The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China Evaluation of Seismic Earth Pressures at the Passive Side Fei SONG 1 and Jian-Min ZHANG 1 PhD. Candidate, Institute of Geotechnical Engineering, School of Civil Engineering, Tsinghua University, Beijing. China Professor, Institute of Geotechnical Engineering, School of Civil Engineering, Tsinghua University, Beijing, China Email: songf5@mails.tsinghua.edu.cn, zhangjm@tsinghua.edu.cn ABSTRACT : Estimation of seismic earth ressures constitutes an imortant subject of research in civil engineering. Previous studies showed strong deendence of earth ressure coefficient on lateral strain constraint of the backfill, i.e., the wall dislacement has a significant effect on seismic earth ressures acting against retaining structures. Besides, exerimental investigations and theoretical analysis indicate that when the backfill is at the assive side, the lanar sliding surface assumtion will overestimate earth resistance for higher wall friction angles, which will make structures such as sheet ile walls and their anchor blocks deending on earth ressures at the assive side for suort underdesgned. In this aer, based on seudo-static analysis and the concet of intermediate soil wedge with curved surface, a new methodology is develoed to evaluate seismic earth ressures at the assive side under any boundary strain constraint for a rigid retaining structure with translational movement. It has the advantage over the Mononobe-Okabe method since it can take into account the effect of wall dislacement on lateral earth ressures and the curved sliding surface is emloyed in the analysis. The aroach can determine the seismic earth ressure coefficient of normally consolidated cohesionless soil under any lateral deformation between the isotroic comression and the assive states. Corresonding comuter rogram is written to calculate the seismic earth ressure of a tyical retaining wall system. The results are comared with those obtained based on the assumtion of a lanar sliding surface. KEYWORDS: Seismic earth ressure, seudo-static analysis intermediate soil wedge; curved sliding surface;
The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China 1.INTRODUCTION Evaluation of seismic earth ressures is of ractical significance for the earthquake resistant design of retaining structures such as retaining walls, sheet ile bulkheads, cofferdams, bridge abutments, and basement walls of buildings. Among the several aroaches which have develoed to solve the dynamic earth ressure roblems (e.g., Okabe, 19; Mononobe and Matsuo, 199; Matsuo and Ohara, 196; Ichihara and Matsuzawa, 1973; Wu and Finn, 1999; Richards et al., 1979&1999), the well known Mononobe-Okabe method based on seudo-static analysis and limit equilibrium theory is still most widely used to determine the seismic earth ressure on a retaining structure due to its definite advantage of simlicity. The method is a modification of Coulomb s wedge theory by taking into account the inertia forces on a sliding soil wedge caused by earthquake accelerations. Its essential effectiveness in estimating the seismic active is confirmed by a number of exeriments, field observations and redictions (Mononobe and Matsuo, 199; Ishii et al. 196; Ohara et al., 197; Seed and Whitman, 197; Sherif et al., 19; Whitman, 199&1991).However, as in Coulomb theory, the failure surface is assumed lanar in the Mononobe-Okabe solution, regardless of the fact that the most critical sliding surface may be curved. Exerimental investigations and theoretical analysis show that the most critical sliding surface is usually curved in the assive earth ressure case and the lanar sliding surface assumtion will seriously overestimate earth resistance esecially for high wall friction angles. The Mononobe-Okabe method usually rovides an unsafe rediction of the seismic assive resistance. Therefore, it is more suitable to emloy the curved sliding surface in the analysis of earth ressure at the assive side (Terzaghi, 193&1967; James and Bransby, 197; Whitman and Christian, 199). Many researchers have done a lot of exerimental and theoretical researches in this field and develoed some theories and methods for the determination of assive earth ressure (Chen and Liu 199; Soubra and Kastner, 1991; Morison and Ebeling 1995; Soubra ; Kumar 1; Subba Rao and Choudhury 5). However, one of the basic requirements of all these theories and methods is that the wall should move sufficiently to create a limit-equilibrium state in the backfill, which means they are only suitable for determining the seismic earth ressure at the limit state. But this condition is not always satisfied. In many engineering ractices, large deformation causing active or assive states cannot occur in the backfill behind a retaining structure so that the earth ressure may fall anywhere between active and assive ressures. Exerimental evidences indicate that wall movement has significant effect on magnitude and distribution of lateral earth ressure (Terzaghi, 193; Matsuo et al., 197; Sherif et al., 19&19; Ishibashi et al., 197; Fang et al., 196&199). So this imortant fact must be taken into consideration in seismic resistant design of retaining structures. Based on the analysis of the strain ath test results, Zhang et al.