SEISMIC PASSIVE EARTH PRESSURE WITH VARYING SHEAR MODULUS: PSEUDO-DYNAMIC APPROACH

Transcription

1 IGC 29, Guntur, INDIA SEISMIC PASSIE EART PRESSURE WIT ARYING SEAR MODULUS: PSEUDO-DYNAMIC APPROAC P. Gos Assistant Professor, Deartment of Ciil Engineering, Indian Institute of Tecnology Kanur, Kanur 2816, India. S. Kolatayar PG Student, Deartment of Ciil Engineering, Indian Institute of Tecnology Kanur, Kanur 2816, India. ABSTRACT: Knowledge of seismic assie eart ressure is ery muc imortant to design te retaining wall in te eartquake rone region. Using seudo-dynamic aroac, a limited number of inestigations ae been rformed to obtain te seismic assie eart ressures considering a constant magnitude of sear modulus trougout te backfill. Truly saking, te sear modulus and tus sear and rimary wae elocities in te soil medium ary wit te det. oweer, no significant attention as been roided by te researcers towards te determination of seismic assie eart ressure beind a non ertical cantileer retaining wall wit arying sear modulus trougout te backfill. Using seudo-dynamic aroac, tis ar resents a study on te seismic assie eart ressure beind a nonertical rigid cantileer retaining wall by considering time and ase difference occeleration in te backfill; and te ariation of sear modulus wit det. Te results are roided in tabular and graical form. 1. INTRODUCTION Te determination of assie resistance on a retaining wall, under bot static and seismic conditions, is ery muc essential as te damage of suc eart retaining structures may lead to significant loss of life and wealt. Seeral inestigations ae been rformed by different researcers to determine te assie eart ressure on a rigid retaining wall under seismic condition. Te early work on eartquake induced lateral eart ressure acting on a retaining wall under bot actie and assie conditions was reorted by Okabe (1926, and Mononobe & Matsuo (1929 using seudo-static aroac. Tis analysis was later recognied as well known Mononobe-Okabe metod (Kramer 1996 to comute te seismic eart ressure. In te seudo-static analysis, te dynamic load induced by an eartquake is considered as time-indendent, wic eentually assumes tat te magnitude and ase occeleration are uniform trougout te backfill. Aart from tis, seudo-static analysis does not consider te ariation of sear modulus trougout te backfill. To oercome tis constraint, Steedman & Zeng (199 used a seudo-dynamic aroac to redict te seismic actie eart ressure beind a ertical cantileer retaining wall. In seudo-dynamic analysis, te time and ase difference due to finite sear wae elocity were considered. Steedman & Zeng (199 considered te effect of non-uniform sear modulus distribution and amlification occeleration on te magnitude octie eart ressure beind a ertical wall. Coudury & Nimbalkar (25 and Gos (27 also used seudo-dynamic aroac to redict te seismic assie eart ressure beind a cantileer retaining wall. In general, te sear modulus and tus te sear and rimary wae elocities in te ground ary wit det wic affects te ase occeleration. oweer, using seudo-dynamic aroac, te seismic assie eart ressure beind a nonertical cantileer retaining wall by considering te influence of arying sear modulus in te backfill, as not drawn muc attention from te researcers. Te resent study exlores te effects of soil friction angle φ, wall inclination θ, wall friction angle δ, oriontal eartquake acceleration coefficient α, ertical eartquake acceleration coefficient α, amlification factor, sear wae elocity s, rimary wae elocity and det exonent causing sear modulus ariation β, on te seismic assie eart ressure using seudo-dynamic aroac. Te limit equilibrium metod, wit a lanar failure surface beind te retaining wall, as been considered to comute te assie resistance of te wall. 2. DEFINITION OF TE PROBLEM A rigid nonertical cantileer retaining wall of eigt is laced wit a dry, coesionless, oriontal backfill as sown in Figure 1. Te wall face (AB on te backfill side is inclined at an angle θ wit te ertical and as a wall friction angle δ. Te objectie is to determine te assie eart 522

