THE EFFECTS OF A SETBACK ON THE BEARING CAPACITY OF A SURFACE FOOTING NEAR A SLOPE

Transcription

1 Journal of GeoEngneerng, Chng-Chuan Huang Vol. 3, and No. Wen-We, pp. 5-3, Kang: Aprl The 008 Effects of a Setback on the earng Capacty of a Surface Footng Near a Slope 5 THE EFFECTS OF A SETACK ON THE EARING CAPACITY OF A SURFACE FOOTING NEAR A SLOPE Chng-Chuan Huang and Wen-We Kang ASTRACT The effects of setback on the ultmate bearng capacty of rgd surface footngs placed on the crest of the slope are evaluated usng a lmt-equlbrum-based method. It s shown that the values of the bearng capacty coeffcent or correcton factor for the ultmate bearng capacty of a footng placed on the crest of a slope can be expressed as lnear functons of the setback-tofootng-wdth rato up to a certan threshold value namely, (b/) t. eyond ths threshold value, the ultmate bearng capacty s the same as those obtaned for a smlar footng placed on sem-nfnte level ground. The value of (b/) t can be unquely related to the nternal frcton angle of the foundaton sol, φ, regardless of the change n the slope angle,. Good agreements are obtaned between the value of correcton factors obtaned heren and those reported n the lterature. It s also shown that a conventonal ndrect approach whch deals wth the effects of a setback usng an equvalent surcharge on the slope surface also works well n the sense that the ndrect approach generate small errors n the correcton factors when compared to those obtaned usng the drect approach such as the one proposed n the present study. Key words: earng capacty, surface footng, slope, lmt equlbrum method.. INTRODUCTION A smple and straghtforward way of mtgatng possble dsasters nduced by the bearng capacty falure of a shallow footng located near a slope s to ncrease the setback of the footng (b) as schematcally shown n Fg.. Regardless of the mportance of setback to practcal applcatons, research nto ths aspect has been very lmted (e.g., Meyerhof, 957; Graham, 988). One of the reasons that analytcal solutons for the ultmate bearng capacty takng nto account the setback of footng (expressed by a setback dstance b ) has rarely been provded s schematcally shown n Fg.. That s, the effect of setback has conventonally only been ndrectly consdered by usng the effect of surcharge, namely, γ D f N q (γ: unt weght of sol, D f : depth of overburden sol, N q : Terzagh s bearng capacty coeffcent for surcharge). In such cases, the ultmate bearng capacty of footng placed on the crest of a slope wth a slope angle namely, q u (>0, b> can be expressed usng Terzagh s bearng capacty formula (Terzagh, 943): qu(> 0, b> = γ Ν γ (> 0, b= + γ Df Νq(>0, b= () n whch : wdth of footng N γ (>0, b= : bearng capacty coeffcent due to self-weght of sols for a footng adjacent to the slope wth a slope angle. Manuscrpt receved January 3, 008; accepted Aprl, 008; revsed Aprl 4, 008. Professor (correspondng author), Department of Cvl Engneerng, Natonal Cheng Kung Unversty, Tanan, Tawan 700, R.O.C. (e-mal: samhcc@mal.ncku.edu.tw). Graduate student, Department of Cvl Engneerng, Natonal Cheng Kung Unversty, Tanan, Tawan 700, R.O.C. N q (>0, b= : bearng capacty coeffcent due to surcharge for a footng adjacent to the slope wth a slope angle. D f : vrtual depth of footng comparable to the footng wth a setback. (D f = b tan s used n the followng). However, ths ndrect approach has some shortcomngs. Frst, the surcharge s non-unform adjacent to the toe of the footng as llustrated by the shaded area n Fg., and the effect Fg. a Fg. c b Potental falure surface Schematc fgure showng the effect of setback on the falure mechansm of footngs placedon the crest of the slope e b d Schematc fgure showng the approxmaton of a footng wth a setback usng a footng wth a unform surcharge.. c b d D f

2 6 Journal of GeoEngneerng, Vol. 3, No., Aprl 008 of non-unformty of surcharge ncreases when the dstance of setback b ncreases. Second, the crust of the sol that provdes overburden pressure to the slope surface s subjected to possble slope nstablty smlar to that of an nfnte slope,.e., a drvng force s actng on the nterface c-d (see Fg. ) whch s not accounted for n Eq. (). A possble way to elmnate the above drawbacks s to use a drect approach to evaluate the effect of a setback of the footng on the bearng capacty coeffcent, N γ usng a correcton factor, η b, as dscussed below. Varous correcton factors have been successfully used n foundaton engneerng practces n evaluatng the effects of slope nclnatons, load nclnatons and load eccentrctes (Meyerhof, 963; Hansen, 970; Vesc, 973). To the best knowledge of the authors, a smplfed equaton expressng the correcton factor η b s yet to be developed.. ANALYTICAL METHODS AND VERIFICATION Fgure 3 schematcally shows the falure mechansm assumed n the present study, whch conssts of a trangular actve wedge under the footng wth a wdth, a transtonal zone bounded by a logarthmc falure lne (defned by the followng equaton) and a passve zone. tan r ro e μ φ = () where r : radus from the toe of the footng r o : radus at the nterface between the actve wedge and the transtonal zone μ : angle between r o and r. In the case of gentle slopes, a passve zone may be fully or partally ncluded n the falure mechansm, dependng on the relatve poston of the slope and footngs. An example s shown by the dotted lnes n Fg. 3. The falure mechansm shown n Fg. 3 has been verfed by Huang and Tatsuoka (994) based on the model test results reported by Huang, et al. (994). The zone bounded by the falure mechansm llustrated n Fg. 3 s dvded vertcally nto slces wth a wdth of 0 mm. It was found that slces wth ths wdth consstently provde accurate results n the sense that further reducng the slce wdth gves no mprovement for the calculated values of q u. Janbu s rgorous slce method (Janbu, 973) whch satsfes force and moment equlbra s used n the present study. It has been shown that ths method s as accurate as other rgorous methods, such as Spencer s and Mogenstern and Prce s slce methods (Fredlund and Krahn, 977) and the generalzed varatonal method (Leshchnsky and Huang, 99). The equlbrum formulaton of the slce method and procedure for dervng ultmate bearng capacty (coeffcents) of footngs are descrbed n Appendx A. Formulatons and verfcatons of ths method have been descrbed n-detal by Huang, et al. (994). In the present study, and also n the study of Huang, et al. (994), a mnmum value of vertcal footng load s searched va an optmzaton of θ and μ,.e., va a systematc search of crtcal falure mechansms. The ultmate bearng capacty of footng, q u s then calculated based on Eqs. (A) through (A3). The bearng capacty coeffcents for a footng adjacent to aslope ( > wth or wthout a set back, namely, N γ(>0, b= or N γ(>0, b>, can be calculated usng Eq. (A4). Note that n Huang s analyses, the exponental functon n Eq. () s replaced by e μ tan η and the value of η s determned va a tral-and-error procedure searchng for an optmzed value of η that generates a mnmum value of N γ. Huang and Tatsuoka (994) found that the optmzed value of η s only slghtly dfferent from the nternal frcton angle of sol φ. Therefore, the dfference n the value of N γ when usng e μ tan φ or e μ tan η n Eq. () s small, as shown n Fg. 4. Ths fgure also shows that the value of N γ obtaned usng the present method s close to the lower bound of varous theoretcal solutons compled by Graham, et al. (988). As has been examned by Huang and Tatsuoka (994) that the assumpton of the thrust heght of nterslce force at /3 of the sde-face of the slce generates acceptable nterslce and slce base forces n the sense that postve nterslce and slce base forces are obtaned along the entre crtcal falure surface. Fgure 5 compares the analytcal values of N γ for a rgd footng placed on the slope wth varous slope angles ( = 0, 30, and 40 ) and setbacks (b/. Comparable results on the N γ vs. b/ relatonshp showng the transton of N γ from b/ = 0 nto b/ = (a case of footng placed on a level ground) can be obtaned. All these curves plateau when the value of b/ s beyond the threshold value of b/, namely (b/) t, ndcatng the dmnshng of the nfluence of footng setback on the ultmate bearng capacty of footng when placed wth a setback-tofootng wdth rato b/ larger than (b/) t. Fgure 5 also shows that n the cases of φ = 40, = 0 and φ = 40, = 40, Meyerhof (957) provded (b/) t 4.5 whch s close to that provded n the present study ((b/) t 5.. It s also seen that except n the case of b/ = 0 for φ = = 30 and φ = = 40 condtons, whch are unlkely to exst n practce, the dfference n the value of N γ between these two analytcal solutons s generally less than 9% ~ 3%. Comparng ths varaton wth those found n other theoretcal approaches, as typcally shown by the shaded area n the logrthmc scale n Fg. 4, a varaton n theoretcal values of N γ of only 9% ~ 3% can be consdered very small. φ φ φ θf θ μ o φ 3. THE EFFECT OF THE SETACK OF A FOOTING PLACED ADJACENT TO THE SLOPE μ φ Fg. 3 Falure mechansm used n the present study o Fgure 6(a) shows an example of the effect of setback (b/) on the bearng capacty coeffcent (N γ ) for a vertcally loaded footng placed near a slope consstng of a cohesonless sol wth φ = 30 and varous slope angles (). A footng wdth = 5 m and a unt weght of sol, γ = 7.6 kn/m 3 are used n the present

3 Chng-Chuan Huang and Wen-We Kang: The Effects of a Setback on the earng Capacty of a Surface Footng Near a Slope 7 study to derve ultmate bearng capacty of footngs. It has been verfed that the values of N γ obtaned heren are not susceptble to the changes n the values of and γ for wde ranges of 0.5 m 0 m and 0 kn/m 3 γ 0 kn/m 3. The effect of b/ on N γ s expressed by usng a correcton factor η b n Terzagh s bearng capacty formula: or qu (>0, b> = γ Νγ (>0, b= ηb (3) Ν q η b = = Ν q γ (>0, b> u(>0, b> γ (>0, b= u(>0, b= (4) n whch, N γ (>0, b> and N γ (>0, b= : bearng capacty coeffcents nduced by the self-weght of sols for a gven slope angle () under b > 0 and b = 0 condtons, respectvely. q u (>0, b= : ultmate bearng capacty for a footng adjacent to the slope wth a gven slope angle () and a setback dstance b = 0. Fgure 6(a) shows that for the cases nvestgated, threshold values of b/, namely, (b/) t can be clearly defned and the effect of b/ on N γ s lmted to the condton of b/ (b/) t. For b/ > (b/) t, the bearng capacty of footng s controlled by a falure mechansm smlar to that of level ground. The values of η b can be related to b/ usng b-lnear curves consstng of a segment wth a slope of S b and a flat segment: N γ ( > 0, b = 0 ) c = 0, φ = 40 o The present study Graham et al. (988) Huang and Tatsuoka (994 ) Zhu (00 Range of theoretcal solutons compled by Graham et al. (988) and η = S (/ b ) + for b/ (/ b ) (5) b b t η = S (/ b )+ for b/ > (/ b ) (6) b b t t Fgures 6(b), 6(c), and 6(d) show smlar analytcal data to those shown n Fg. 6(a), except that Fgs. 6(b), 6(c), and 6(d) are for φ = 35, 40, and 45, respectvely. Smlar conclusons to those of Fg. 6(a) can be drawn, except that the values of (b/) t tend to ncrease wth the ncrease n φ. Fgure 7 shows that the value of (b/) t can be expressed as a lnear functon of φ, as follow: (/ b)= t b+ b φ (7) Slope angle, ( ο ) Fg. 4 Comparsons of analytcal values of N γ for a rgd footng adjacent to the slope obtaned n varous studes (compled from Graham, et al., 988 ) n whch, b = 8. and b = 0.33 (/degree) In Fg. 7, the values of (b/) t provded by Meyerhof (957) based on the results shown n Fg. 5 are also plotted. The smlarty between these two analytcal results s clear. Fgure 8 shows the value of S b representng the slope of the lnes for the η b vs. b/ relatonshp whch can be expressed usng the followng equatons. The use of ths equaton s based on the result of a comparatve study on varous types functons, such as, polynomnal, logarthmc, and exponental functons. The comparatve study shows that curve-fttngs usng Eq. (8) generate relatvely great values of correlaton coeffcent compared wth the cases usng other types of functon. Sb = x (tan (8) ) z n whch, x and z are functons of φ as shown n Fgs. 9 and 0: x = x+ x φ (9) n whch, x = and x = 0.03 (/degree) The value of z n Eq. (8) can be expressed as (see Fg. : Fg. 5 Comparsons of theoretcal values of N γ obtaned by Meyerhof (957) and the present study z = z+ z φ ( n whch, z = 0.35 and z = 0.04 (/degree).