(199) established the relation between the lateral earth ressure coefficient and the strain increment ratio and develoed a new theory for determining the lateral earth ressure between the active and assive states. By emloying the concet intermediate soil wedge which deends on mobilized frictional resistance, Zhang et al. extended Mononobe-Okabe method to new earth ressure formulas for determining the dynamic lateral earth ressure under any lateral deformation. The method has undoubted theoretical basis and clear hysical concets and is easy for alication because of its simlicity. However, as has been discussed reviously, the intermediate soil wedge with the lanar sliding surface assumtion will result in higher earth resistance at the assive side and the overestimation may be very serious esecially when the soil-wall interface is rough. Structures such as sheet ile walls and their anchor blocks deend on earth ressures at the assive side for suort. If the suorting ressures are unconservative, the structure may be underdesigned. In the resent technical note, the revious intermediate soil wedge deending on mobilized frictional resistance with the lanar sliding surface (Zhang et al, 199) is modified to one with the sliding surface as a combination of a logarithmic siral and a straight line when the wall movement renders the backfill at the assive side. On the basis a new seudo-static methodology of analysis is develoed to determine lateral earth ressures for any intermediate state form isotroic comression to assive conditions for a rigid retaining structure with translational movement. A research comuter rogram is written to calculate the seismic earth ressure coefficient and the results are analyzed and comared with those obtained using a lanar sliding surface.
The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China. FORMULATION OF THE METHOD The backfill soil may extend or comress with the wall dislacement, which makes the soil under different strain constraints. Roscoe (197) emhasized the imortance of strain influence on earth ressures. The model test results of James and Bransby (197) indicated that the mobilized wall fricion angle and internal friction angle of the soil are different under different wall movements, leading to the change of earth ressures acting on retaining structures. The deendence of the coefficient of lateral earth ressure on the strain increment ratio was studied and on the basis the formation mechanism of earth ressure was discussed by Zhang et al. (199). Lateral strain arameter R is used to reresent the lateral deformation of the soil, which is caused by the wall dislacement. The relation between the soil lateral strain arameter R and the wall dislacement can be estimated by the formulas roosed by Zhang et al. (199): a Δ ( Δ a Δ ) R = Δ a (.1) 1 ( Δ < Δ ) a Δ 3 ( Δ Δ ) R = Δ (.) 3 ( Δ > Δ ) In the above equations wall dislacement is rescribed minus when its direction is away from the backfill while ositive when its direction is toward the backfill. a and reresent wall dislacements required to develo active and assive earth ressures resectively and can be estimated by the available exerimental results (Terzaghi 193, Mastuo et al. 197; Sherif et al. 19 & 19, Fang et al. 196 & 199, Terashi et al.1991, Ishihara et al. 1995). a and are constants changing within the ranges: < a < 1 and < < 1 and both are recommended to take around.5. When the wall dislacement makes the backfill soil fall between active and isotroic comression states, R changes within the range:-1 R 1. While for an intermediate stress state between isotroic comression and assive stress states, there is 1 R 3. The mobilized wall friction angle δ mob changes with the lateral strain arameter R and can be estimated by the following equations suggested by Zhang (199): k1 1 R δ mob = δ a ( 1 R 1) (.3) k R 1 δ mob = δ ( 1 R 3) (.) Where k 1 and k are exonents determined by tests and they can be assigned a value of unity, i.e., k 1 =k =1, if the change in δ mob with R is assumed linear. In the equations δ a and δ reresent wall friction angles mobilized at active and assive states resectively. Zhang et al.(199) investigated the deendence of the lateral earth ressure coefficient on the strain increment ratio based on a series of strain ath tests and established the relation between the earth ressure coefficient K and the lateral strain arameter R. K K = ( 1 R 1 1 (1 K ) R ) (.5) K 1 K = 1+ ( R 1) ( 1 R 3) (.6) In which K is the earth ressure coefficient at rest and K is the assive earth ressure coefficient. When R-value is 1 or 3, the backfill is at the active or assive state and the corresonding earth ressure coefficient
The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China is the active earth ressure coefficient K a or the assive one K. When R value is 1, the soil is at the isotroic comression stress state and the earth ressure coefficient is 1 for isotroic soils. Based the equation (5) and (6) Zhang et al.(199) extended Mononobe-Okabe theory to evaluate the seismic earth ressure for retaining walls under any lateral dislacement. However, as has been discussed, for the stress state of backfill soil between isotroic comression and assive states, with the increase of wall dislacement and mobilized wall friction angle it is more rational to emloy the intermediate sliding wedge with the curved surface for the evaluation of seismic earth ressure. Therefore, a modified method using curved sliding surfaces based on the seudo-static concet is adoted for the comutation of the seismic earth ressure coefficients at the assive side. The soil-wall and the dynamic intermediate soil wedge at the assive side for the analysis is given in Fig. 1. A dry, homogeneous, and isotroic cohesionless backfill with surcharge is assumed in the analysis. It s required to estimate the magnitude of seismic earth ressure against a rigid retaining structure of vertical height H with an inclination α to the vertical, as shown in Fig. 1, in the resence of horizontal earthquake acceleration kh. g and vertical earthquake acceleration k v. g. The ground surface is horizontal. The mobilized wall friction angle δ mob increases with the increase of wall dislacement. Horizontal and vertical seismic coefficients and k v can be determined based on an equivalent seismic coefficient (Zhang et al. 199) for considering the non-uniform seismic acceleration distribution with height of the backfill soil. Investigations show that there is a considerable reduction in the shearing resistance of the soil when the average ground acceleration exceeds a certain critical value (Okamoto, 1956; Richards, 199). In the resent analysis it is assumed that the basic soil arameters: unit weight γ and internal friction angle φ are not affected by the occurrence of an earthquake. So the seismic earth ressure coefficients resented in this technical note is only alicable for earthquake acceleration magnitude less than such critical values. Fig. 1. Dynamic intermediate soil wedge with curved surface If the wall moves toward the backfill while its dislacement is not large enough and does not reach, the shear strength of the backfill soil is not able to be fully mobilized, and therefore, a assive soil wedge cannot be formed behind the wall. However, in rincile, a soil wedge roducing the maximum lateral earth ressure against the wall exists for any given level of wall dislacement. This soil wedge at the assive side is therefore called an intermediate sliding wedge with a curved surface, as shown in Fig. 1. The sliding surface of the intermediate soil wedge OABCO is assumed to be of a comosite shae comrising an arc of the logarithmic siral AB and a straight line BC near the ground. In this case, mobilized internal friction angle of soil on the comosite sliding surface and wall friction angle mobilized are designated by φ mob and δ mob. The seismic earth ressure coefficient at the assive side can be evaluated by the equation (.6). However, as is ointed out, the seismic assive earth ressure coefficient K in equation (.6) shouldn t be determined by Mononobe-Okabe theory based on lanar sliding surface assumtion, but should be estimated by the methods based on curved sliding surface. The method roosed by Kumar () is modified and extended to include the earth ressure roduced by the following three arts: (Ⅰ) vertical inertial body force kv. W, where W is the weight of soil wedge OABCO; (Ⅱ) surcharge q. OC; (Ⅲ) seudo-static forces kh. q. OC and k v. q. OC. The detailed derivation is not given here due to the limitation of the article length. And it should be noted that the wall friction angle δ in the method should be substituted by the mobilized wall friction angle δ mob estimated by equation (.). The seismic assive earth coefficient K in equation (.6) can be obtained by this modified method. The method develoed in this article is suitable for the estimation of seismic earth ressures at the assive side acting on rigid retaining structures against isotroic normally consolidated cohesionless soil.
The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China 3. DISCUSSION AND COMPARISON OF RESULTS Corresonding comuter code is written based on the methodology develoed above. The variation of seismic earth ressure coefficient at the assive side with horizontal seismic coefficient and wall friction angle forϕ =5, k v = and α= is resented in Fig. 6 5 3 1 =.1 =. =.3 =. =.5 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 Normalized wall dislacement ( Δ / Δ ) (a) δ = 11 1 =.1 9 =. 7 6 =.3 =. =.5 5 3 1 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 Δ / Δ (b) δ =φ/3 1 1 1 6 =.1 =. =.3 =. =.5 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 Δ / Δ (c) δ =φ/3 Figure Seismic earth ressure coefficients at the assive side It can be seen from Fig that when the wall dislacement toward the backfill soil makes it fall between isotroic comression and assive states (1 R 3 ),with the increase of wall dislacement level the shear strength of soil and wall friction angle is increasingly mobilized, leading to the increase of the seismic earth ressure coefficient. When = /9, R=1, the backfill is at the isotroic comression state and the earth ressure coefficient is 1 and when =, R=3, the backfill is at the assive state and the earth ressure coefficient is the assive earth ressure coefficient based on comosite sliding surface. Seismic earth ressure coefficient at the assive side decreases with increase in and decrease in δ. In order to study the effect of the shae of the sliding surface on the evaluation of seismic earth ressure coefficients, a comarison is made of seismic earth ressure coefficient values at the assive side obtained from resent study and those based on the lanar sliding surface with resect to different horizontal seismic coefficients forϕ =5,α=, k v =, which is shown in Fig. 3.
