DETERMINATION OF POINT OF APPLICATION OF SEISMIC ACTIVE THRUST ON RETAINING WALL

Transcription

1 4 th International Conference on Earthquake Geotechnical Engineering June 5-8, 007 Paer No. 149 DETERMINATION OF POINT OF APPLICATION OF SEISMIC ACTIVE TRUST ON RETAINING WALL Deeankar COUDURY 1, Sanjay NIMBALKAR ABSTRACT Deign of retaining wall need the comlete knowledge octive earth reure. Under earthquake condition, the deign require ecial attention to reduce the devatating effect of thi natural hazard. But under eimic condition, mot of the available literature tarting from the ioneering work of Mononobe-Okabe (196, 199) [ee Kramer, 1996] to the latet work of Choudhury and Singh (006) give the eudo-tatic analytical value of the active earth reure a an aroximate olution to the real dynamic nature of the comlex roblem. In thi eudo-tatic method, only the magnitude of the eak ground acceleration at a articular intant of time are aumed to act with the inertia comonent of the oil ma along with other tatic force. ence, the total active thrut obtained by thi analyi i analogou to the tatic active thrut analyi with different magnitude, and the variation of earth reure along the height of the retaining wall remain linear though the comutation of correct oint olication of total eimic active thrut i imortant for the deign uroe of the wall. But there i no coe to find out the oint olication of eimic active thrut by eudo-tatic aroach but to aume it to act at one-third height from the bae of the wall. Moreover, in thi eudo-tatic method, neither the time deendent effect olied earthquake load nor the effect of hear and rimary wave aing through the oil media are conidered. Correcting thee error, in recently develoed eudodynamic method onalyi, all thee variation are conidered to comute eimic active earth reure [Steedman and Zeng, 1990; Choudhury et al., 006; Choudhury and Nimbalkar, 006]. But the comlete olution to comute the oint olication of eimic active thrut which i very imotant for deign of the wall i till carce. In thi aer, a comlete cloed-form olution for comuting the oint olication of eimic active thrut uing limit equilibrium method onalyi with eudo-dynamic aroach i adoted. In the reent work the effect of variation orameter like oil friction angle, wall friction angle, time eriod of earthquake ground motion, eimic hear and rimary wave velocitie of backfill oil, oil amlification and eimic eak horizontal and vertical ground acceleration on the eimic active earth reure are tudied. A exected, the reent reult how that with eimicity, the oint olication of eimic active thrut varie a lot, comared to a contant value mentioned in eudo-tatic method onalyi. It i found that the oint olication of eimic active thrut i above one-third from the bae of the wall comared to the tatic value of one-third from the bae of the wall. Alo thi location hift away from the bae of the wall with increae in interfacial oil-wall friction. It i alo found that the reent reult give higher deign value of the eimic active earth reure coefficient comared to the conventional euo-tatic Mononobe-Okabe value. The non-linearity of the eimic active earth reure ditribution increae with eimicity comared to a linear eudo-tatic eimic active earth reure ditribution. Alo the reent reult obtained by cloed-form olution are comared and found to match well with a very few available dynamic centrifuge tet reult [Steedman and Zeng, 1991] in literature. Keyword: eudo-dynamic, oil-wall interface, body wave, amlification, time eriod 1 Aitant Profeor, Deartment of Civil Engineering, Indian Intitute of Technology Bombay, Powai, Mumbai , INDIA. dc@civil.iitb.ac.in Reearch Scholar, Deartment of Civil Engineering, Indian Intitute of Technology Bombay, Mumbai, INDIA.