4 8 Journal of GeoEngneerng, Vol. 3, No., Aprl 008 (a) φ = 30 Fg. 7 Relatonshps between (b/) t and nternal frcton angle of sol, φ (b) φ = 35 Fg. 8 Relatonshps between S b and slope angles, for φ = 35 (c) φ = 40 Fg. 9 Values of coeffcent x as functon of φ (d) φ = 45 Fg. 6 Correcton factors η b for ultmate bearng capacty of footngs placed near the slope wth varous φ and slope angles Fg. 0 Values of coeffcent z as functon of φ

5 Chng-Chuan Huang and Wen-We Kang: The Effects of a Setback on the earng Capacty of a Surface Footng Near a Slope 9 4. COMPARATIVE STUDY ON THE EFFECT OF SETACK Table summarzes the emprcal equatons for η b dscussed earler. Correcton factors η b calculated usng emprcal Eqs. (5) ~ ( are compared wth the analytcal value of η b provded by Meyerhof (957), Graham, et al., (988) and the present study, as shown n Fg.. In general, values of η b obtaned n the present study agree well wth those reported by Graham, et al., (988) for the case of φ = 45 and = 6 ; they also agree well wth those reported by Meyerhof (957) for the case of φ = 40 and = 0. Fgure shows a comparson between the values of η b obtaned n two ways. The frst are based on the drect approach as proposed n the present study and the second are those based on the ndrect approach usng syntheszed bearng capacty coeffcents N rq (>0, b= whch takes nto account the combned effect of N γ and N q expressed by: or N γq( > 0, b> qu( > 0, b> = γ q γ D N = + γ γ D = N + u( > 0, b= f q( > 0, b= f γ> ( 0, b= 0 ) Nq( > 0, b= () D f Nγq( > 0, b> = Nγ( = gγ + Nq( = gq () n whch, N γq (>0, b> : bearng capacty coeffcent of footng on the crest of a slope (b > 0, >, takng nto account the combned effect of self-weght and surcharge of sols. N γ (= : N γ for a footng placed on level ground N q (= : N q for a footng placed on level ground g γ : correcton factor for N γ(= the effect of sloped ground g q : correcton factor for N q(= the effect of sloped ground D f : vrtual depth of footng embedment correspondng to a certan footng setback (D f = b tan s used n the present study) Table Correcton factors for the setback of footng adjacent to the slope under vertcally and statcally loaded condtons Number of equatons Equaton Parameters 5 η b = S b (b/ for / < (b/) t See below 6 η b = S b (b/) t See below for / (b/) t 7 (b/) t = b + b φ b = 8. b = 0.33 (/degree) 8 S b = x z See below 9 x = x + x φ 0 z = z + z φ x = x = 0.03 (/degree) z = 0.35 z = 0.04 (/degree) The values of N γq(=30, b=) and N γq(=30, b ) shown n Table for Vesc (973) and Hansen (97 are calculated usng Eq. () and bearng capacty coeffcents suggested n the respectve studes. For the value of η b calculated usng ndrect approaches, η b s defned as follows: N N D N η b = = + N N N γq( > 0, b> γ( > 0, b= f q( > 0, b= γ> ( 0, b= γ> ( 0, b= γ> ( 0, b= D N = + N f q( = q γ= ( g g γ Values of η b n Table and Fg. were calculated usng the followng equatons suggested by Hansen (97: and N =.5 [N ] tan γ ( = q( = φ (4) 5 gγ = g q = ( 0.5 tan ) (5) Values of η b n Table and Fg. were also calculated usng equatons suggested by Vesc (973): and N = [N +] tan γ ( = q( = φ (6) gγ = g q = ( tan ) (7) Fg. Comparsons of analytcal values of η b obtaned n varous analytcal studes Fg. Comparsons of η b based on analytcal values of N γ and emprcal values of N γq (3)

6 30 Journal of GeoEngneerng, Vol. 3, No., Aprl Table Comparsons the values of N γ for φ = 40 N γ (= N γ (=30, b= or N γ (=30, b=) N γq (=30, b=) N γ (=30, b=) or N γq (=30, b=) The present study Hansen ( Vesc (973) Graham, et al. (988) Zhu ( Kumar and Mohan Rao (003) Corrected values of N γ (= usng g γ expressed by Eq. (5) or Eq. (7) Analytcal values of N γ based on drect approach for the effect of b/ N γq (>0, b> calculated usng Eq. () and the equatons of N γ (=, N q( = g γ and g q suggested by varous authors, Eqs. (4) ~ (8) N γq (>0, b> calculated usng Eq. () and values of N γ (>0, b= and N q (>0, b= reported by Kumar and Mohan Rao (003) n Eqs. (), (3), (4), and (6) tan Nq( e π φ φ = = tan 45 + (8) It can be seen n Eq. (3) that the accuracy of calculated values of η b based on Hansen s and Vesc s solutons are not nfluenced by the values of g γ and g q, because both proposed g γ = g q as shown n Eqs. (5) and (7). The values of N γq (>0, b= for Zhu (00 and Kumar and Mohan Rao (003), as summarzed n Table, are calculated