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i) Professor, Department of Civil Engineering, National Cheng Kung University, Taiwan (samhcc＠mail.ncku.edu.tw).The manuscript for this paper was received for review on April 10, 2008; approved on January 29, 2009.Written discussions on this paper should be submitted before November 1, 2009 to the Japanese Geotechnical Society, 4-38-2, Sengoku,Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

249

SOILS AND FOUNDATIONS Vol. 49, No. 2, 249–258, Apr. 2009Japanese Geotechnical Society

EFFECTS OF HORIZONTAL AND VERTICAL SEISMIC LOADS ONTHE BEARING CAPACITY OF A SURFACE FOOTING ADJACENT TO A SLOPE

CHING-CHUAN HUANGi)

ABSTRACT

The correction factor (hie) for the ultimate bearing capacity of a footing placed adjacent to a slope, taking into accountthe combined eŠect of horizontal and vertical seismic loads, represented by horizontal and vertical seismic coe‹cients,kh and kv, respectively, was derived using a modiˆed Janbu's slice method. The in‰uence of slope angles `a' on thevalues of `hie' are studied here. It was found that the values of hie can be expressed as an exponential function of `kh/(1－kv)' and `a', with a measurable interdependency between `a' and `hie'. The in‰uence of `a' on the value of `hie' in-creases as the input value of `kh/(1－kv)' increases. Equations derived based on the analytical results are proposed toaccount for this eŠect. Based on the analyses of 11 near-fault seismographers obtained in the 1999 Chi-Chi earthquakein Taiwan, a ratio between the vertical and the horizontal seismic coe‹cients, l, of between ±0.25 is suggested for in-cluding the combined eŠect of vertical and horizontal seismic forces in evaluating the seismic bearing capacity of foot-ings located in near-fault areas.

Key words: bearing capacity, earthquake resistant, inclined load, shallow foundation, slope (IGC: E3/E6/E8/H1/H9)

INTRODUCTION

The bearing capacity of footings adjacent to a slopehas been studied in countries with mountainous terrains(Kusakabe et al., 1981; Sawada et al., 1994; Huang et al.,1994; Huang and Tatsuoka, 1994). Conventional analyti-cal solutions for the seismic bearing capacity of a footinghave been derived mainly for rigid strip footings placedon the surface of a level ground (Sarma and Iossifelis,1990; Budhu and Al-Karni, 1993; Richards et al., 1993;Dormieux and Pecker, 1995; Richards et al., 1996; Zhu,2000; Kumar and Mohan Rao, 2002, 2003; Kumar andKumar, 2003; Choudhury and Subba Rao, 2005). Studieson the seismic bearing capacity for a rigid surface footingadjacent to a slope subjected to seismic loadings are verylimited (Sawada et al., 1994; Choudury and Subba Rao,2006; Huang and Kang, 2008). The importance of theseismic bearing capacity of a footing situated on a slopehas been noted in the post-earthquake investigation ofsome severely damaged soil retaining walls on hillsides(Huang, 2000; Kato, 2001; Fang et al., 2003; Huang andChen, 2004; Huang, 2005). The ultimate bearing capacityof a rigid, rough strip surface footing (qu(aÀ0, bÀ0, khÀ0, kvÀ0)) adja-cent to a cohesionless slope subjected to vertical andhorizontal seismic loadings, represented by vertical andhorizontal seismic coe‹cients, kv and kh, respectively,can be expressed using a modiˆed Terzaghi's bearingcapacity equation (Terzaghi, 1943) as follows:

qu(aÀ0, bÀ0, khÀ0, kvÀ0)＝12･B･g･Ng(a＝0, b＝0, kh＝0, kv＝0)･hs･hie (1)

where,B: width of the footingg: unit weight of the foundation soil

Ng(a＝0, b＝0, kh＝0, kv＝0):bearing capacity coe‹cient due to the self-weightof soils for a surface footing placed on a levelground (a＝0, a: slope angle) subjected to a stati-cally vertical load (b＝0, b: inclination angle of thefooting load).

hs: correction factor for a slanted surface with a slopeangle, a.

hie: correction factor for the combined seismic eŠectincluding an inclined loading and inertial forces offailure soil mass.