2 INTRODUCTION Evaluation of eimic active earth reure i eential for the afe deign of rigid retaining wall in the eimic zone. In the at, many reearcher have develoed everal method to determine the eimic active earth reure on a rigid retaining wall. The ioneering work on earthquake-induced lateral active earth reure acting on a retaining wall were reorted by Okabe, 196 and Mononobe and Matuo, 199. Thi eudo-tatic aroach following the Coulomb tatic earth reure analyi i known a Mononobe-Okabe method (ee Kramer, 1996) to comute eimic earth reure. Recent work of Richard et al. 1999, Saran and Guta, 003, Choudhury and Singh, 006 and few other alo conidered the eudo-tatic method to comute eimic active earth reure behind a retaining wall. Green and Ebeling, 00, 003, Green et al., 003 had erformed numerical analyi uing by uing geotechnical oftware FLAC. owever regarding the ditribution of dynamic active earth reure with deth, many author have commented favorably on the magnitude of the Mononobe- Okabe lateral earth force but have diagreed on the oint olication, which indicate the uncertainty over the ditribution of eimic active earth reure with deth. There i no jutification in the extended Coulomb aroach adoted by Mononobe-Okabe for the eimic cae by auming the linear active earth reure ditribution with deth. To overcome thi drawback, the time and hae difference due to finite hear wave roagation behind a retaining wall wa conidered uing a imle and more realitic way of eudo-dynamic method, rooed by Steedman and Zeng, Again Steedman and Zeng, 1991 comared the theoretical reult with the centrifuge model tet reult to validate the eudo-dynamic method. Steedman and Zeng, 1990 conidered in their analyi a vertical rigid retaining wall uorting horizontal backfill for a articular value of oil friction angle (φ) and a articular value of eimic horizontal acceleration (k h g, where g i the acceleration due to gravity) only. But the effect of variou arameter uch a wall friction angle (δ), oil friction angle (φ), time eriod of earthquake ground motion, eimic hear and rimary wave velocitie of backfill oil (V and V ), oil amlification factor (f) and eimic eak horizontal and vertical ground acceleration (k h g and k v g) on the eimic active earth reure behind a rigid retaining wall by the eudo-dynamic method didn t get any attention till today. ence in thi aer, a comlete cloed-form olution for comuting the oint of alication of eimic active thrut uing limit equilibrium method onalyi with eudo-dynamic aroach i adoted. METOD OF ANALYSIS Similar to the eudo-dynamic aroach which conider finite hear wave velocity within the backfill material a rooed by Steedman and Zeng, 1990, here alo it i aumed that the hear modulu (G) i contant with the deth of retaining wall throughout the backfill. Conider the fixed bae vertical rigid retaining wall AB of height a hown in Figure 1. The wall i uorting a coheionle backfill material with horizontal ground. In the reent tudy, both the hear wave velocity, V = (G/ρ) 1/, where, ρ i the denity of the backfill material and rimary wave velocity, V = (G(-ν)/ρ(1-ν)) 1/, where ν i the oion ratio of the backfill are aumed to act within the oil media due to earthquake loading. In the reent analyi, the ueroition of the hear and rimary wave i unlikely becaue the oil media i conidered a a homogeneou, emi-infinite layer intead o bounded layer. ence for uch an unbounded homogeneou oil layer, it i unlikely to form the tanding wave from the ueroition of the hear and rimary wave in uch continuou media (Kramer, 1996). For mot geological material, V /V = 1.87 (Da, 1993). The eriod of lateral haking, T = π/ω (Kramer, 1996), where ω i the angular frequency i conidered in the analyi. A laner ruture urface inclined at an angle, α with the horizontal i conidered in the analyi.

3 A C a h = f k h g z Q ha Q va W a z h δ P ae CB dz α φ F a V, V a h = k h g Figure 1. Model retaining wall conidered for comutation of eimic active earth reure. Both the body wave (i.e. hear and rimary wave) roagating through the backfill oil are conidered to be influenced by the oil amlification and act within the backfill oil due to the earthquake loading. The bae of the wall i aumed to be ubjected to the harmonic horizontal eimic acceleration with amlitude a h (= k h g, where g i the acceleration due to gravity) and the harmonic vertical eimic acceleration with amlitude a v (= k v g). The exact nature of the oil amlification i deendent on everal factor, uch a, the geometry and rigidity of the adjacent tructure, the tiffne and daming of the oil, the deth of the oil layer and o on. Again, imilar to the aumtion made by Steedman and Zeng, 1990, here alo the linear variation of the horizontal and vertical eimic acceleration from the inut acceleration at the bae to the higher value (deending uon the oil amlification) at the to of the retaining wall, uch that k h(z = ) = f k h(z = 0) and k v(z = ) = f k v(z = 0), where f i a contant and i termed a amlification factor, i conidered. To obtain the critical deign value of the eimic earth reure, it i aumed that both the horizontal and vertical vibration tart at exactly the ame time without any hae hift between thee two vibration, which i the wort oible combination of earthquake loading. Referring to Figure 1, the horizontal and the vertical acceleration at any deth z and time t, below the to of the wall can be exreed a, z ( z) ah(,) z t = 1+ ( f 1 ) kh..in g w t V (1) z ( z) av( z, t) = 1+ ( f 1 ) kv. g.inw t V () The ma o thin element of the wedge at deth z (Figure 1) i, mz ( ) γ z = dz (3) gtanα where, γ i the unit weight of the backfill. The total horizontal inertia force acting within the failure zone can be exreed a,

4 h Q () t = m(z)a (z, t)dz (4) 0 h TV γ kh Qh( t) = [ πcow ζ + TV(in wζ in wt) ] 4π tanα TV γ kh( fa 1) + 3 π ( π cow ζ +TV inw ζ ) + ( TV ) (co wt co wζ ) 4π tanα (5) where, λ = TV i the wavelength of the vertically roagating hear wave and ζ = t-/v. Now, the total vertical inertia force acting within the failure zone can be exreed a, Q ( t ) = m(z)a (z, t)dz (6) v 0 v TV γ kv Qv() t = πco ωψ TV (in in ) ωψ ωt 4π tanα + TV γ k ( f 1) 4π tanα v + π 3 ( π coωψ + TVin ωψ ) + ( TV) (coωt co wψ ) (7) where, η = TV, i the wavelength of the vertically roagating rimary wave and ψ = t /V. Though the horizontal acceleration i acting from left to right and vice-vera and the vertical acceleration i acting from to to bottom and vice-vera, only the critical combination of direction of Q h (t) and Q v (t) are conidered, which give the maximum eimic active earth reure for the deign of the wall. The ecial cae o rigid wedge i given, in the limit a γ a h a h lim ( Q h ) max = = W = k v gtanα g h W (8) γ av av lim ( Qv ) max = = = k v vp gtanα W W (9) g which i equivalent to the eudo tatic force aumed in the Mononobe-Okabe method. The total (tatic and eimic) active thrut, P ae (t) can be obtained by reolving the force on the wedge a hown in Figure 1) and conidering the equilibrium of the force. ence P ae (t) can be exreed a follow, Win( α φ) + Qh( t)co( α φ) + Qv( t)in( α φ) Pae() t = co( δ+ φ α) (10) where, W i the weight of the failure zone, φ i the oil friction angle and δ i the wall friction angle. The eimic active earth reure coefficient, K ae i defined a,