usng Eq. () and values of N γ (>0, b= and N q (>0, b= reported n the respectve studes. Note that the present study s not ntended to examne n-detal the background leadng to the wdely varated theoretcal and/or emprcal solutons of N γ and N γq provded by varous methods. In stead, the present study focuses on the comparson of η b provded by varous analytcal and/or emprcal methods. It s well-known that theoretcal solutons of N γ (for dentcal values of φ and ) can span a wde range of ±00% from the averaged theoretcal values of N γ obtaned usng varous methods. Ths fact s exemplfed n Fg. 4 and has also been dscussed by Tatsuoka, et al., (989) and Huang and Tatsuoka (994). The essence of Table s that theoretcal solutons of N γ for the level ground ( = and that for = 30 derved here are always close to the averaged values of N γ obtaned usng varous methods. It seems that values of N γ(=30, b=) and N γ(=30, b=) obtaned here are somewhat greater than those obtaned by other emprcal solutons. The greater values of N γ(=30, b=) and N γ(=30, b=) for about 5% ~ 30% compared to other solutons ndcate that composed values of N γq(=30, b=) and N γq(=30, b=) based on varous emprcal approaches are somewhat conservatve. For the theoretcal solutons provded n the frst two columns of Table, the ones provded n the present study devate from the averaged values wthn +7% and 0%. Ths s consdered small when compared wth the range of ± 00% for varous theoretcal solutons as dscussed prevously. Fgure shows that the values of η b calculated usng the ndrect approach, as shown n Eq. (3), and the drect approach as proposed heren are comparable for b/ 4, suggestng that conventonal ndrect approaches also work well as an alternatve analytcal approach. 5. CONCLUSIONS In ths study, a lmt equlbrum method ncorporated wth Janbu s slce method s used to evaluate the effect of setback on the ultmate bearng capacty of a surface footng placed near the shoulder of a slope. Ths method elmnates possble shortcomngs that may be assocated wth conventonal approaches whch employ overburden pressure and Terzagh s bearng capacty coeffcent for surcharge, N q, to ndrectly evaluate the effect of setback. The analytcal results show that the bearng capacty of the footng ncreases almost lnearly wth an ncrease n setback dstance up to certan threshold values denoted by a dmensonless setback-to-footng wdth rato, (b/) t. eyond these threshold values, the ultmate bearng capacty remans constant lke that of a footng placed on a sem-nfnte level ground. The results show that the value of (b/) t s a lnear functon of the nternal frcton angle of the foundaton sol (φ) regardless the value of slope angle () ranged between 0 and 35. The results also show that the gradent (S b ), characterzng the lnear relatonshp between the effect of setback (n terms of η b ) and the normalzed setback (b/), can be expressed as functons of φ and. The correcton factors, η b whch were obtaned n the present study are comparable wth those obtaned n two prevously documented analytcal studes. It s also shown that an ndrect approach whch takes nto account the effect of setback on the ultmate bearng capacty of footng by usng equvalent surcharges on the slope surface also works well n the sense that the ndrect approach examned heren generates comparable values of η b as those provded by straghtforward analytcal solutons derved n the present study. APPENDIX A FORMULAS AND PROCEDURES FOR CALCULATING SEISMIC EARING CAPACITY OF FOOTINGS ADJACENT TO THE SLOPE The sol mass confned by the slp lnes shown n Fg. A- s dvded nto n vertcal slces (wdth of slce, s = 0. m) wth base nclnatons ( =,, n) n the present study. These slces are grouped nto two categores, namely, slces subjected to the footng load, P and Q, at ther top surfaces and the slces wthout footng load at ther top surfaces. The followng assumptons are made.. Assume nter-slce thrust heghts to be /3 of the nter-slce heghts and calculate θ for all slces (see Fg. A- for the defnton of θ ).. The ultmate footng loads P f and Q f are unformly dstrbuted on the surface of the slces located drectly under the footng,.e., P f = P f = = P fm = P f, and Q f = Q f = = Q fm = Q f (m: number of slces drectly subjected to the footng load s n f ; n f = 50 n the present study). 3. Q f = P f tan β, β: angle of load nclnaton on the footng base.

7 Chng-Chuan Huang and Wen-We Kang: The Effects of a Setback on the earng Capacty of a Surface Footng Near a Slope 3 Q P β Pf No. T o E o No. sec + ( + Δ ) tan φ sec C W P T D = + tan φ tan (A9) G = ( W ΔT) tan (A f W E n T n No. n h e W s H h q P Q No. T T Δ T E E ΔE θ G = ( P + W ΔT ) tan + Q (A) H f tan tan φ = tan β+ (A) + tan φ tan h N S Takng the moment equlbrum about the center of the slce base yelds the followng equaton: Fg. A- Schematc fgure of forces and boundary condtons used n the slce method Accordng to: () the force equlbrum n horzontal and vertcal drectons for slce Nos. n, () Mohr-Coulomb s falure crteron, S = (S f / F s ) = (N tan φ) / F s (assumng cohesonless sols and no pore water pressure; F s : safety factor aganst shear falure; n F s =.0 n the present study), and (3) E = En Eo, ultmate footng load on the surface of the slce, P f and the dfferental between nter-slce forces, ΔE (= E E ) can be obtaned as: f nf E E + D + D + G + G P = n nf n o f f nf + nf + n f H f n (A) ΔE = A + (A) The ultmate bearng capacty (q u ) and the bearng capacty coeffcent (N γ ) can be obtaned usng: q u N γ n Eq. (A), Pf nf = (A3) q u = (A4) γ A= Sf sec (A5) =( P+ W ΔT) tan + Q (A6) C + ( W + P ΔT) tanφ sec S f = tan φ tan + Fs ΔT = T T n Eq. (A), D f sec + ( Δ ) tanφ sec C W T = + tan φ tan (A7) (A8) ΔT ΔE h + Q hq T = E tan θ ΔE tan θ + + s (A3) Assumng that the wdth of slce s small, ΔT 0 and ΔE 0, we can rewrte Eq. (A3) as: Q hq T = E tan θ + (A4) s In the present study, Eq. (A4) nstead of Eq. (A3) s used. The computer algorthm for calculatng ultmate footng load P f s as follows:. Assume ΔT = 0.. Calculate the frst approxmated value of P f usng Eq. (A). 3. Calculate ΔE and E ( =,,, n) usng Eq. (A) 4. Calculate T ( =,,, n) usng Eq. (A4). 5. Calculate an mproved value of P f usng Eq. (A). 6. Repeat Steps (4) to (6) untl the convergence of P f s acheved (the convergence crteron for P f was New P f Old P f / (Old P f ) 0.3% n the present study.) NOTATIONS A : term used n Eq. (A5) (N/m) : wdth of footng (m) b: dstance of setback (m) s : wdth of slce (m) : term used n Eq. (A6) (N/m) (b/) t : threshold value of b/ that tranform from a near-slope bearng capacty nto a level-ground bearng capacty (dmensonless) C : Cohesve shear resstance for slce No. (N/m) D f : vrtual depth of footng embedment (m) D f : term used n Eq. (A9) (N/m) D : term used n Eq. (A8) (N/m) ΔE : dfferental value of E (N/m) E ( = 0... n): horzontal nterslce force (N/m) G f : term used n Eq. (A (N/m) G : term used n (A) (N/m)

8 3 Journal of GeoEngneerng, Vol. 3, No., Aprl 008 g q : correcton factor on N q (= for the effect of sloped ground (dmensonless) g γ : correcton factor on N γ (= for the effect of sloped ground (dmensonless) H f : term used n Eq. (A) (N/m) h : arm of rotaton for horzontal nter-slce force (m) h q : arm of rotaton for horzontal sesmc force at the top of slce (m) n f : number of slces drectly under the footng N : normal force on the slce base (N/m) N q : Terzagh s bearng capacty coeffcent for footng embedment or surcharge (dmensonless) N γ : Terzagh s bearng capacty coeffcent for self-weght of sols (dmensonless) N q (>0, b= : bearng capacty coeffcent due to surcharge for a footng adjacent to the slope wth a slope angle (dmensonless) N γ(= : N γ for a footng placed on a horzontal ground (dmensonless) N q(= : N q for a footng placed on a horzontal ground (dmensonless) N γ (>0, b= : N γ for a footng adjacent to the slope wth a slope angle (dmensonless) N γ (>0, b> : N γ for a footng placed on the crest of a slope wth a setback (b > (dmensonless) N γq (> : bearng capacty coeffcent of footng adjacent to the slope (b = 0, >, takng nto account the combned effect of self-weght and surcharge of sols (dmensonless) P f ( =... m): footng load on the top of slce No. ; m : number of slces drectly subjected to the footng load (N/m) P : vertcal load at the top of slce No. (N/m) P f : total footng load (N/m) Q f ( =... m): horzontal force excerted by the footng (N/m) Q : horzontal load at the top of slce No. (N/m) q u : Ultmate bearng capacty of footngs (N/m ) q u(>0, b= : ultmate bearng capacty for a footng adjacent to the slope wth a gven slope angle, (N/m ) r: radus of a logrthmc spral from the toe of footng (m) r o : radus of a logrthmc spral at the nterface between the actve wedge and the transtonal zone (m) S b : gradent of the lnear η b vs. b/ relatonshp (dmensonless) S f ( =... m): shear force at the base of slce No. drectly subjected to the footng load (N/m) S : shear force at the base of slce No. (N/m) ΔT : dfferental value of T (N/m) W : self-weght of slce No. (N/m) : slope angle (degree) : base nclnaton for slce No, (degree) β: footng load nclnaton (degree) φ: nternal frcton angle of sols (degree) γ: unt weght of sols (kn/m 3 ) η: parameter dctatng the curvature of a log-spral (degree) η b : correcton factor on N γ for a footng setback (dmensonless) μ: angle between r o and r for a log-spral determng the transtonal zone bounded by a log-spral (radan) θ : thrust lne nclnaton for slce No. (degree) REFERENCES Fredlund, D. G. and Krahn, J. (977). Comparson of slope stablty methods of analyss. Canadan Geotechncal Journal, 4(3), Graham, J., Andrews, M. and Shelds, D. H. (988). Stress characterstcs for shallow footngs n cohesonless slopes. Canadan Geotechncal Journal, 5(), Hansen, J.. (97. A Revsed and Extended Formula for earng Capacty. Dansh Geotechncal Insttute, Copenhagen, ul. 8, pp. Huang, C. C. and Tatsuoka, F. (994). Stablty analyss for footngs on renforced sand slopes. Sols and Foundatons, 34(3), 37. Huang, C. C., Tatsuoka, F. and Sato, Y. (994). Falure mechansms of renforced sand slopes loaded wth a footng, Sols and Foundatons, 34(), Janbu, N. (973). Slope Stablty Computatons, Embankment-Dam Engneerng, Casagrande Volume, John Wley and Sons, Kumar, J. and Mohan Rao, V.. K. (003). Sesmc bearng capacty of foundatons on slopes, Geotechnque, 53(3), Leshchnsky, D. and Huang, C. C. (99). Generalzed slope stablty analyss: Interpretaton, modfcaton, and comparson, Journal of Geotechncal Engneerng, ASCE, 8(, Meyerhof, G. G. (957). The ultmate bearng capacty of foundatons on slopes. Proc. 4th Internatonal Conference on Sol Mechancs and Foundaton Engneerng, London,, Meyerhof, G. G. (963). Some recent research on the bearng capacty of foundatons, Canadan Geotechncal Journal, (), 6 6. Tatsuoka, F., Huang, C. C., Mormoto, T. and Okahara, M. (989). Dscusson on stress characterstcs for shallow footngs n cohesonless slopes. Canadan Geotechncal Journal, 6(4), Terzagh, K. (943). Theoretcal Sol Mechancs, New York, Wley. Vesc, A. S. (973). Analyss of ultmate loads of shallow foundatons, Journal of the Sol Mechancs and Foundaton Dvson, 99(), pp Zhu, D. (00. The least upper-bound solutons for bearng capacty factor Nγ. Sols and Foundatons, 40(), 3 9.