displacement analysis in earth slope design under seismic ... · c.n.r., istituto di tecnica delle...

Displacement analysis in earth slope

design under seismic conditions

A.L. Simonelli

Gruppo Nazionale per la Difesa dai Terremoti,

C.N.R., Istituto di Tecnica delle Fondazioni,

Napoli, Italy

ABSTRACT

During the past years experience has shown that the traditional pseudo-staticdesign methods applied to earth slopes under seismic conditions are ratherineffective. Alternative design methods can be based on the evaluation of theearth slope displacements induced by real earthquake excitations. In the firstpart of the paper a brief illustration of the different approaches is presented;then the Code pseudo-static method and a displacement analysis procedure areapplied, for the sake of simplicity, to the typical case of an indefinite slopein cohesionless soil, and to a set of accelerometric data coming from the 1980Irpinia earthquake. The comparison points out noticeable differences in theresults, confirming that the adopted pseudo-static method overestimates theearthquake effects; in conclusion a simple design procedure is outlined whichtakes into account the results of displacement analyses.

INTRODUCTION

The slope behaviour under seismic conditions has always been studied bymeans of classical limit equilibrium methods in which dynamic forces aretransformed into equivalent static forces. Most of the National Design Codessuggest the application of such methods; hence the values of seismic coeffi-cients are given, as a function of site seismic vulnerability, in order todetermine the equivalent pseudo-static action. Nevertheless, the choice of theseismic coefficient values gives rise to some uncertainties, considering thatthey significantly affect the results of the analyses. Further uncertainties areconnected to the values to be adopted for the safety factor. The Italian SeismicCode demands a safety factor SF=1.3 for static actions while it does notimpose a definite pseudo-static safety factor value (PSF) for seismic loading.

An alternative approach consists in the analysis of slope displacementsinduced by actual seismic excitation, represented by earthquake accelerogram

Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509

494 Soil Dynamics and Earthquake Engineering

recordings. The amount of displacement is therefore calculated and may becompared to a definite limit allowable displacement.

DISPLACEMENT ANALYSIS

The displacement analyses methods are based on the well known model of arigid block sliding on a plane surface, proposed by Newmark [1] in 1965, andsubsequently re-elaborated by many researchers in order to study the behaviourof dams, slopes and earth retaining walls (e.g. Simonelli and Viggiani [2]).These methods define an acceleration limit value (Kc-g) of the "surface", upto which the "block" moves together with the surface, without any relativedisplacement; when the acceleration becomes greater than the critical valueKc-g, the inertial force induced on the block (Kc-W, where W is the blockweight) causes the sliding between the block and the surface; the latter willmove more rapidly than the first one, with subsequent accumulation of relativedisplacements; this phenomenon will then stop every time the relative velocitybetween the block and the surface becomes zero again. Critical accelerationcoefficient Kc can be determined resolving the pseudo-static limit equilibriumequation, imposing PSF=1. The calculated displacements are thereforefunctions of both the limit acceleration Kc-g and the characteristics of theaccelerometric waveforms.

Research on displacement analysis has been focused on several aspects:the proper determination of the limit acceleration, for different geometricaland/or mechanical soil characteristics (e.g. Sarma [3], Cividini et ai. [4],Crespellani et al. [5]); the influence of single parameters of accelerometricwaveforms on displacements, in order to simply characterise the effectivenessof seismic motion data (e.g. Sarma [3,6], Ambraseys and Menu [7]); the defi-nition of seismic input motion for given regions and for given earthquakecharacteristic ranges (e.g. Ambraseys and Menu [7], Battistella and Faccioli[8]).

Critical acceleration coefficientWith reference to the scheme in Figure 1, the critical acceleration value Kc-gcan easily be determined; in this case a significative expression of coefficientKcis:

Kc =(SF-1) - - (1)1-tancpianp

where SF is the static safety factor. It is obvious that Kc can be defined onlyfor SF> 1; its value depends on soil strength parameters 0 and c (which isinside SF equation) and on slope geometrical characteristic B.Equation (1) can also be applied to different geometrical conditions and tosaturated soil, if both the safety factor SF and the equivalent inclination B of


Soil Dynamics and Earthquake Engineering 495

N=Wcos(B)-KWsin(B)

PSF = Kc W cos(B) + W sin(B) = C + [W cos(B) - Kc W sin(B)] tan(<£)

Figure 1 - Rigid block sliding on a plane surface: determination ofcritical acceleration coefficient Kc

the actual slope are properly defined (e.g. Franklin and Chang [9]). The ex-pression changes, if the arising of pore pressure during the seismic event hasto be taken into account [2].

Cohesionless soil In the particular case of cohesionless soil, which has beenwidely studied, the static safety factor is SF = tan(<£)/tan(B) and Equation (1)simplifies as follows

Kc = tan(cp-p) 0)

in this case Kc is a function of the difference between <t> and 6, not of theirabsolute values; this fact allows very simple and effective representations ofdisplacement analysis results.