5 K ae P ae = (11) γ Subtituting for Q h (t) and Q v (t) in the equation (8), an exreion for K ae in term of Q h (t), Q v (t) and W can be derived a, K ae ( ) k ( TV ) ( ) k TV h S v + m+ ( ) ( ) ( ) ( ) 1 in α φ co α φ in α φ = m 1 tanα co δ+ φ α π tanα co δ+ φ α π tanα co δ+ φ α ( ) k ( f 1) TV co ( α φ) k ( f 1) TV in ( α φ) h S v + m π tanα co ( δ+ φ α) π tanα co ( δ+ φ α) m 4 (1) where, ( TV ) ( ) t t t S m = π coπ + inπ inπ 1 T TV T TV T t TV t t m = π co π + in π in π T TV T TV T ( ) ( ) t TV t TV t t S S m = π π co π in π + coπ co 3 + T TV T TV π T T TV t TV t TV t t m = π πcoπ in π + coπ co 4 + T TV T TV π T T TV From equation (1), it i een that K ae i function of the dimenionle arameter /TV, /TV, t/t, f and the wedge angle α. The maximum value of K ae i obtained by otimizing K ae with reect to t/t and α. It i found that K ae i a function of /TV, /TV, which i the ratio of time for a hear wave and rimary wave to travel the full height of the wall to the eriod of lateral haking and amlification factor f. Uing excel readheet tool SOLVER, otimization of K ae i done. Total eimic active thrut can alo be defined a, Pae = Pa + Pahd + Pavd (13) where, P a i the reure acting on the retaining wall due to vertical weight of the wedge, P ahd i the reure acting on the wall due to horizontal inertia of the wedge and P avd i the reure acting on the wall due to vertical inertia of the wedge.

6 And the eimic active earth reure ditribution can be obtained by differentiating the total active thrut with reect to the deth of the wall a given by, ae Pae() t ( t) = (14) z khγ z co( α φ) z γ z kvγ z z in( α φ) ae() t = f in w t + + f in w t tanα co( δ + φ α) V tanα tanα V co( δ + φ α) k co( ) ( TV ) hγ α φ TV z TV z z + ( f 1) cow t inw t + cow t co tanα co( δ + φ α) π V πz V π z wt (15) V in( ) TV k ( TV ) v z TV γ α φ z z + ( f 1) co w t in w t + co w t co wt tanα co( δ + φ α) π V πz V π z V Equation (1) and (15) for the eimic active cae of earth reure match exactly with thoe obtained by Choudhury and Nimbalkar, 006 for a ecific cae of f = 1.0, and that of Steedman and Zeng, 1990 for a ecific cae of k v = 0. The acting oint of P ad, d above the bae can be found by taking moment about the bae of the wall. Then, if M d i the dynamic comonent of the bending moment d M d( z = ) ad co δ ( z) dz = P coδ = (16) P coδ ad 0 ad where, m = λ k h co(α-φ) and n = η k v in(α-φ). Thu, d i the function of /TV, /TV, nd t. RESULTS AND DISCUSSIONS In the cae of coheionle oil, to avoid the henomenon of hear fluidization (i.e. the latic flow of the material at a finite effective tre) for the certain combination of k h and k v (Richard et al., 1990) the value of φ conidered in the analyi are to atify the relationhi given by, φ >tan 1 k h 1 kv (17) Reult are reented in the tabular and grahical form for normalized eimic active earth reure along the normalized deth of the wall. Variation orameter conidered i a follow: φ = 0 0, 30 0, 35 0 and 40 0 δ = -0.5φ, 0,0.5φ and φ. k h = 0.0, 0.1, 0., 0.3. k v = 0.0k h, 0.5k h and k h. f = 1.0, 1., 1.4, 1.6, 1.8 and.0 Seimic active earth reure coefficient (K ae ) Figure how the effect of oil amlification on the eimic active earth reure coefficient (K ae ) for different value of k h with k v = 0.5k h, φ = 35 0, δ = φ/, /TV = 0.3, /TV = From the lot, it may be een that the eimic active earth reure coefficient (K ae ) increae with increae in oil amlification factor and the rate of increae i more for higher value of k h. For examle, when k h

