EVALUATION OF SEISMIC ACTIVE EARTH PRESSURE USING HORIZONTAL SLICE METHOD AND LOG-SPIRAL FAILURE SURFACE

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1 EVALUATIO OF SEISMIC ACTIVE EARTH PRESSURE USIG HORIZOTAL SLICE METHOD AD LOG-SPIRAL FAILURE SURFACE S. BAISHYA orth Eastern Regonal Insttute of Sene and Tehnology (ERIST), Arunahal Pradesh, Inda A. SARKAR & A. K. DEY Former under-graduate student, ERIST, Arunahal Pradesh, Inda ABSTRACT Evaluaton of atve earth pressure n sesm ondton s one of the mportant problems of geotehnal earthquake engneerng. In ths study the stat and sesm atve earth pressure exerted by a homogeneous bakfll behnd a rgd retanng wall s evaluated usng Horzontal Sle Method (HSM). Both ohesonless and ohesve sols are onsdered usng - sol n the formulaton. The falure surfae s assumed to be a logarthm spral whh represents a more realst falure surfae ompared to the ommonly used planar surfae. The sesm nerta fore s onsdered by usng pseudo-stat horzontal aeleraton oeffent, leadng to smple formulaton of an otherwse omplex phenomenon. Important parameters n the sesm earth pressure problem vz., depth of tenson rak, surharge over the bakfll, sesm horzontal aeleraton, batter angle of the wall, and, the shear strength parameters of the bakfll are nluded n the study. Earth pressure s expressed n terms of dmensonless fators termed as Earth Pressure Fators (EPF) whh makes t possble to express both stat and sesm earth pressure by smple equaton resemblng the famlar bearng apaty equaton. The effay of HSM s frst verfed usng a planar wedge falure for whh results are known from prevous studes. The method s then extended for logarthm spral falure surfae. The rtal falure surfaes are dentfed by usng dfferent ntal rad of the spral n order to optmze the EPFs produng maxmum atve earth-pressure. The advantage of HSM s exploted to fnd the varaton of sesm earth pressure along the heght of the retanng wall. The results are presented n the form of user frendly non-dmensonal desgn harts. Sesm Earth-Pressure, Horzontal Sle Method, Pseudostat. ITRODUCTIO Evaluaton of sesm atve earth pressure behnd retanng walls s one of the mportant problems of geotehnal earthquake engneerng. The varous methods used for the purpose are pseudostat, pseudodynam and dynam analyss. Pseudostat methods are usually preferred due to smplty. Among the lmt equlbrum methods used for pseudostat analyss of sesm earth pressure, the most well known s the Mononobe-Okabe (MO) method. Although extensvely used, reent studes (Gazetas, et. al., 2004; Psarropoulos, et.al., 2005) h ndated that the method overestmates the earth pressure, espeally n the upper half of the bakfll. In addton, the varaton of pressure along the bakfll heght s not avalable. Another smple and user frendly method proposed by Prakash and Saran (Prakash and Saran, 966; Saran and Prakash, 968), here referred to as PSM, expresses the

2 atve earth pressure n terms of non-dmensonal fators termed here as Earth Pressure Fators (EPFs) smlar to Terzagh's bearng apaty fators. They h onsdered - sol, effet of wall frton, pseudostat nerta fore and h separated the ontrbutons of surharge, oheson and frtonal resstane of sol to the atve earth pressure. Ths leads to smple formulaton of total earth pressure. However, varaton of earth pressure along the bakfll heght s not avalable. In the reent past a new lmt equlbrum method, known as Horzontal Sle Method (HSM), based on subdvson of the falure wedge nto horzontal sles s proposed (Shahghol et. al., 200; our et. al., 2006). The method s based on equlbrum of fores and moments atng on the horzontal sles. The method s sutable for analyss of homogeneous and non-homogeneous bakflls/slopes and for renfored bakflls. In ths study, frst the HSM s used to fnd EPFs usng a trangular falure wedge and results are ompared to those obtaned by usng full trangular wedge as per PSM. After verfaton of effay of HSM, t s extended for fndng EPFs by assumng log-spral falure surfaes. In both the ases the EPFs are optmsed by onsderng dfferent tral falure surfaes n order to produe maxmum atve earth pressure. Results are presented n the form of user frendly desgn urves. 2. VERIFICATIO OF HSM USIG PLAAR FAILURE WEDGE 2. Earth Pressure usng Whole Wedge (PSM) Fg. 2. shows a falure wedge formed by a planar falure surfae n the φ bakfll behnd the retanng wall havng batter angle arryng a surharge q. Weght of the wedge s W, angle of wall frton s sol-wall adheson s a, and the depth of tenson raks s H. Ths depth an be expressed n terms of rak depth fator f H / H. The atve earth pressure on the wall s P and makes an angle wth the normal to the wall-bakfll nterfae. The horzontal nerta fore n sesm ondton s replaed by a pseudostat fore k h W where kh s the horzontal oeffent of aeleraton. Fg. 2. Planar Falure Wedge and Fores Atng on the Wedge By resolvng the fores n vertal and horzontal dretons and smplfyng, the earth pressure an be expressed as (Prakash and Saran, 966; Saran and Prakash, 968):