The correction factor for the ultimate bearing capacityof footing placed adjacent to a slope, hs, has been studiedby Hansen (1970), Vesic (1973), and Huang and Chen(2004). Empirical equations for calculating the value of hs

are well-established (Hansen, 1970; Vesic, 1973). Ingeneral, independent correction factors can be applied toTerzaghi's bearing capacity equation, as the hs or hie usedin Eq. (1), to account for various eŠects. A correctionfactor to account for the bearing capacity decreases whenthe footing is located near to a slope can be expressed asfunctions of the distance between the toe of the footingand the crest of the slope (or the `setback' from theslope), as those proposed by Kusakabe et al. (1981),

250250 HUANG

Sawada et al. (1994), and Huang and Kang (2008a). Dis-cussions regarding the in‰uence of a setback to the bear-ing capacity of footing situated on a slope is beyond thescope of the present study.

The following empirical equation has been proposedby Budhu and Al-Karni (1993) to account for the com-bined eŠect of horizontal and vertical seismic loadings,based on a limit equilibrium method for a rough rigidfooting placed on a cohesionless level ground:

hie＝Ø1－ 23

kv» exp Ø－5.3 k1.2h

1－kv» (2)

Zhu (2000) has proposed the following empirical equa-tion for the correction factor, hie based on the upperbound solution for a rigid rough footing placed on thesurface of a cohesionless level ground:

hie＝Ø1－ kh

1－kv»

5.5

(3)

It should be noted that both Eqs. (2) and (3) were der-ived for the case of a footing placed on a level ground, sothe in‰uence of `a' on the value of hie is unknown. There-fore, the use of hie values derived from Eqs. (2) and (3) toevaluate the value of qu(aÀ0, bÀ0, khÀ0, kvÀ0) using Eq. (1) isbased on an assumption that values of hie are independentof the slope angle `a'. To eliminate the possible short-comings associated with this assumption, Huang andKang (2008b) derived a general equation of hie for thebearing capacity of a footing placed adjacent to a slope(aÀ0) subjected to a horizontal seismic force on the foot-ing load (represented by the angle of load inclination, b＝tan－1 (kh)) and an inertial force of the soil mass (khÀ0)based on a pseudo-static-based approach modiˆed fromthe Janbu's slice method (Janbu, 1973) originally formu-lated by Huang and Tatsuoka (1994), expressed as:

hie＝exp (－t･kh) (4)

where, `t' is a curve ˆtting parameter which can be ex-pressed as:

t＝t1･a＋5.82 (5)t1＝0.044(1/degree) (6)

Huang and Kang (2008) compared the analytically de-rived values of hie based on Eqs. (4)–(6) with those report-ed in the literature for some typical conditions, as shownin the columns of kv＝0 in Table 1, and found that thediŠerence in the values of hie derived by Huang and Kang(2008b) and those reported by Zhu (2000) are negligiblysmall. It should be noted that all theoretical solutionsshown in Table 1, except the one for Huang and Kang(2008), were for the case of a level ground (a＝0). Asshown in Eqs. (4)–(6), the value of hie is a function of `a'.However, the inter-dependency between `a' and `hie' hasnever been investigated. The present study is a follow-upof that performed by Huang and Kang (2008b), aiming toextend Eqs. (4)–(6) to account for both horizontal andvertical seismic loadings on the value of hie, and at thesame time, to explore the interdependence between `a'and `hie'

Although analytical and/or empirical equations, asshown in Eqs. (1) through (3), have been derived for eval-uating the combined eŠect of vertical and horizontal seis-mic loads, the design outcome can greatly vary due to thechoice of the input value of `kv/kh' or `l' in the designand/or analysis. Seed and Whitman (1970) suggested thatkv＝0 (or kv/kh＝l＝0) is a reasonable assumption for thepractical design of conventional gravity soil retainingwalls using pseudo-static methods. Wolfe et al. (1978)also suggested to use kv＝0 in the practical seismic designof geosynthetic-reinforced soil retaining walls. Tatsuokaet al. (1998) performed pseudo-static analyses on a 5.8 m-high geosynthetic-reinforced soil (GRS) retaining wall(namely, the Tanata wall) located in the severely shakenarea during the 1995 Hyogoken Nambu earthquake (ML

