SEISMIC STABILITY ANALYSIS OF FOOTING ADJACENT TO SLOPES BY SLIP LINE METHOD

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China SEISMIC STABILITY ANALYSIS OF FOOTING ADJACENT TO SLOPES BY SLIP LINE METHOD ABSTRACT : M. Jahanandish and M.R. Arvin Assciate Prfessr, Dept. f Civil Engineering, Shira University, Shira, Iran PhD Student, Internatinal Institute f Earthquake Engineering and Seismlgy (IIEES), Tehran, Iran Email: Jahanand@shirau.ac.ir, m.arvin@iiees.ac.ir Many structures have t be placed n slping grunds by shallw ftings. Evaluatin f seismic stability and bearing capacity f such ftings is a prminent matter that rarely is cnsidered by researches. In general there are tw slpes net t a fting. Apart frm symmetric cases, ne f the tw sides f fting is mre likely t fail (first side) and mst researches ignre the effect f the ther side f fting (secnd side) in their analysis. In this paper stress characteristic methds (slip line methd) has been emplyed t shw the effect f secnd side n the bearing capacity f fting under bth static and dynamic lads. Bth hrintal and vertical earthquake cefficients have been impsed t the system. It is shwn that slpe f the ther side aggravates the stability f fting in cmparisn with fting placed between tw symmetric slpes. Besides the present study has taken in t accunt the inclinatin f lad n fting. Results shw that as the inclinatin f lad n fting increases, the bearing capacity f fting decreases. KEYWORDS: Seismic cefficients, slpe, first side, secnd side, partial mbiliatin, bearing capacity

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China. INTRODUCTION In practice very ften fundatins have t be placed n tp r n the surface f slpes. Structures such as retaining walls, bridge abutments and transmissin twers usually are put n slpes by shallw ftings. Besides in hilly regins there is n way t escape frm putting fundatins n tp f slps. The prblem f evaluating static bearing capacity f these kinds f ftings has been discussed in varius studies by limit state methds (Meyerhf, 957, 963; Hansen, 970; Vesic, 973; Reddy, 976; Saran et al., 989; Narita, 990). Just small number f researches can be fund in evaluating seismic stability f fting n slpes. Zhu (000) applied upper bund apprach t estimate Nγ fr fting n slping grund during earthquake. Using methd f characteristics, Kumar (003) evaluated bearing capacity factrs f fting n slpes under earthquake bdy frces. Mst studies incrprate single-side mechanism f failure t asses bearing capacity f fting and ignre the effect f secnd side f the fting. Budhu & Al-karani (993) emplyed limit equilibrium methd t estimate seismic bearing capacity f ftings n hrintal grund, cncerning a bth-side failure mechanism. Saran. et al. (989) utilied bth-side failure mechanism t find Nc, Nq and Ng fr fting n the surface, n tp r net t tp f slpes. They als cnsidered partial mbiliatin f strength parameters in the side f fting which was less susceptible t failure than the ther side. The matter f partial mbiliatin f strength parameters which is shwn by a cefficient between er and unity (m) was als used t find seismic bearing capacity f fting n hrintal grund (Budhu & Al-karani, 993). Using stress characteristic methd (SCM), Kumar (003) studied the static and seismic bearing capacity f fting n tp f slpes and determined Ng frm bth-side mechanism and Nc and Nq frm single-side mechanism. In this study, assuming bth-side mechanism, static and seismic bearing capacity f ftings n slpes are evaluated using SCM. Bth hrintal and vertical seismic cefficient is taken in t accunt t d a pseud static analysis n a c φ sil. It is assumed that at a side f fting which is less likely t cllapse (secnd side); just a prtin f strength is mbilied. In additin, angles f slpes net t fting are varied t evaluate the effect f secnd side f fting n the bearing capacity... Definitin f the prblem This study aims at finding the ultimate bearing capacity f a hrintal fting with width B in the presence f vertical and hrintal earthquake acceleratin K v g and K h g. Generally tw slping grund with different inclinatin angle (i and i) are suppsed n each side f fting (Fig ). Arbitrary surcharges, q and q, with different inclinatin angle t nrmal n surface, j and j can be applied n slpes. In determinatin f bearing capacity f fting, inclinatin f lad n fting is als incrprated, That is bearing capacity has been determined by this methd while tan (H/V) #0. Fig. Gemetry and lading cnditin f a fting n tp f slpe