2 ressure coefficient and distribution by knowing te assie resistance P r unit lengt of te wall in te resence o sinusoidal base saking subjected to linearly arying oriontal and ertical accelerations wit amlitudes of ( ( f 1 + a 1 α g and ( ( 1 + fa 1 α g, resctiely, were is any det below te ground surface and g is te acceleration due to graity. Te arameters sown in Figure 1 are considered as ositie and te unit weigt of te soil is taken as γ. P δ Fig. 1: Failure Mecanism and Associated Forces 3. ANALYSIS A Q W θ B d α Q A lanar failure surface (BC in Fig. 1 at an angle α, wit te oriontal as been considered in tis analysis. Te assie trust, P makes an angle δ, wit te normal to te wall face AB. ere, te failure mecanism as been soled using seudo-dynamic analysis to comute te assie resistance. Te seudo-dynamic analysis, wic considers finite sear and rimary wae elocities, can be deelod by taking into account te ariation of sear modulus and amlification of acceleration in te backfill and tus bot ase and magnitude of te accelerations ary. Te resent analysis considers bot sear ( s and rimary wae elocity ( acting witin te backfill during eartquake in te direction as sown in Figure 1. Te analysis includes a riod of lateral saking T, wic can be exressed as, 2π T (1 ω were, ω is te angular frequency. In ractice, bot sear and rimary wae elocities ary wit det as a result of non-uniform sear modulus distribution in te backfill, generally in sands, and te ariation of sear modulus wit det can be exressed as, G G β β 1 (2 φ R s, C a α g a α g a α g a α g were G is a constant and is te det below te ground surface. Te sear wae elocity may ten be deduced as a function of det, s 1/ 2 β / 2 G ρ (3 were, ρ is te mass density of backfill. Primary wae elocity can be calculated as 1.87 s wic is alid for most of te geological materials (Das, Te time increment for te assage o sear wae from te base to a det will ten be Δt s ( d s ( ( ρ G 1/ 2 / ( ( 1 / 2 1 / 2 β β 1 β / 2 Similarly, te time increment for te assage o rimary wae from te base to a det will be, Δ t ( d ( ( ρ / G ( ( 1 β /2 1 β/2 β / 2 oweer, as te sear modulus aries wit det, it is necessary to define an aerage magnitude of sear and rimary wae elocities as. β /2 G sag (1 β / 2 ts ( ρ Δ, ag 1.87 Te ariation occeleration troug a soil layer wit arying sear modulus also dends on different factors suc as daming and te interaction of reflected, refracted and surface waes in te icinity of structure. For a sinusoidal base saking, te oriontal and ertical accelerations at any det below te ground surface and time t can be exressed as, (, 1 ( 1 ( α + a t g f a (, 1 ( 1 ( α + a t g f a ( sag ρ / G β t 1 β /2 ( ( 1 /2 1 β /2 ( ρ / G β t β / 2 ( ( 1 /2 1 β /2 Te mass of te small saded art of tickness d (Fig. 1 is gien by, γ ( ( 1+ tanθ m( d (9 g Te total weigt of te failure wedge W is gien by, 2 ( 1 tan tan W γ + α θ (1 2 Te oriontal inertia force exerted on te small element due to oriontal eartquake acceleration can be exressed as m(α (,t. Terefore, te total oriontal inertia force Q (t acting in te failure wedge is gien by te integral Q ( t m a, t d (11 ( ( (4 (5 (6 (7 (8 523

3 Similarly, te total ertical inertia force Q (t acting in te failure wedge is gien by, Q ( t m a, t d (12 ( ( For alues of β in te range of interest, Equations (11 and (12 can be soled analytically only for and 1 and must be soled numerically for intermediate alues of β. Te total assie resistance P can ten be determined by taking te oriontal as well as te ertical force equilibrium of te failure wedge and is gien by, P ( t W cos sin( α + φ ( α + φ + δ θ ( cos( α + φ + ( sin( α + φ cos( α + φ + δ θ Q t Q t (13 Te first term on te rigt and side gies te static assie resistance; wereas te last term redicts te dynamic assie resistance due to eartquake loading. Te direction of bot oriontal and ertical inertia forces as sown in Figure 1 causes te most critical effect on te wall under seismic condition. Te seismic assie eart ressure coefficient can ten be defined as, K ( t 2P 2 (14 γ It can be obsered tat K is a function of α, t/t, /T s, /T and β. /T s is te ratio of time taken by te sear wae elocity to trael te full eigt to te riod of lateral saking T and /T is te ratio of time taken by te rimary wae to trael te full eigt to T. Te otimiation as been done wit resct to α and t/t to get te minimum alue of K. During otimiation, te alues of α and t/t ae been aried in te ranges of 9 and 1 resctiely. Te assie eart ressure distribution beind te wall can be determined by taking artial deriatie of P wit resct to and exressed as, (, t P (, t γ sin( α + φ ( 1+ tanθ cos α φ δ θ αγ + + a ( 1 tanθ 1 ( f 1 cos( α + φ t s cos ( α + φ+ δ θ αγ + + a ( 1 tanθ 1 ( f 1 sin( α + φ t cos + + ( α φ δ θ ( + + (15 Te first term on te rigt and side gies te assie eart ressure under static condition and te second and tird terms reresent te dynamic assie eart ressure due to oriontal and ertical eartquake accelerations resctiely, under te influence omlification of excitation and ariation of sear modulus wit det. ere, te distribution of assie eart ressure is found to be nonlinear in nature wic is not te case in seudo-static analysis. 4. RESULTS Te comutations ae been rformed by writing a comuter code in MATLAB. To find te minimum alue of K, te magnitudes of α and t/t ae been aried indendently at interals of.1 and.1 resctiely. 4.1 Seismic Passie Eart Pressure Coefficient Te ariations of seismic assie eart ressure coefficient K wit canges in α for different alues of θ and are resented in Figure 2 for φ 3, δ.5φ, α.5α,.5, /T s.3 and /T.16. It can be seen tat te magnitude of assie eart ressure coefficient decreases continuously wit an increase in te magnitude of α. It can also be obsered tat te alue of K decreases wit increase in wall inclination θ. Te lowermost line in Figure 2 is for θ 1 ; wereas te urmost line is for θ 1. It as been seen tat iger te amlification factor, te lesser is te magnitude of K. (a (b K K θ (lower most α θ (lower most α Fig. 2: ariation of Passie Pressure Coefficient K wit α for φ 3, δ.5φ, α.5 α,.5, /T s.3 and /T.16. (a 1. (b