The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China 36 3 Planar surface δ= Planar surface δ=φ/3 Planar surface δ=φ/ Planar surface δ=φ/3 Curved surface δ= Curved surface Curved surface Curved surface δ=φ/3 δ=φ/ δ=φ/3 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 Normalized wall dislacement ( Δ / Δ ) 36 3 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 Δ / Δ 36 3 (a) =.1 (b) =. 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 Δ / (c) =.3 (d) =. Δ 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 1/9 /9 3/9 /9 5/9 6/9 7/9 /9 1 Δ / Δ / Δ (e) =.5 Fig 3 Comarison of seismic earth ressure coefficients for φ=5, α=, k v = 3 Δ It can be seen from Fig.3 that when δ =, the results obtained by both methods are identical because in this case the focus of the log siral locates in the infinite distance from the to of the wall and the curvature of the sliding surface is very small and so the curved surface can be regarded nearly as lanar. However, when δ is not zero, the error resulted from lanar intermediate sliding wedge gradually increases when the shear strength of the soil and wall friction angle are increasingly mobilized with the increase of wall dislacement. When the
The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China wall friction angle and the wall dislacement are small, i.e., δ <φ/ or << /9, the results based on lanar sliding surface assumtion can satisfy the recision request of the engineering. When the wall friction angle and the wall dislacement are large, the backfill soil aroaches the assive limit state, and it is reasonable for the engineers to divide the calculation results based on the lanar sliding surface by a roer safety factor in ractical design. The seismic earth ressure coefficients after reduction is similar to the ones obtained on the basis of curved or comosite sliding surface.. CONCLUSIONS A new methodology is develoed in this technical note to evaluate the seismic earth ressure coefficients for normally consolidated cohesionless backfill under any lateral deformation between the isotroic comression and assive states. The resent method has the advantage over the method roosed by Zhang et al. (199) since it is based on the concet of intermediate soil wedge with curved sliding surface. Calculation results show that when the wall friction angle and wall dislacement are large, the seismic earth ressure coefficients at the assive side obtained by the resent method are areciably smaller than those got based on lanar surface assumtion, which means that the error due to lanar assumtion is always on the unsafe side while the roosed method is more reasonable for this case. By the comarison with results based on sliding wedge with lanar surface, it is found that when the wall friction angle and wall dislacement is large, it s reasonable to divide the calculation results based on the lanar sliding surface by a roer safety factor in engineering ractices. ACKNOWLEDGEMENT The authors wish to thank the science fund for doctoral rogram of university from the Ministry of Education of China (No. 736) and National Basic Research Program of China (973 Program) (No. 7CB711) for their financial suort. REFERENCES Chen, W. F. and Liu X. L. (199). Limit analysis in soil mechanics, Elsevier, Amsterdam, The Netherlands. Fang, Y. S. and Ishibashi, I. (196). Static earth ressures with various wall movements. Journal of geotechnical engineering 1:3, 317-333. Fang, Y. S., Chen T. J. and Wu, B. F. (199). Passive earth ressures with various wall movements. Journal of Geotechnical Engineering :, 137-133. 1)Ichihara, M. and Matsuzawa, H. (1973): Earth ressure during earthquake, Soils and Foundations 13:,75-6. Ishii, Y., Arai, H. and Tsuchida, H. (196). Lateral earth ressure in an earthquake. Proceedings of the nd World Conference on Earthquake Engineering, Tokyo, 1: 11-3. Ishibashi, I. and Fang, Y. S. (197): Dynamic earth ressures with different wall movement modes. Soils and Foundations 7:, 11-. Ishihara, K., Arakawa, T., Saito, T., Hada, M. and Huang, Y. (1995). Study on the earth ressure by using a large-size soil box with a movable wall. Proceedings of the 3th Jaan National Conference on Soil Mechanics and Foundation Engineering, 1717-17. James, R. G. and Bransby, P. L. (197). Exerimental and theoretical investigation of a assive earth ressure roblem. Geotechnique :1, 17-37. Kumar, J., and Subba Rao, K.S. (1997). Passive ressure coefficients, critical failure surface and its kinematic admissibility. Geotechnique, 7:1,15-19. Kumar, J. (1).Seismic assive earth ressure coefficients for sands. Canadian Geotechnical Journal 3:,76-1. Matsuo, H. and Ohara, S. (196). Lateral earth ressure and stability of quay walls during earthquakes. Proceedings of nd World Conference on Earthquake Engineering, Tokyo, 1:5-13. Matsuo, M., Kenmochi, S. and Yagi, H. (197). Exerimental study on earth ressure of retaining wall by field
The 1 th World Conference on Earthquake Engineering October -17,, Beijing, China tests. Soils and foundations 1:3, 7-1. Mononobe, N., and Matsuo, H. (199). On the determination of earth ressure during earthquakes. Proceedings of the World Engineering Congress 9,177-15. Morrison, E. E., and Ebeling, R. M. (1995). Limit equilibrium comutation of dynamic assive earth ressure. Canadian Geotechnical Journal, 3:3, 1~7. Ohara, S., Maehara, H. and Nagata, H. (197). On seismic active earth ressures (in Jaanese). Tsuchi-to-Kiso1:, 7-35. Okamaoto, S. (1956). Bearing caacity of sandy soil and lateral earth ressure during earthquakes. Proceedings of the 1st World Conference on Earthquake Engineering, California, 1-6 Okabe, S. (19): General theory on earth ressure and seismic stability of retaining wall and dam. Journal of the Jaan Society of Civil Engineering 1:6, 77-133. Richards, R. Jr., and Elms, D. G. (1979). Seismic behavior of gravity retaining walls. Journal of the Geotechnical Engineering Division15:, 9-6. Richards, R., Elms, D.G., and Budhu, M. (199). Dynamic fluidization of soils. Journal of Geotechnical Engineering, 1:5, 7-759 Richards, R. Jr., Huang, C., and Fishman, K. L. (1999). Seismic earth ressure on retaining structures. Journal of Geotechnical and Geoenvironmental Engineering 5:9,771-77. Roscoe, K. H., (197). The influence of strains in soil mechanics, Geotechnique, :,9-17. Seed, H. B. and Whitman, R.V. (197). Design of earth retaining structures for dynamic loads. Proceedings of Secialty Conference on Lateral stress in the Ground and Design of Earth Retaining Structures, 13-17. Sherif, A., Ishibashi, I. and Lee C. D. (19). Earth ressures against rigid retaining walls. Journal of geotechnical engineering 1:GT5, 679-695. Sherif, M. A. and Fang Y. S. (19). Dynamic earth ressures on walls rotating about the to, Soils and foundations :,19-117. Soubra, A.H., Kastner,R.(1991). Passive earth ressure coefficients in seismic areas by the limit analysis method, Soil Dynamics and Earthquake engineering V, 15-6. Soubra, A.H. (). Static and seismic assive earth ressure coefficients on rigid retaining structures, Canadian Geotechnical Journal 37:, 63-7. Subba Rao, K.S. and Choudhury, D. S. M. (5). Seismic assive earth ressures in soils. Journal of geotechnical and geoenvironmental engineering. 131:1,131-135. Terashi, M., Kitazume,M. and Matsunaga, Y. (1991).Centrifuge modeling on assive earth ressure of sand. Proceedings of the 6th Jaan National Conference on Soil Mechanics and Foundation Engineering, 157-159. Terzaghi, K. (193). Large retaining wall tests, Engineering News Record, 136-1. Terzaghi, K. (193). Theoretical Soil Mechanics, Wiley, New York. Terzaghi, K. and Peck, R.B. (1967): Soil mechanics in engineering ractice, Wiley, New York. Whitman, R. V. (199). Seismic design and behavior of gravity retaining walls, Proceedings of ASCE secialty Conference on Design and Performance of Earth Retaining Structures, Ithaca, N.Y., 17-. Whitman, R. V. and Christian, J. T. (199): Seismic resonse of retaining structures, Symosium on Seismic Design for World Port, Port of Los Angeles, Los Angeles, California. Whitman, R. V. (1991). Seismic design of earth retaining structures, Proceedings of International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, Rolla, Missouri, 1767-177 Wu, G. and Finn, W.D.L. (1999): Seismic lateral ressures for design of rigid walls, Canadian Geotechnical Journal, 36: 59-5 Zhang, J. M., Shamoto, Y. and Tokimatsu, K. (199a). Evaluation of earth ressure under any lateral deformation. Soils and Foundations 3:1, 15-33. Zhang, J. M., Shamoto, Y. and Tokimatsu, K. (199b).Seismic earth ressure theory for retaining walls under any lateral dislacement. Soils and Foundations, 3:,13-3. Zhang, J.-M., Y Shamoto & K Tokimatsu (199c): Earth ressures on rigid walls during earthquakes, ASCE Geotechnical Secial Publication 75:, 153-17.