Displacement equationSlope displacement values are obtained by the integration of the followingdifferential equation

/ rt \(3)

coscp

where ii(t) is the relative acceleration and a(t) is the base acceleration (i.e. theacceleration time-history). For the case examined above (i.e. unsaturatedcohesionless soil, Kc = tg(0-B)) relative accelerations, hence velocities anddisplacements, essentially depend on the difference (</>-#), being the influence



of the term (l/cos<£) negligible for the usual range of friction angle <£ values(as the applications will confirm in the next paragraph).

APPLICATION TO THE IRPINIA REGION

Displacement analysis method for cohesionless unsaturated soil will be appliedto a real set of accelerometric data, relative to the Irpinia region in SouthernItaly. This region was struck by a strong earthquake, Magnitude M=6.9, onNovember 23, 1980. After this event the regional seismic classification wasupdated, according to the Italian Code seismic categories listed in Table 1.

Table 1 - Italian seismic classification categories

SEISMICZONE

IIIIII

S

1296

Ksism

0.100.070.04

S: seismic levelKsism: seismic coefficient

Seismic motion dataDuring the 1980 earthquake, accelerometric records were obtained at differentsites, by means of the ENEL accelerographic recording network (Berardi et al.[10]). Table 2 reports, for 15 sites, the maximum acceleration values A^^ ofthe two horizontal components, the epicentral distance and the seismicclassification. Almost all of the sites belong to the II seismic category zone,whose seismic coefficient is Ksism=0.07.The highest values of A^^ have been recorded in five sites (no. 3, 6, 7, 9 and12) quite close to the epicenter (d<50Km); in Figure 2, the horizontalacceleration time history components are plotted for each of these sites. Themaximum acceleration values range between 0.11-g and 0.30-g; as regardsSturno, where the recorded acceleration values are significantly higher thanthose at any other location, some doubts on the proper operation of theaccelerometer have been raised.The long duration of the ground motion is a relevant characteristic of therecords, which is probably due to a sequence of at least two successive shocks,as clearly appears from the accelerogram waveforms. High values of energycontents measured by means of Arias intensity parameter (I*) have been foundfor Calitri and Sturno accelerometric records (I > lOOcm/s); this fact is partlydue to the rather low dominant frequencies, generally ranging between 2 and3 Hz (see Simonelli and Viggiani [2]).

Evaluation of suffered displacementsThe displacements have been computed for each of the accelerometric timehistories, and for a wide set of friction angle </> and slope inclination 15 values.



Table 2 - Recording sites: seismic classification and main groundmotion characteristics

Id. no.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

SITE

Arienzo

Auietta

Bagnoli Irpino

Bisaccia

Bovino

Brienza

Calitri

Garigliano

Mercato S.Severino

Rionero in Vulture

S.Severo

Sturno

Torre del Greco

Trlcarico

Vieste

Seismiczone

II

II

II

I

I

II

II

II

III

I

II

II

II

II

II

Amax(g)NS

0.027

0.056

0.137

0.094

0.045

0.215

0.157

0.039

0.108

0.098

0.025

0.219

0.061

0.047

0.036

Amax(g)EW

0.037

0.059

0.172

0.080

0.047

0.157

0.171

0.033

0.139

0.095

0.021

0.296

0.040

0.035

0.031

d(Km)

79

32

29

34

57

48

27

137

50

41

104

38

80

77

144Id. no.: site identification numberd : distance from the focus

All the results obtained confirm the significative dependence of displacementsfrom (0-ft) value; on the contrary, for constant (<M), the influence of thesingle parameters is negligible, according to Equations (2) and (3). This factis evidentiated in Figure 3, where displacements caused by Bagnoli Irpinoaccelerogram (EW component) are plotted in two different ways against <t> andft values.

The effect of accelerogram maximum value A^^ on computeddisplacements is shown in Figure 4.a). No simple correlation between A^^and displacement can be individuated considering the extreme variability of theobtained data; it is clear that other accelerogram waveform characteristicsstrongly influence the amount of displacement, particularly the low frequencyaccelerogram components and the long duration, prevailing on the role of A^^parameter. On the contrary, if we observe the displacement values at each site(computed for each of the two horizontal accelerogram components), theinfluence of A^^ is noticeable, since the other characteristics of the twowaveforms are quite similar (Figure 4.b)).



BAGNQLI IRPINQ

EWcomp. - Amax=0.172g j l_ NScomp. - Amax=0.137g

(9/10);

0•1•2

Ik,,,, „„...y-""- ' i

ii

20 40 60 80Time (s)

20 40 60 80Time (s)

BRIENZA !