4 4.2 Seismic Passie Eart Pressure Distribution Te normalied assie eart ressure distribution is sown in Figure 3 for different alues omlification factor wit φ 3, δ.5φ, θ 1, α.2, α.5α,.5, /T s.3 and /T.16. It can be obsered from te figure tat te alue of assie eart ressure decreases marginally wit increase in te magnitude of and te difference in ressure becomes a maximum at te base of wall for different alues of Te ariation of /γ wit canges in / for different alues of φ and δ wit θ 1, α.2, α.5α,.5, /T s.3, /T.16 and 1.4 is resented in Figure 4. It can be seen from te figure tat te magnitude of assie eart ressure increases wit increase in te alue of φ; and for a articular alue of φ, te magnitude of assie eart ressure increases wit an increase in te alue of δ. / /γ Fig. 3: Normalied Passie Eart Pressure Distribution for Different alues of fa (φ 3, δ.5φ, θ 1, α.2, α.5α,.5, /T s.3 and /T.16 / δ δ.5φ δ φ /γ Fig. 4: Normalied Passie Eart Pressure Distribution for Different alues of φ and δ (θ 1, α.2, α.5α,.5, /T s.3, /T.16 and COMPARISON For nonertical retaining wall, te resent alues of K can only be comared wit te existing alues roosed by arious researcers for, as te alues of K for β > are scarce in te literature. For, te obtained alues are exactly same as tose obtained by Gos (27 for different alues of θ. oweer, te comarison of te resent alues of K for different alues of β is sown in Table 1 for a range of arious arameters. It as been seen tat tere is no significant difference in te alues of K wit cange in β for low alues of /T s. Te same obseration was also made by Steedman & Zeng (199 for ertical wall under actie case. Table 1: alues of Passie Eart Pressure Coefficient K for φ 3, δ.5φ, α.2, α.5α, 1.4 K /T s /T θ CONCLUSIONS Using te seudo-dynamic aroac, te effects of te soil friction angle, wall inclination, wall friction angle, oriontal and ertical eartquake acceleration, amlification of ibration, sear and rimary wae elocity wit teir ariation along te det of te backfill on te seismic assie eart ressure beind a nonertical cantileer retaining wall ae been exlored. It as been found tat te magnitude of seismic assie eart ressure decreases wit an increase in te wall inclination θ, amlification factor and te sesmic acceleration coefficients α and α. Te assie eart ressure sows little ariation wit te cange in magnitude of te det exonent β. Te nonlinearity of seismic assie eart ressure distribution increases wit an increase in seismicity, wic causes te oint olication of te total assie trust to be sifted. Due to te use of more rational seudo-dynamic aroac oer te seudo-static aroac, te resent alues of K and attern of assie eart ressure distribution could be used by te design engineers to design and assess te beaiour of nonertical cantileer retaining wall under seismic condition. 525

5 REFERENCES Coudary D. and Nimbalkar S. (25. Seismic Passie Resistance by Pseudo-dynamic Metod, Geotecnique, 55(9: Das B.M. (1993. Princiles of Soil Dynamics, PWS- KENT Publising Comany, Massacusetts. Okabe S. (1926. General Teory of Eart Pressure, Journal of te Jaanese society of Ciil Engineers, 12(1: 311. Gos P. (27. Seismic Passie Eart Pressure Beind Nonertical Retaing Wall Using Pseudo-Dynamic Analysis, Geotecnical and Geological Engineering, 25: Kramer S.L. (1996. Geotecnical Eartquake Engineering, Prentice all, Englewood Cliffs, N.J. Mononobe, N. and Matsuo. (1929. On te Determination of Eart Pressure during Eartquakes, Proceedings of te World Engineering Conference, ol. 9, Steedman R.S. and Zeng X. (199. Te Influence of Pase on te Calculation of Pseudo-static Eart Pressure on a Retaining Wall, Geotecnique, 4: