2011) 1041–1051 1051

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