3 2 P γh + qh H a (2.) where,,, a are the Earth Pressure Fator (EPF)s and are gven by a 2 f + 0.5tanα + tanθ+ 0.5 f tanα osθ +φ+ khsnθ +φ snβ +δ f + tanα + tanθ osθ + φ+ khsnθ + φ snβ + δ osβseα+ osφseθ sn β +δ (2.2a) (2.2b) (2.2) β φ+ θ + α (2.2d) For a partular wall-bakfll system, Eqn. 2. shows that n order to maxmze P the EPFs and are to be maxmzed and a s to be mnmzed, by onsderng dfferent values of nlnaton of falure plane, θ. The earth pressure n stat ondton an be obtaned by puttng k h 0 n Eqn. 2.2a-b. a s ndependent of kh as oheson s not affeted by nerta fore. 2.2 Earth Pressure usng Horzontal Sle Method Fg. 2.2 shows the same falure wedge explaned n Se. 2. but sled horzontally. The -th sle has an area A, thkness t, top wdth b weght W, and, earth pressure P. Fg. 2.2 Horzontal Sles n Planar Falure Wedge and Fores Atng on the Wedge Consderng equlbrum of the -th sle the earth pressure atng on t an be expressed as Psn β + δ γa osθ + φ+ khsnθ + φ qb osθ + φ+ k snθ + φ t tanθ tanαsn θ + φ+ θ + φ + (2.3) h 2os where b s the wdth of topmost sle. Summng up suh fores for all the sles and based on geometry of sles, the total earth pressure an be expressed agan as gven by Eqn. 2.. The

4 expressons for and oheson n ths ase s gven by a are found to be same as those gven by Eqn. 2.2a and 2.2b. EPF for tanθ f + tanαsn θ + φ+ 2osθ + φ snβ + δ EPFs usng PSM and HSM are alulated for dfferent nlnatons of falure plane and are shown n Fg Ths shows that EPFs alulated by both the methods are dental. (2.4) Fg. 2.3 EPFs usng HSM and PSM φ 20 0,α 0,k 0 h 0, f In order to maxmze earth pressure, the EPFs are alulated for eah value of φ by varyng the nlnaton of falure plane and optmal values of EPFs alulated by both HSM and PSM are seleted. Fg. 2.4 shows suh values of for dfferent values of kh for a pratal range of φ, whle Fg. 2.5 shows values of a for all values of k h. Fg. 2.3 through Fg. 2.5 learly shows that the EPFs alulated by usng HSM are dental to those obtaned by usng PSM. Ths further shows that HSM an relably be used for alulaton of sesm atve earth-pressure EPFS USIG HORIZOTAL SLICE METHOD AD LOG-SPIRAL FAILURE SURFACE Fg. 3. shows a falure wedge formed by a log-spral falure surfae n a φ sol. The varous quanttes assoated wth the bakfll and the wall are kept same as desrbed n Se. 2.. The equaton of log-spral s taken as Rθ R exp tanφ θ φ (3.) where Rθ s the radus vetor makng angle θ wth the horzontal at the orgn. R s the ntal radus of the spral at the surfae of the bakfll. The orgn s taken at pont X 0,Y 0 from the top of the wall

5 at the toe sde. The spral s normal to the bakfll surfae and passes through the toe of the wall. Fg. 2.4 Values of for dfferent values of φ 4 0, f 0 α Fg. 2.5 Values of for dfferent values of φ 4 0, f 0 a α Referrng to Fg. 3., the -th horzontal sle has a thkness t, and the rad vetors at ts top and bottom are R and R+ respetvely, makng respetve angles θ and θ+ wth horzontal at the orgn. The varous fores atng on the -th sle are shown n Fg By resolvng the fores atng on the -th sle, the earth pressure atng on t an be expressed as

6 P γa where, os θ + φ+ khsnθ + φ snβ + δ * sn θ + θ + φ L + at snβ + δ qb tanαsn osθ + φ+ khsnθ + φ snβ + δ θ + φ osθ + φ snβ + δ (3.2) θ 0.5 θ + θ (3.3a) β α +θ + φ (3.3b) Fg. 3. Horzontally Sled Falure Wedge formed by Log-Spral Falure Surfae Usng a oheson, Eqn. 3.2 an be expressed as P γa where + qb L a a (3.4) θ + φ+ khsnθ +φ snβ +δ os a (3.5a) a θ + φ+ khsnθ + φ snβ + δ os (3.5b) θ + φ tanαsnθ + φ sn β + δ sn θ * + θ + φ t os. (3.5) sn β + δ L By summng up ontrbuton of eah sle to earth pressure and usng propertes of the log-spral, the the total earth pressure and EPFs for total earth pressure an be expressed as

7 Fg. 3.2 Fores atng on -th Sle 2 P γh + qh H a (3.6) A (3.7a) H b (3.7b) H Rtoe R (3.7) Hsnφ 2 a a Here, A area of whole falure wedge and Rtoe s the radus of spral at the toe of the wall. It s mportant to note that n order to maxmze the total earth pressure, t s neessary to onsder a number of falure surfaes assumng dfferent values of ntal radus R. The value of R and orrespondng angle of spral are to be obtaned by tral and error. toe θ toe In ase of tenson rak of depth H of tenson raks, θ r, an be obtaned as f H, the angle made by the radus vetor at the bottom level θ r ln tanφ f H + snφ +φtanφ lnsn θ R r (3.8) By solvng Eqn. 3.8, the orrespondng radus vetor zone L r, whose ontrbuton to a Rr and the length of ar n the tenson rak should be negleted n suh ases, s obtaned as L r R r R snφ (3.9)

8 4. RESULTS AD DISCUSSIO Fg. 4. through 4.3 shows the values of EPFs orrespondng to those falure surfaes that produe maxmum total earth pressures. Fg. 4. Values of for dfferent values of φ 8 0, f 0.0 a α Fg. 4.2 Values of for dfferent values of φ 4 0, f 0.0 α Fg. 4.3 Values of for dfferent values of φ 8 0, f 0.0 a γ α

9 Fgs. 4.2 and 4.3 shows that atve earth pressure nreases wth nrease n horzontal aeleraton oeffent k. Ths nrease for surharge omponent.e. however dereases wth nrease n φ value of sol. h Use of HSM s advantageous to fnd the varaton of earth pressure along the heght of wall whh s not avalable n other pseudostat methods lke MO and PSM. Pressures at the mddle of horzontal sles are omputed by usng Eqn. 3.4 and are normalsed wth the maxmum pressure ntensty under stat ondton. Suh varatons of normalsed pressure wth wall heght are shown n Fgs. 4.4 to 4.6, 0 for a 0 m hgh wall wth batter angle α 4. These Fgures show that the varaton of pressure along wall heght s more or less parabol. Presene of surharge adds a omponent of earth pressure that remans more or less onstant wth heght, whle presene of oheson leads to negatve earth pressure at the top porton of the wall. In all ases, sesm earth pressure s hgher than the stat pressure. 0 0 Fg. 4.4 Varaton of Earth Pressure along Wall Heght φ 5 α 4,q 0, 0, f Fg. 4.5 Varaton of Earth Pressure along Wall Heght φ 5 α 4,q 0, 0, f 0

10 0 0 Fg. 4.6 Varaton of Earth Pressure along Wall Heght φ 5 α 4,q 0, 30, f 0 5. SUMMARY AD COCLUSIO In ths study the Horzontal Sle Method s used to analyse earth pressure exerted by φ sol under sesm ondton usng pseudostat approah. Use of realst Log-Spral falure surfae s made n the analyss. The total earth pressure s expressed n terms of non-dmensonal Earth Pressure Fators (EPFs) separatng the ontrbutons of frtonal resstane, surharge and oheson. Use of HSM gves, n addton to total earth pressure, the varaton of earth pressure along the wall heght. The results of ths study s useful n analyss of sesm atve earth pressure behnd retanng walls. ACKOWLEDGEMET Authors are grateful to the Slab Consortum, Frane ( for provdng the powerful free software SCILAB, whh s used for omputatons n ths work. REFERECES Gazetas, G., Psarropoulos, P.., Anastasopoulos, I. and Gerolymos,. (2004). Sesm Behavour of Flexble Retanng Systems subjeted to Short Duraton Moderately Strong Extaton. Sol Dynams and Earthquake Engneerng. 24, our, H., Fakher, A. and Jones, C.J.F.P. (2006). Development of Horzontal Sle Method for Sesm Stablty Analyss of Renfored Slopes and Walls. Geotextles and Geomembranes. 24, Prakash, S. and Saran, S. (966). Stat and Dynam Earth Pressures behnd Retanng Walls. Pro. 3rd Symposum on Earthquake Engneerng. Unv. of Roorkee, Roorkee,, Psarropoulos, P.., Klonars, G. and, Gazetas, G. (2005). Sesm Earth Pressure on Rgd and Flexble Retanng Wall. Sol Dynams and Earthquake Engneerng. 25, Saran, S. and Prakash, S. (968). Dmensonless Parameters for Stat and Dynam Earth Pressure for Retanng Walls. Journal of Indan Geotehnal Soety. :3, Shahghol, M., Fakher, A. and Jones, C.J.F.P. (200). Horzontal Sle Method of Analyss. Geotehnque. 5:0,