7 = 0.3, eimic active earth reure coefficient, K ae increae by, 0.1% when change from 1.0 to 1., 16.7% when change from 1. to 1.4, 14.3% when change from 1.4 to 1.6, 1.53% when change from 1.6 to 1.8, and 11.13% when change from 1.8 to.0. Thu, the reent tudy reveal the ignificant influence of oil amlification on the eimic active earth reure coefficient. Normalied eimic active earth reure ditribution Figure 3 how the effect of oil amlification on normalized ditribution of the eimic active earth reure with k h = 0.3, k v = 0.15, φ = 35 0, δ = φ/, /TV = 0.3, /TV = Seimic active earth reure how marginal increae near the to of the retaining wall and relatively more increae at the bottom of the wall with the increae in oil amlification factor,. At the bottom of the retaining wall, the eimic active earth reure increae by about 5.79 % when change from 1.0 to 1., 5.47 % when change from 1. to 1.4, 5.18 % when change from 1.4 to 1.6, 4.93 % when change from 1.6 to 1.8, and 4.7 % when change from 1.8 to.0. While at the mid-height of the wall, eimic active earth reure increae by about 6.43 % when change from 1.0 to 1., 6.04 % when change from 1. to 1.4, 5.7 % when change from 1.4 to 1.6, 5.4 % when change from 1.6 to 1.8, and 5.1 % when change from 1.8 to.0. It alo how the non-linear ditribution of the eimic active earth reure under different oil amlification factor. K ae k v = 0.5k h, φ = 35 0, δ = φ/, /TV = 0.3, /TV = 0.16 = 1.0 = 1. = 1.4 = 1.6 = 1.8 = k h Figure. Effect of oil amlification on eimic active earth reure coefficient (K ae ) for different value of k h with k v = 0.5k h, φ = 30 0, δ = φ/, /TV = 0.3, /TV = 0.16.

8 k h = 0.3, k v = 0.15, φ = 35 0, δ = φ/, /TV = 0.3, /TV = 0.16 z/ = 1.0 = 1. = 1.4 = 1.6 = 1.8 = ae /γ Figure 3. Effect of oil amlification on normalized eimic active earth reure ditribution with k h = 0.3, k v = 0.15, φ = 30 0, δ = φ/, /TV = 0.3, /TV = Point of location of dynamic thrut increment ( d ) The value of oint of location of dynamic active thrut increment ( d ) are given in Table 1 for different value of k h, k v, δ and φ. It i evident from Table 1 that, the magnitude of the oint of location of dynamic active thrut increment ( d ) increae with the increae in both the horizontal and vertical eimic acceleration. For examle, when k h change from 0.0 to 0., for the cae of δ = 0 0, φ = 30 0 and k v = 0.0, the value of d i increaed by 0.7 %. Again for the cae of δ = 15 0, φ = 30 0 and k v = 0.5k h, the value of d i increaed by 5.8 %. Table 1. Point of location of dynamic active thrut increment ( d ) for different value of k h, k v,, δ and φ with /λ = 0.3, /η = 0.16, f = 1. k h φ δ k v k v k v k v (- indicate that reult are not available due to nonconvergence of the olution)