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China. STRESS CHARACTERISTICS METHOD (SCM) Stress characteristics methd is a well-knwn apprach t slve stress bundary value prblem n their limit state. This methd assumes equilibrium f stresses everywhere in the system while cmpnents f stress at every pint f the bdy satisfy the yield criterin. Using SCM, Sklvski (960), prvided the slutins f different prblems under plane strain cnditin and C et. al. studied the ai-symmetric case. A typical sil element has been shwn in Fig (a) and crrespnding Mhr circle f stress is drawn n Fig (b). On assumptin that sil beys Mhr-Culmb Yield criterin, tw failure surfaces that are depicted by lines SCL (+) and SCL (-) n Fig (b), can be recgnied n stress space where P is the ple. SCL (+) and SCL (-) are called psitive and negative stress characteristics lines respectively and they make angle µ = ( π / 4) ( φ / ) with majr principal stressσ. Figure (a) A typical sil element under seismic lads. (b) Mhr circle f stress f element shwn n part a. It is pssible t relate nrmal and shear stresses t S andψ. S is the mean stress and ψ is the angle that majr principal stress makes with psitive directin f ais (Eqn..). σ = S + ( S.sinφ + c.csφ).cs ψ σ = S ( S.sinφ + c.csφ).cs ψ τ = ( S.sinφ + c.csφ).sin ψ (.) Equilibrium equatins fr element in Fig (a) can be stated as: σ τ τ + σ + = = f f (.) In Eqn 4., f = K h γ and f = ( K v )γ. Substituting Eqn.. in t Eqn.. and slving the resulting set f equatins, a cuple f failure directins will be fund (Eqn..3 and Eqn..4).

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China d / d = tan( ψ + µ ) (.3) d / d = tan( ψ µ ) (.4) Eqn..3 and Eqn..4 indicate psitive and negative stress characteristics directin respectively (Fig 3). As mentined earlier µ = ( π / 4) ( φ / ). Figure 3 Psitive and negative stress characteristic lines (SCL) Further result f Eqn.. and Eqn.. is that the fllwing equatins are gverning equatins alng psitive and negative characteristics lines respectively. ds + ( S tanφ + c) dψ = f (tanφ d d) + f φ φ c c ( S c tanφ) d d + d d ds ( S tanφ + c) dψ = φ φ c c ( S c tanφ) d d d d f (tanφ d + d) f (tanφ d + d) + (tanφ d d) (.6) (.5) Equatins.3 t.6 are a set f partial differential equatins and can be slved by an apprpriate numerical apprach such as finite difference methd. suppse A and B are tw arbitrary pints n the bundary f slpe (Fig 3); having quantity f variables,, S and ψ n A and B and slving fr equatins.3 t.6 the same variables will be fund n pint C inside the slpe bdy. Then pint C becmes a knwn pint and can be used t find stress state f ther pints n the bdy at limit state. In ther wrds, by applicatin f equatins f SCM alng characteristic lines and having stress bundary cnditins, stress field at limit state can be calculated. 3. SOLUTION PROCEDURE Every fundatin n slpe and in D cases is under the influence f tw slpes, ne n the right side and anther n the left side. A side which is mre likely t be run by fting is called first side and anther side is named secnd side. Mathematically fr an istrpic and hmgeneus sil mass, if an arbitrary pint M is chsen n each slpe, the ne which has the greatest algebraic value f the fllwing equatin (Eqn. 3.) is mre susceptible t failure and s is the first side. M tan ( SC) = (3.) B M

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China In Eqn. 3., B is the width f fting. Based n the assumed inclinatin angle f applied lad n fting, it is pssible t find angles α and α f rigid wedge ABC under the fting (Fig 4). The calculated α and α must satisfy the relatinα + α = π + φ. Internal angle β f wedge ABC is assumed t be frmed frm intersectin f a pair f psitive and negative slip lines, therefre its value is equal t π φ. After determinatin f internal angles f rigid wedge ABC, the crdinate f verte C, can be calculated readily. Figure 4 Frces diagram f rigid wedge under the fting. Bundary cnditins n slpes are knwn. S using equatins.3 t.6 and slving fr active and transitin nes, stresses and then tangential and nrmal frces (T, T, N, N) n sides AC and BC f rigid wedge under the fting will be emerged. Satisfying equilibrium equatins f rigid wedge alng and directins, give the values f H and V respectively. H and V must be checked t see if tan (H/V) is identical t inclinatin f assumed lad n fting. If it was the case, the slutin ends and if nt, cefficient f partial mbiliatin f strength (m) n the secnd side f fting must be changed. By try and errr, the eact value f m is fund s that tan (H/V) becmes equal t lad inclinatin n fting. Assumed inclinatin angle f lad n fting must be smaller than frictin angle between sil and fting, therwise H and V have t be determined based n the frictin angle sil-fting interface. 4. RESULTS T evaluate the effect f angle i (angle f secnd side slpe) n vertical bearing capacity f fting, certain 3 slpe and sil type ( i = 30, B = m, γ = 8 KN m, c = 40 KPa, φ = 30 ) have been assumed and calculated under varius hrintal seismic cefficients. 7 5 Kh=0 Kh=0. BthSide Mechanism SingleSide Mechanism V/Bc 3 Kh=0. Kh=0 Kh=0. 9 Kh=0. 7-30 -0-0 0 0 0 30 40 50 i (deg) Figure 5 Effect f secnd side slpe n bearing capacity f fting.

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China As depicted results n Fig 5 shw, fr all cases, increase in i, leads t decrease f vertical bearing capacity V/Bc (c is chesin f sil). Here i decreases when the algebraic value f Eqn. 3. at secnd side ges dwn. The vertical bearing capacity tends t increase, as the algebraic value f Eqn. 3. at secnd side appraches the ne f first side. Effect f hrintal seismic cefficient n bearing capacity f ftings has als been assessed. T achieve this, different Kh are applied n a fting with certain value f i and i. Results (Fig 6) indicate that by increasing Kh (tward first side), vertical bearing capacity decreases, but the rate f descending is lwer fr larger first side angle. T evaluate and cmpare the effects f bth vertical and hrintal seismic cefficients n bearing capacity f ftings, fr sme Kh s (Kh=0.05, 0., 0.5, 0., 0.5), varius Kv's are applied and prblem is slved (i and i values are cnstant). Results shw that increasing Kv cntrary t gravity directin leads t decrease in bearing capacity fr each value f Kh (Fig 7). In additin, if certain increase in Kv and Kh are applied t system, rate f decrease f bearing capacity due t Kh is larger than the ne f Kv (Fig 7). 4 36 3 6 V/Bc 6 6 0 0.05 0. 0.5 0. 0.5 0.3 0.35 0.4 0.45 Kh Figure 6 Effect f hrintal earthquake cefficients n bearing capacity 3 f fting ( i = 0, B = m, γ = 0 KN m, c = 50 KPa, φ = 30 ). i=0 0 0 30 40 50 8 7 Kh=0.05 6 Kh=0. V/Bc 5 Kh=0.5 4 Kh=0. 3 Kh=0.5 0 0.05 0. 0.5 0. 0.5 0.3 0.35 0.4 0.45 0.5 Kv Figure 7 Effect f hrintal and vertical earthquake cefficients n bearing capacity f fting. 3 ( i = 45, i = 0, B = m, γ = 0 KN m, c = 50 KPa, φ = 30 )