＝7.2, ML: the magnitude of the earthquake on theRichter scale) using representative horizontal and verticalground acceleration records obtained from the JapanMeterological Agency (JMA)-Kobe station with a peakhorizontal ground acceleration, PHGA＝0.821g (g:gravitational acceleration). They found that the use ofthe most critical value of av/ah (＝kv/kh) obtained fromthe real-time horizontal and vertical ground accelerationrecords results in only a 1z decrease of the safety factor(or critical seismic coe‹cient) of the wall, compared tothat using kv＝0. They concluded that the eŠect of thevertical seismic load component on the GRS wall back-ˆlled with cohesionless soils is very small. Ling andLeshckinsky (1998) used a high value of kv/kh＝0.9,which is the ratio between the maximum vertical groundacceleration (PVGA) and PHGA found at diŠerent timesof the ground acceleration records, to explain the behav-ior of the Tanata wall. However, their results werechallenged by Huang and Wang (2005) based on a `multi-wedge' analysis taking into account the stabilizing eŠectsof facing and passive resistance at the toe of the wall. Hu-ang and Wang (2005) performed a Newmark-type (New-mark, 1965) displacement analysis using real-time valuesof horizontal and vertical ground accelerations identicalto those used by Tatsuoka et al. (1998), namely, theJMA-Kobe records, and found that a small value of kv/kh

(or av/ah)＝0.058 is adequate for explaining the behaviorof the Tanata wall. This supports the conclusion made byTatsuoka et al. (1998) mentioned above. Huang et al.(2003) used kv/kh＝0.2, based on the horizontal and verti-cal ground acceleration records of station (TCU 052) ob-tained in the 1999 Chi-Chi earthquake (ML＝7.3), to ana-lyze the seismic displacement of some geosynthetic-rein-forced walls located in a near-fault area in central Tai-wan. Good agreements between the observed and calcu-lated wall displacements were obtained. Huang and Chen(2004) and Huang (2005) analyzed two near-fault leaning-type soil retaining walls situated on the slope which wereseverely damaged by the 1999 Chi-Chi earthquake andfound that the assumption of kv＝0 in these analysesgenerated negligible errors. With the exception of thestudy by Ling and Leshchinsky (1998), all the above-men-tioned studies suggested kh/kvÃ0.2 for properly describ-ing the behavior of soil retaining walls and/or footings

251

Fig. 1. Failure mechanism used in the present study

Fig. 2. Comparisons of analytical values of Ng for a rigid footing ad-jacent to the slope obtained in various studies (compiled fromGraham et al., 1988)

Fig. 3(a). Comparisons between analytical values of hie and those cal-culated using empirical equations for q＝309(after Huang andKang, 2008b)

251EFFECTS OF HORIZONTAL AND VERTICAL SEISMIC LOADS

subjected to seismic loads.

METHODOLOGY AND VERIFICATION

The slice method proposed by Janbu (1973), in whichforce and moment equilibria are satisˆed for every slicewithin the sliding mass, is modiˆed to take into accountthe pseudo-static seismic force represented by thehorizontal and vertical inertial forces induced by earth-quake ground excitations (represented by `kh･Wi' and`kv･Wi'; kh and kv: horizontal and vertical seismiccoe‹cients, respectively; Wi: self-weight of slice i), asshown in Fig. A-1 of APPENDIX A. A failure mechan-ism, as shown in Fig. 1 is used, which consists of a trian-gular failure wedge deˆned by a trial-and-error angle, uB,and a constant angle of 909-q for the intersection of thetwo slip lines at the bottom of the triangular wedge; atransitional zone described by a logrithmic spiral with aradius r (＝ro exp (m･tan q) where, ro: starting radius ofthe logrithmic spiral, m: angle of rotation as deˆned inFig. 1); and a passive zone consisting of slip lines with459-q/2 to the horizontal. In the case of a slope, the pas-sive zone may be partially developed (the absent part isshown by the dotted lines in Fig. 1) or not available. Theonly trial-and-error parameter used in the trial-and-errorsearch of the critical failure surface is, therefore, thevalue of uB. In the present study, a small increment of uB,namely DuB＝0.19, is used. This method is a modiˆedversion of that developed by Huang and Tatsuoka (1994)in analyzing a series of loading tests on reinforced andunreinforced sand slopes (Huang et al., 1994) and is iden-tical to that used in the analytical studies performed byHuang and Kang (2008b). Details of the derivation of theultimate bearing capacity of a strip footing based onmodiˆed Janbu's slice method is described in APPEN-DIX A.

Figure 2 shows a comparison of the analytical valuesof Ng(aÀ0, b＝0, kh＝0, kv＝0) for a footing adjacent to a slope un-der static conditions (kh＝0) obtained in various studies.It can be seen that the values of Ng obtained using themethod described above fall within the shaded zone com-piled by Graham et al. (1988), and are also comparable tothose provided by Huang and Tatsuoka (1994) and Zhu(2000). Figures 3(a)–(c) show good agreement betweenthe analytical values of hie and those of the formulae(Eqs. (4)–(6)), both derived in a previous study (Huangand Kang, 2008b) for q＝309, 359, and 409, respectively.

Figures 4(a) and (b) show comparisons of hie obtainedin various studies for q＝309and 409, respectively. Notethat kv＝0 and a＝09are used in this comparison becauseof the limited data available. Figures 4(a) and (b) allshow that the values of hie based on Eqs. (4)–(6) are com-parable with those obtained in the literature, and at thesame time, provide values of hie near to the average orlower limit of the solutions obtained in the literature.

ANALYTICAL RESULTS ON THE EFFECT OFVERTICAL SEISMIC LOADING

Figures 5(a) and (b) show the analytical results of hie

252

Fig. 3(b). Comparisons between analytical values of hie and those cal-culated using empirical equations for q＝359(after Huang andKang, 2008b)

Fig. 3(c). Comparisons between analytical values of hie and those cal-culated using empirical equations for q＝409(after Huang andKang, 2008b)

Fig. 4(a). Comparisons of analytical values of hie obtained in variousstudies for q＝309and a＝09conditions

Fig. 4(b). Comparisons of analytical values of hie obtained in variousstudies for q＝409and a＝09conditions

252 HUANG

for a＝09and 259, respectively, as a function of `kh/(1－kv)' under the condition of q＝309. A reason for choos-ing `kh/(1－kv)' as a key parameter in extending the for-mula expressed in Eq. (7) is that the value of `kh/(1－kv)'represents a pseudo-static gravitational state for the soilmass subjected to horizontal and vertical seismic load-ings, as used in the Mononobe-Okabe seismic earth pres-sure theory (Mononobe, 1924; Okabe, 1924). The valuesof hie can be described using a unique function, as shownbelow regardless of the magnitudes of kv/kh:

hie＝exp Ø－t･kh

1－kv» (7)

or

hie ＝exp Ø－t･kh

1－l･kh»

where, `t' is a curve ˆtting parameter, and t＝6.69 and8.19 for a＝09and 259, respectively.

Figures 6(a) and (b) show similar data to those shownin Figs. 5(a) and (b), respectively, except that Figs. 6(a)and (b) are for the case of q＝409. For q＝409, the valueof hie is also a unique function of kh/(1－kv), as shown inFigs. 5(a) and (b), except that in the cases of Figs. 6(a)and (b), t＝6.16 and 8.30, for a＝09and 359, respec-tively.

Figure 7 shows that the curve ˆtting parameter `t' usedin Eq. (7) is a unique function of the slope angle `a' and isindependent of q within the range of 309ÃqÃ409, as ex-

253

Fig. 5(a). Comparisons between analytical values of hie and those cal-culated using empirical equations for q＝309and a＝09

Fig. 5(b). Comparisons between analytical values of hie and those cal-culated using empirical equations for q＝ 309and a＝259

Fig. 6(a). Comparisons between analytical values of hie and those cal-culated using empirical equations for q＝409and a＝09

Fig. 6(b). Comparisons between analytical values of hie and those cal-culated using empirical equations for q＝409and a＝359

253EFFECTS OF HORIZONTAL AND VERTICAL SEISMIC LOADS

pressed by:

t＝t1･a＋6.22 (8)t1＝0.055(1/degree) (9)

Figures 8 shows a comparison between the analyticallyderived values of hie and the analytically derived values ofhie using Eqs. (7)–(9). A 1:1 relationship is valid for atotal of 147 cases with a 95z conˆdence interval of±0.033 and a high correlation coe‹cient R2＝0.997. Thevalidity of the proposed analytical method and the calcu-lated values of hie are examined again using Table 1 forthe case of q＝309and a＝09. For various cases of kh

(＝0.1 and 0.2) and l (＝0, 0.25, 0.5 and 1.0), good agree-ments between values of hie obtained here and thosereported by various researchers can be seen, except for

the cases of kh＝0.2 and kv＝0.5 and 1.0 kh. Analyticalresults shown in Table 1 also suggest that in the case of kh

＝0.2, the in‰uence of kv on the value of hie is signiˆcant,e.g., the diŠerence in the value of hie between kv/kh＝0and kv/kh＝1.0 is as large as 75z. However, this is nottrue in the case of kh＝0.1, for which the diŠerence in hie

between kv/kh＝0 and kv/kh＝1.0 is only about 19z.

MEASURED VERTICAL GROUND ACCELATIONSIN THE 1999 CHI-CHI EARTHQUAKE

Figures 9(a)–(c) show an example of a seismographerrecorded at a near-fault station (CHY 028) during the1999 Chi-Chi earthquake (ML＝7.3, ML: magnitude onthe Richter scale) in Taiwan (Huang and Chen, 2004; Hu-

254

Fig. 7. Relationships between coe‹cient `t' and slope angle `a'

Fig. 8. Comparisons between the analytical values of hie and the calcu-lated ones using empirical Eqs. (7)–(9)

Table 1. Comparisons of analytical values of hie for q＝309and a＝09obtained in various studies (values in the parentheses are calculated usingthe empirical Eqs. (4)–(6))

hie (kh＝0.1) hie (kh＝0.2)

kv＝0 kv＝0.25 kh kv＝0.5 kh kv＝1.0 kh kv＝0 kv＝0.25 kh kv＝0.5 kh kv＝1.0 kh

Huang and Kang(1) (2008b) 0.56 (0.56) 0.51 (0.53) 0.49 (0.52) 0.47 (0.50) 0.30 (0.31) 0.27 (0.27) 0.23 (0.25) 0.17 (0.21)

Budhu and Al-Karni (1993)(2) — — 0.44 0.41 — — 0.16 0.12

Zhu (2000)(3) 0.56 0.55 0.54 0.52 0.29 0.27 0.25 0.21

Choudhury and Subba Rao (2005) — — 0.42 0.39 — — 0.14 0.10

(1): calculated using Eqs. (4)–(6)(2): calculated using Eq. (2)(3): calculated using Eq. (3)

Fig. 9. Observed values of (a) horizontal ground accelerations, ah (b)vertical ground acceleration, av and (c) l at station CHY028 in the1999 Chi-Chi earthquake (after Huang, 2005)

254 HUANG

ang, 2005, 2006). In Figs. 9(a)–(c), only a portion of thehorizontal ground accelerations, ahÆ0.2g, is focused on

because of its potential for substantial damage to earthstructures; it is indicated by the solid symbols on thehorizontal ground acceleration record shown in Fig. 9(a).

255

Fig. 10(a). Schematic ˆgure showing the positions of Chelungpu faultand 11 near-fault seismographs

Fig. 10(b). Measured values of l for 11 near-fault seismographs in the1999 Chi-Chi earthquake

Fig. 11. In‰uences of slope angle (a) to the the correction factor `hie'for the case of q＝409and kv＝0.5 kh

255EFFECTS OF HORIZONTAL AND VERTICAL SEISMIC LOADS

The corresponding vertical ground accelerations (av) arealso shown by the solid symbols in Fig. 9(b). The ratiosbetween these av and ah are expressed as l (＝av/ah) andare presented in Fig. 9(c). It can be seen that the values ofl are widely spread between lmax (＝0.906) and lmin

(＝－0.607), while the extreme values of l are never inphase with the peak values of ah, as shown in Figs. 9(a)and (c). At the time of peak horizontal ground accelera-tions, such as 0.4664g, 0.7649g, 0.674g, and 0.6191g att＝35.56 sec., 36.185 sec., 37.64 sec. and 38.73 sec., re-spectively, in the parentheses in Fig. 9(a), the corre-sponding values of l, namely lpeak, ranged between 0.169and －0.022, as indicated in Fig. 9(c). Similar investiga-tions are performed for 11 near-fault seismographers (seeFig. 10(a)) recorded in the 1999 Chi-Chi earthquake. Theresults are summarized in Fig. 10(b). In this ˆgure, thevalues of l deˆned in three diŠerent ways are plotted as afunction of the distance from the Che-Lung-Pu fault,which triggered the 1999 Chi-Chi earthquake. The three l

values of lmax, lmin and lpeak have been discussed previous-ly in Figs. 9(a)–(c). It can be seen that the values of lmax

and lmin are widely scattered, similar to those shown inFig. 9(c). However, the values of lpeak fall between －0.25and ＋0.25, suggesting that at the time of peak horizontalground acceleration for the pulses with ahÀ0.2g, peakhorizontal and vertical ground accelerations are never inphase. A value of l＝0.25 (＋for upward) can be used asthe worst combination of vertical and horizontal groundaccelerations on the seismic stability of earth structures.This contradicts l＝0.9 as used by Ling and Leshchinsky(1998) which was obtained using l＝PVGA/PHGA.

Figure 11 shows two typical examples for the eŠect ofslope angles `a' and kh (＝ 0.1 and 0.2) on the values of hie

for kv＝0.5 kh conditions. Linear relationships betweenhie and a can be seen. It should be noted that there are nodata reported in the literature that can be used for com-parison because the in‰uence of slope angle `a' to theseismic bearing capacity correction factor, hie, has neverbeen explored before. Therefore, the present study pro-vides useful analytical tool and equations in facilitatingseismic ultimate bearing capacity evaluations for a foot-ing adjacent to a slope. It should be noted that among thecases examined, the in‰uence of `a' on the value of hie ismeasureable, e.g., an increase in `a' from 09to 309results in decreases of hie for about 15z and 34z, respec-tively, in the cases of kh＝0.1 and kh＝0.2. The in‰uenceof slope angles to the correction factor, hie, should betaken into account in evaluating the ultimate bearing offooting adjacent to a slope using Eqs. (7)–(9) proposedhere.

CONCLUSIONS

Ultimate bearing capacity of a rigid strip footingplaced adjacent to a slope, taking into account the com-bined eŠect of horizontal and vertical seismic eŠects on

256256 HUANG

the soil mass and the footing load, was analyzed using amodiˆed version of Janbu's slice method. The in‰uenceof slope angles (a) on the value of the correction factorfor seismic bearing capacity of footings (hie) was investi-gated. It was found that the correction factor hie whichtakes into account both horizontal and vertical seismicforces in the seismic bearing capacity of a rigid surfacefooting adjacent to a slope, can be expressed as an ex-ponential function of `kh/(1－kv)' and `a'. A good agree-ment between the analytical values of hie obtained in earli-er studies and those of the present study were found forthe case of seismic bearing capacity of a footing placed ona level ground. It was also found that the in‰uence ofslope angles (a) on the value of hie is measurable. The in-‰uence is more signiˆcant in the case of high input valueof kh, such as kh＝0.2. Therefore, it is recommended touse proposed Eqs. (7)–(9) to account for this eŠect.Another major factor in‰uencing values of hie, namely,the ratio between vertical and horizontal seismiccoe‹cients, l, was examined based on a comprehensiveliterature review on some post-earthquake studies and onan analysis of 11 near-fault seismographers obtainedfrom a major earthquake (the 1999 Chi-Chi earthquake,ML＝7.3) in Taiwan. Values of `l' between ±0.25 can beused for near-fault soil structures when evaluating thecombined eŠect of vertical and horizontal seismic load-ings on the ultimate bearing capacity of strip footingsplaced adjacent to a slope.

NOTATION

ah: horizontal ground acceleration (g)ahmax: maximum horizontal ground acceleration

(g)av: vertical ground acceleration (g)B: width of the footing (m)Bs: width of slice (m)Ci: Cohesive shear resistance for slice No. i

DEi: diŠerential value of Ei (N/m)Ei (i＝0-n): horizontal inter-slice force (N/m)

g: gravitational acceleration (＝9.8 m/s2)hei: arm of rotation for seismic force (m)hi: arm of rotation for horizontal inter-slice

force (m)hqi: arm of rotation for horizontal seismic

force at the top of slice (m)kh: horizontal seismic coe‹cients (dimension-

less)kv: vertical seismic coe‹cient (dimensionless)

ML: magnitude of earthquake on the Rictherscale (dimensionless)

nf: number of slices directly under the footing(dimensionless)

Ng(a＝0, b＝0, kh＝0, kv＝0): bearing capacity coe‹cient due to the self-weight of soils for a surface footing placedon a level ground subjected to a staticallyvertical load (dimensionless)

Ni: normal force on the slice base (N/m)Pf: total footing load (N/m)

Pfi (i＝1-m): footing load on the top of slice i; m: num-ber of slices directly subjected to the foot-ing load (N/m)

Pi: vertical load at the top of slice i (N/m)Qfi (i＝1-m): horizontal force excerted by the footing

(N/m)Qi: horizontal load at the top of slice i (N/m)qu: ultimate bearing capacity of footing (N/

m2)ro: starting radius of the logrithmic spiral (m)

Sfi (i＝1-m): shear force at the base of slice i directlysubjected to the footing load (N/m)

Si: shear force at the base of slice i (N/m)t: curve ˆtting parameter (dimensionless)

DTi: diŠerential value of Ti (N/m)Wi: self-weight of slice i (N/m)a: slope angle (degree)ai: base inclination for slice i (degree)b: footing load inclination (degree)q: internal friction angle of soil (degree)g: unit weight of the foundation soil (N/m3)

hie: correction factor for the combined eŠectof inclined loading and horizontal inertialforce of failure soil mass (dimensionless)

hs: correction factor for slanted grounds(dimensionless)

l: ratio of vertical and horizontal seismiccoe‹cients or ground accelerations(dimensionless)

lmax, lmin: maximum and minimum values of l, re-spectively (dimensionless)

lpeak: value of l at peak horizontal ground ac-celeration for a certain pulse

m: angle of rotation for the transitionalfailure zone conˆned by the log spiral(degree)

uB: angle between the footing base and a slipline passing through the heel of footing(degree)

ui: thrust line inclination for slice i (degree)

ABBREVIATION LIST

JMA Japan Meterological Angency

REFERENCES

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4) Dormieux, L. and Pecker, A. (1995): Seismic bearing capacity offoundation on cohesionless soil, J. Geotech. Engrg., ASCE, 121(3),300–303.

5) Fang, Y. S., Yang, Y. C. and Chen, T. J. (2003): Retaining wallsdamaged in the Chi-Chi earthquake, Canadian Geotechnical Jour-nal, 40(6), 1142–1153.

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Fig. A–1. Schematic ˆgure of forces and boundary conditions used inthe slice method

257EFFECTS OF HORIZONTAL AND VERTICAL SEISMIC LOADS

6) Graham, J., Andrews, M. and Shields, D. H. (1988): Stress charac-teristics for shallow footings in cohesionless slopes, Can. Geotech.J., 25(2), 238–249.

7) Hansen, J. B. (1970): A revised and extended formula for bearingcapacity, Danish Geotech. Inst., Copenhagen, Bul., 28.

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APPENDIX A : FORMULATING SEISMICBEARING CAPACITY OF FOOTINGS ADJACENTTO A SLOPE SUBJECTED TO HORIZONTAL ANDVERTICAL LOADS

The soil mass conˆned by the slip lines shown in Fig. 1is divided into n vertical slices (width of slice, Bs＝0.1 m)with base inclinations ai (i＝1, ... n) in the present studyas shown in Fig. A–1. These slices are grouped into twocategories, namely, slices subjected to the footing load, Pi

and Qi, at their surface and the slices without footing loadat their surface. The following assumptions are made.

1. Assume inter-slice thrust heights to be 1/3 of the in-ter-slice heights (Hi) and calculate ui (ui＝tan－1 (hi－1

－hi/Bs); hi＝(Hi/3)－(Bs･tan ai/2); hi＝arm of ro-tation for horizontal inter-slice force Ei) for allslices (see Fig. A–1 for the deˆnition of ui)

2. The ultimate footing loads Pfi and Qfi are uniformlydistributed on the surface of the slices located di-rectly under the footing, i.e., Pf1＝Pf2＝...＝Pfm＝Pf, and Qf1＝Qf2＝...＝Qfm＝Qf (m: number ofslices directly subjected to the footing load is `nf'; nf

＝50 in the present study)

258258 HUANG

3. Qf＝Pf･tan b, b: angle of load inclination on thefooting base.

According to : (1) the force equilibrium in horizontaland vertical directions for slice Nos. 1-n, (2) Mohr-Coulomb's failure criterion, Si＝Ni･tan q (assuming co-hesionless soils and no pore water pressure), and (3) Sn

1 Ei

＝En－Eo, ultimate footing load on the surface of theslice, Pf and the diŠerential between inter-slice forces,DEi (＝Ei－Ei－1) can be obtained as:

Pf＝

En－Eo＋nf

S1

Dfi＋n

Snf＋1

Di＋nf

S1

Gfi＋n

Snf＋1

Gi

nf

S1

Hfi

(A1)

DEi＝－Ai＋Bi (A2)

in Eq. (A2),

Ai＝Sfi･sec ai (A3)

Bi＝[Pi＋(1－kv)･Wi－DTi]･tan ai＋Qi＋Wi･kh (A4)

Sfi＝Ci＋[(1－kv)･Wi＋Pi－DTi]･tan q･sec ai

1＋tan q･tan ai

Fs

(A5)

in Eq. (A1),

Dfi＝Ci･sec ai＋(Wi－DTi)･tan q･sec2 ai

1＋tan q･tan ai(A6)

Di＝Ci･sec ai＋[(1－kv)･Wi＋Pi－DTi]･tan q･sec2 ai

＋tan q･tan ai(A7)

Gfi＝[(1－kv)･Wi－DTi]･tan ai＋Wi･kh (A8)

Gi＝[Pi＋(1－kv)･Wi－DTi]･tan ai＋Qi＋Wi･kh (A9)

Hfi＝tan b＋tan ai－tan q

1＋tan q･tan ai(A10)

Taking the moment equilibrium about the center of theslice base yields the following equation:

Ti＝Ei･tan ui－DEi･tan ui＋DTi

2

＋－DEi･hi＋Qi･hqi＋kh･Wi･hei

Bs(A11)

Assuming that the width of slice is small, DTiª0 and DEi

ª0, we can rewrite Eq. (A13) as:

Ti＝Ei･tan ui＋Qi･hqi＋kh･Wi･hei

Bs(A12)

The ultimate bearing capacity (qu) and the bearing capaci-ty coe‹cient (Ng) can be obtained as:

qu＝Pf･nf

B(A13)

Ng＝2･qu

g･B(A14)

In the present study, Eq. (A12) instead of Eq. (A11) isused.

The computer algorithm for calculating ultimate foot-ing load Pfi is as follows:(1) Give an initial trial-and-error value of uB(≒209) to

determine the shape of triangular wedge below thefooting.

(2) Construct a logrithmic spiral as shown in Fig. 1.(3) Divide the area conˆned by the failure surface into

vertical slices with a uniform width of Bs＝0.1 m asschematically shown in Fig. A–1.

(4) Assume DTi＝0.(5) Calculate the ˆrst approximated value of Pf using

Eq. (A1).(6) Calculate DEi and Ei (i＝1, 2, ..., n) using Eq. (A2).(7) Calculate Ti (i＝1, 2, ..., n) using Eq. (A12).(8) Calculate an improved value of Pf using Eq. (A1).(9) Repeat steps (6)–(9) until the convergence of Pf is

achieved (the convergence criterion for Pf was 0.3zin the present study).

(10) Give a new trial-and-error value of uB (＝old uB＋DuB; DuB＝0.19).

(11) Repeat steps (2)–(10).(12) Find a minimum value of Pf among all trial-and-er-

ror failure mechanisms.(13) Calculated values of qu and Ng according to Eqs.

(A13) and (A14), in which g＝20 kN/m3 and B＝5 mwere used.