9 COMPARISON OF RESULTS Table how the comarion of the reult of oint olication of the total eimic active/aive thrut with the already ublihed reearch (Mononobe-Okabe, 196, 199, Steedman and Zeng, 1990, Richard and Elm, 1979) and the current available deign recommendation in code (IS , Eurocode ). From Table, it i clear that the reult of oint olication of the total eimic active/aive thrut, comuted by the eudo-dynamic method comare well with that obtained by Steedman and Zeng, 1990 for the active cae due to imilarity of the method and the difference i attributed to the conideration of the vertical eimic acceleration in the reent tudy. owever, the other value are obtained by uing eudo-tatic method with different theorie which conider only linear ditribution of the eimic earth reure. ence the reent olution give a logical formulation of the roblem and reentation of the reult which i required for the deign uroe. Table. Comarion of oint olication of total active thrut obtained by the reent tudy with available method for = 10 m, φ = 34 0, δ = 17 0, γ = 17.3 kn/m 3, k h = 0.3, k v = 0.3, f = 1.0 Method Point olication of total active thrut (h) Mononobe-Okabe (196,199) Steedman and Zeng (1990) Richard and Elm (1979) IS 1893 (1984) Eurocode 8 (1998) Preent tudy CONCLUSIONS In the reent analyi, eudo-dynamic method which conider the time, hae change and amlification effect in body wave viz. hear and rimary wave roagating through the backfill of the retaining wall, i alied to determine eimic active earth reure coefficient, earth reure ditribution and the oint of location of eimic active thrut. Effect of variou arameter uch a wall friction angle (δ), oil friction angle (φ), hear wave velocity (V ), rimary wave velocity (V ), both the horizontal and vertical eimic acceleration (k h g and k v g) and amlification factor (f) on eimic earth reure behind rigid retaining wall ha been tudied. Non-linearity of the eimic earth reure ditribution increae with eimicity, which lead to the hifting of the oint olication of total active/aive thrut required for the deign uroe. But the conventional eudo-tatic aroach give only linear earth reure ditribution irreective of tatic and eimic condition leading to a fixed oint olication i.e. 1/3 rd from the bae of the wall, which i a major drawback in the analyi. Again, comarion of oint olication of total active thrut i done with available method in the literature. REFERENCES Choudhury, D., Nimbalkar, SS. and Mandal, JN. Influence of oil-wall interface friction on eudodynamic earth reure, Proc. of 8th U.S. National Conference on Earthquake Engineering (8NCEE), Aril 18-, 006, San Francico, CA, USA. Choudhury, D. and Nimbalkar, SS. Peudo-dynamic aroach of eimic active earth reure behind retaining wall, Geotechnical and Geological Engineering, Sringer, The Netherland, Vol. 4, No. 5, , 006. Choudhury, D. and Singh, S., New aroach for determination of tatic and eimic active earth reure, Geotechnical and Geological Engineering, Sringer, The Netherland, Vol. 4, No. 1, , 006.

10 Choudhury, D., Sitharam, TG and Subba Rao, KS. "Seimic deign of earth retaining tructure and foundation", Current Science, India, Vol. 87, No. 10, , 004 Eurocode 8. Deign roviion for earthquake reitance of tructure, Green, RA, and Ebeling, RM. Seimic analyi of cantilever retaining wall, US Army Technical Reort, ERDC/ITL-TR-0-3, 00. Green, RA and Ebeling, RM. Modeling the dynamic reone of cantilever earth-retaining wall uing FLAC, Proceeding of the 3rd International Symoium on FLAC and FLAC3D: Numerical Modeling in Geomechanic, October 003, Sudbury, ON, Canada, Green, RA, Olgun, CG, Ebeling, RM and Cameron, WI., Seimically induced lateral earth reure on a cantilever retaining wall, Earthquake Engineering, ASCE, 133, , 003. IS Indian Standard Criteria for Earthquake Reitant Deign of Structure, Part V fourth reviion, Kramer, SL. Geotechnical Earthquake Engineering, New Jerey, Prentice all, Richard, R. Jr., and Elm, DG. Seimic behavior of gravity retaining wall. J. Geotech. Engrg., ASCE, Vol. 105(4), , Richard, R., Jr., Elm, DG. and Badhu, M., Dynamic fluidization of oil. J. Geotech. and Geoenvir. Engrg., ASCE, Vol. 116(5), , Richard, R. Jr., unag, C., and Fihman, KL. Seimic earth reure on retaining tructure. J. Geotech. And Geoenvir. Engrg., ASCE, Vol. 15(9), , Saran, S. and Guta, RP. Seimic earth reure behind retaining wall. Indian Geotechnical Journal, Vol. 33, No. 3, , 003. Steedman, RS. and Zeng, X. The influence of hae on the calculation of eudo-tatic earth reure on a retaining wall, Geotechnique, Vol. 40, No. 1, , Steedman, RS. and Zeng, X. Centrifuge modeling of the effect of earthquake on free cantilever wall, Centrifuge, 91, Balkema, Rotterdam, The Netherland, , 1991.