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China Results f present study are cmpared t thse f ther researches. Methd f Saran et al. (989) is mre similar t present study since bth studies, cnsider partial mbiliatin f strength parameters at secnd side. Results shw that in all cases, calculated Nγ f present study is smaller than that assessed by the ther researches. Results f Meyerhf (957) are clser t values estimated by the present study (Table ) and the wrst difference is with Saran et al. (989). In Table 4., De is the distance between fting crner and tp f slpe. Nc values fr varius internal frictin angles and in different first side angle (i) are fund and cmpared t Hansen slutins (Fig 8). At i=0 bth study give the same result while fr ther values f i, Hansen slutins are always greater than present study. T make precise cmparisn, the value f i is suppsed t be er fr all cases. Table 4. Cmparisn f present study t thse f ther researches (Saran, 989). φ i De/B Meyerhf (957) Miun (960) Siva Reddy and Mgaliah (975) Chen (975) Saran (989) Present Study 30 5 0 0 3.76 5.5 0.55 30 0 0 7.5 8-0.6 7.475 30 30 0 3. - 5.0-6.4.73 40 0 0 34 44-55 53.47 33.6 40 0 55 - - - 85.98 50.065 40 0 70 - - -. 63.933 40 30 0 0 - - 9.5 5.37 4.7 40 30 40 7 - - 6. 6.785 90 80 70 60 50 Present Study (fi=0) Present Study (fi=0) Present Study (fi=40) Hansen (fi=0) hansen (fi=30) Present Study (fi=0) Present Study (fi=30) Hansen (fi=0) Hansen (fi=0) Hansen (fi=40) Nc 40 30 0 0 0 0 0 0 30 40 50 60 i (deg) Figure 7 Cmparisn f Nc value f present study with Hansen slutin. ( i 0, B = m, K v = 0, K = 0) = h

The 4 th Wrld Cnference n Earthquake Engineering Octber -7, 008, Beijing, China 5. CONCLUSIONS The results f the present study can be summaried as fllws: - Increase in secnd side slpe angle, leads t decrease f static and seismic vertical bearing capacity f fting n slpes. - As the hrintal seismic cefficient in directin f first side slpe increases, regardless f first and secnd side slpe angles, vertical bearing capacity decreases. 3- Bigger vertical seismic cefficient, cntrary t gravity directin, results in smaller vertical bearing capacity f fting. 4- Partial mbiliatin f strength parameters r m clses t unity as first and secnd side slpes tend t be symmetric with respect t ais. 5- Bearing capacity f fting resulted frm this study is always smaller than thse frm ther studies. This fact is due t partial mbiliatin f strength parameter in the secnd side f fting. REFERENCES Budhu, M. and Al-Karni, A. (993). Seismic bearing capacity f sils. Ge technique 43:, 8 87. C, A. D., Easn, G. and Hpkins, H. G. (96). Aially symmetric plastic defrmatin in sils. Philsphical Hansen, J. B. (970). A revised and etended frmula fr bearing capacity. Geteknisk Inst., Bull 8, 5. Kumar, J., and Kumar, N. (003). Seismic bearing capacity f rugh ftings n slpes using limit equilibrium. Getechnique 53:3, 363 369. Meyerhf, G. G. (957). The ultimate bearing capacity f fundatins n slpes. Prc. 4th Int. Cnf. Sil Mech. Fund. Engng, Lndn, 384 386. Meyerhf, G. G. (963). Sme recent research n the bearing capacity f fundatins. Can. Getech. J :, 6 6. Narita, K. and Yamacuchi, H. (990). Bearing capacity analysis f fundatins n slpes by use f lg-spiral sliding surfaces 30:3, 44-5. Saran, S., Sud, V. K., and Handa, S. C. (989). Bearing capacity f ftings adjacent t slpes. J. Getech. Eng 5:4, 553 573. Siva Reddy, A. and Mgaliah, G., (975). Bearing Capacity f Shallw Fundatins n Slpes. Indian Getechnical Jurnal, 5:4, pp. 37-53. Sklvski, V. V. (960). Statics f sil media. Lndn, Butterwrth Scientific Publicatins. Transactin f Ryal Sciety f Lndn, Series A, 54, -45. Vesic, A. S. (973). Analysis f ultimate lads f shallw fundatins. J. Sil Mech. Fund. Div., ASCE 99:, 45 73. Zhu, D. (000). The least upper-bund slutins fr bearing capacity factr Ng. Sils Fund 40:, 3 9.