EW comp. - Amax=0.157 g NS comp. - Amax=0.215 g

20 40 60 80Time (s)

20 40 60 80Time (s)

a z(9/10) 1 .

MERCATO S.SEVERINO

EW comp. • Amax=0.139 g NS comp. - Amax=0.108 g

20 40 60 80Time (s)

20 40 60 80Time (s)

20 40 60 80Time (s)

Figure 2 - Irpinia 1980 earthquake: main accelerometric records


499

80

Soil Dynamics and Earthquake Engineering

Figure 3 - Effect of friction angle <£ and slope inclination 8 oncalculated displacement D (Bagnoli I.- EW comp. accelerogram): Dversus 15, for constant <£ values (on the left); D versus d>, for constant(<M) values (on the right)

The comparison between displacement analysis results and SeismicCode design indications has been performed for the five sites where thestrongest earthquake motions were recorded (Figure 5).The displacements caused by the main accelerometric component are plottedagainst (</>-B) in the upper part of the diagrams. In the lower part of the samediagrams the values of PSF, calculated for the proper site seismic category, arereported. It is obvious that displacements are greater than zero only when (0-fl)values are so low, that the correspondent Kc are lower than the site A . Fora further decrease of (0-B) values, displacements rapidly increase with increa-sing gradients. The diagrams show that, with the only exception of Sturno, thedisplacements are rather small (just a few centimeters) in correspondence tovery low (</>-B) values, whose PSF would be unacceptable; for example, in thediagrams the PSF corresponding to (<£-B) of slopes which suffered a 10 cmdisplacement are indicated. On the contrary, according to the conventionallyadopted PSF value (equal to 1.3), only much greater (<£-B) values (incrementedof about 5°) would be considered safe. In conclusion the Code pseudo-staticapproach seems to overestimate the real effect of the 1980 earthquake.

DESIGN CRITERIA

Definition of local seismic input motionOne of the main problems for the application of the displacement analysismethod as a design tool is the definition of the site seismic input motion. It hasbeen clearly demonstrated that the site maximum acceleration alone is notsufficient to characterise the expected seismic motion; it would be necessaryto associate a proper value range of the expected waveform characteristics to



100D

(cm)

a)

80

60 -

40 -

20 -

0.05

D(cm)

100

80

b)

0=35°25"

0.10 0.15 0.20 0.25 0.30Amax (g)

STURNO XCAUTRI

60

40

BAGNOU I. /20 f-

MERCATO S.S. BRIENZA

0.05 0.10 0.15 0.20 0.25 0.30Amax (g)

Figure 4 - Effect of maximum acceleration A ax on displacements D :a) D versus accelerogram A , for (<£-B)=2 and <j> = 25°,35°; b) D versusaccelerogram A , for each site

The real waveform effect on displacement can be directly accounted forby defining the correlation functions of actual displacements with critical acce-leration ratio Kc/A^ax (Ambraseys and Menu [7]). These Authors analysed aset of near-field strong motion data, relative to 11 earthquakes in the surface-wave Magnitude range Ms = (6.9±0.3), and defined the correlation equationsfor several cases.This approach has been applied to Irpinia region, by utilising the 1980earthquake accelerograms. In Figure 6 the computed displacements are plottedversus Kc/A x ratio, together with the upper bound curve, whose equation is


Soil Dynamics and Earthquake Engineering

70D 60

(cm) so40302010

0.8

1PSF

12

1.4

501

D(cm)

PSF

70605040302010

0.8

1

12

1.4

\\f-35" IBAGNOU IRPINO EW comp.j

>*\

**'«•.,

:.J

p T L

BRIENZA NS Comp.

D(cm)

PSF

/o605040302010

0.8

1

12

1.4

\\0-3s- ICAUTRI EW comp.|

((P~I3) !

r s L

70

D *>(cm) 50

40302010

0.8

PSF'

MERCATO S.SEVERINO EW Comp.|

«25\

70D 60

(cm) 50 -4030 -20 -10

0.8

PSF

\0«35« STURNO EW Comp Figure 5 - Comparison between slopedisplacements D caused by 1980earthquake and pseudo-static safetyfactors PSF according to SeismicCode, versus slope (<t>-K)



10D

(cm)10

10

10

10

10 0.2 0.4

Figure 6 - Permanent displacements vs. critical acceleration ratio andupper bound Mine (Irpinia 1980). Estimated regressions from Ambraseysand Menu (1988): mean (A -line) and confidence limits for 97.5%line)

JCr2.82- 3.67 (-££-) (4)

where Dmax is the maximum permanent displacement. In the same figure theregression curves elaborated by Ambraseys and Menu are reported, relative tomean displacement (A line) and to confidence limit of 97.5% (A line).It is obvious that the correlation function defined for the Irpinia region has tobe validated by the acquisition of ulterior experimental data. However, thegood agreement with the A^ curve encourages its utilization.

Displacement and safety factorsFor any site, the expected maximum acceleration, together with the regionalcorrelation function between maximum displacement and critical accelerationratio (Equation (4)), can be utilised either to predict the behaviour of a givenslope, or as a design tool. In the first case, for an assigned slope the ratio Kcover site A^^ can be calculated; then the maximum displacement that theslope can suffer has to be determined from Equation (4). Inversely, for anyimposed displacement, the correspondent critical acceleration ratio has to becalculated from Equation (4); then, given site A^x* Kc can be determined,hence slope (<t>-K) design value.



0.35 r

Figure 7 - Design Chart: slope (0-B) vs. site A^ for given displacementD; Code pseudo-static factors PSF for seismic categories (I,II,III) and staticfactor FS vs. (<£-fl) and <t> values

The design procedure can be simply applied by means of the DesignChart in Figure 7. In the upper part of the figure slope (<£-B) value can bedetermined in function of site A^x and displacement values. Several iso-displacement lines are reported, in particular the 3, 5 and 10 cm ones, whosevalues can be considered as tolerable displacements, the choice depending onthe particular design problem.The results of the design procedure based on allowable displacements can thenbe compared to the Code design method, by utilising the lower part of Figu-re 7, where the pseudo-static factors (for the three seismic categories) and thestatic factor are plotted, versus (<£-B) value and friction angle 0.

In the figure the design procedure is illustrated for Bagnoli Irpino andCalitri sites, assuming, for instance, A^x = 0.172-g, which is the maximumvalue of the recorded accelerograms. Assigned an allowable displacement aslow as 3 cm, (0-6) value little greater than 6° would result. On the contrary,



in the range <£ = 25*4-35°, the correspondent II category PSF values areabout 1.1, which can not be considered acceptable; the III category PSF valuesare about 1.2, while the static SF value range is about 1.27-r 1.37. Therefore,the application of the present Seismic Code method appears to be too conser-vative; a slight improvement in the design could be obtained lowering theseismic category; moreover, it can be observed that even slopes designed forstatic actions would behave quite satisfactorily, suffering displacements rangingfrom about 3 to 10 cm. It can be easily verified that the result of thecomparison does not substantially change even if reasonable higher A xvalues are adopted.

ConclusionsThe comparison between the displacement analysis and the Seismic Codedesign procedures points out that the latter is too overconservative.The illustrated Design Chart, based on the evaluation of the maximumexpected displacement, allows a more rational earth slope design.Moreover, according to the indications of the Chart, a traditional pseudo-staticapproach could still be utilised, but either adopting lower PSF values orreducing the seismic classification.

ACKNOWLEDGEMENTSThe author is grateful to his friend Elios Fortunato who actively helped himin the elaboration of data and figures.

REFERENCES

1. Newmark,N.M. "Effect of earthquakes on dams and embankments"Geotechnique, 15, pp. 139-159, 1965

2. Simonelli,A.L. and Viggiani,C. "Some remarks on retaining wall designunder seismic conditions" Proceedings of Tenth World Conference onEarthquake Engineering, Madrid, Spain, 1992

3. Sarma,S.K. "Seismic stability of earth dams and embankments" Geotech-nique, 25, pp.743-761, 1975

4. Cividini,A., Pergalani,F. and Petrini,V. "La risposta dei versanti adazioni sismiche attraverso un modello semplificato" Ingegneria sismica,3, pp.28-44, 1991

5. Crespellani,T., Ghinelli,A., Madiai,C. and Vannucchi,G. "Analisi distabilita dei pendii natural! in condizioni sismiche" Rivista Italiana diGeotecnica, 2, pp.49-74, 1990

6. Sarma,S.K. and Yang,K.S. "An evaluation of strong motion records anda new parameter Ag$" Earthquake Engineering and Structural Dynamics,15, pp. 119-132, 1987

7. Ambraseys,N.N. and MenuJ.M. "Earthquake-induced ground displace-ments" Earthquake Engineering and Structural Dynamics, 16, pp.985-1006, 1988

8. Battistella,C. and Faccioli,E. "Studi di pericolosit& sismica ed instability



del pendii in zone appenniniche" Atti delle Conferenze di Geotecnica diTorino, XV Ciclo, Torino, Italy, 1991

9. Franklin,A.G. and Chang,F.K. "Earthquake resistance of earth androckfill dams; permanent displacements of embankments by Newmarksliding block analysis" Miscellaneous Paper 5.77.77, 17.5. ArmyEngineers Waterways Experiment Station, Vicksburg, 1977

10. Berardi, R., Berenzi,A. and Capozza,F. "Campania-Lucania earthquakeon 23 November 1980. Accelerometric recordings of the main quake andrelating processing" Atti Convegno Annuale Progetto FinalizzatoGeodinamica, Udine, Italy, 1981


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