Upper bound seismic rotational stability analysis of gravity retaining walls considering embedment depth

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1 J. Cent. South Univ. (05) : DOI: 0.007/s Upper bound seismic rotational stability analysis of gravity retaining walls considering embedment depth LIU Jie( 刘杰 ),, HUNG Da( 黄达 ), 3, YNG Chao( 杨超 ), SUN Sha( 孙莎 ) 4. School of Civil Engineering, Chongqing University, Chongqing , China;. School of Civil Engineering, Hebei University of Engineering, Handan , China; 3. Key Laboratory of New Technology for Construction of Cities in Mountain rea of Ministry of Education, Chongqing , China; 4. College of erospacing Engineering, Chongqing University, Chongqing , China Central South University Press and Springer-Verlag Berlin Heidelberg 05 bstract: Stability analysis of gravity retaining wall was currently based on the assumption that the wall had no embedment depth. The effect of earth berm was usually neglected. The present work highlighted the importance of embedment depth when assessing the seismic stability of gravity retaining walls with the pattern of pure rotation. In the framework of upper bound theorem of limit analysis, pseudo-static method was applied into two groups of parallel rigid soil slices methods in order to account for the effect of embedment depth on evaluating the critical acceleration of wall-soil system. The present analytical solution is identical to the results obtained from using limit equilibrium method, and the two methods are based on different theory backgrounds. Parameter analysis indicates that the critical acceleration increases slowly when the ratio of the embedment depth to the total height of the wall is from 0 to 0.5 and increases drastically when the ratio exceeds 0.5. Key words: gravity retaining wall; embedment depth; seismic rotational stability; upper bound analysis; parallel rigid soil slices Introduction Seismic stability analysis is important to the design of gravity retaining walls, especially in earthquake-prone region. Under earthquake, when gravity retaining wall moves away from the backfill (i.e. towards earth berm which in this case would be in passive condition), passive thrust due to the earth berm acts on the wall surface. This passive thrust can impose restrictions on sliding and tilting of the gravity retaining walls. lthough earth berm has the favorable effect of increasing seismic stability of gravity retaining wall, the research work was replete in which most of the researchers assumed that the gravity retaining wall has no embedment depth (i.e. earth berm is neglected) [ 7]. For example, the tilting moment-balance method (namely conventional limit equilibrium approach) was used to analyze the seismic rotational stability of gravity retaining walls without embedment depth [ 7]. However, there were some studies in which the researchers had tried to understand the effect of embedment depth on cantilevered walls [8 ]. In these studies, the retaining structure (cantilevered wall) was considered a weightless sheet model. For the seismic rotational stability analysis of gravity retaining wall, its size and weight must be took into account due to personal dimension, which is clearly different from cantilevered wall. The seismic rotational stability of retaining structures were generally evaluated through the factor of safety based on tilting moment-balance method (i.e. limit equilibrium method) [ 3]. By this conventional approach, the magnitude and the distribution of the seismic earth pressure acting on the retaining structure must be analyzed before calculation of factor of safety. In recent years, the upper bound method was employed to evaluate seismic stability of retaining structures [ 5]. Unlike the tilting moment-balance method of analysis, the earth pressure acting on wall needs not be considered using the upper-bound method. In the framework of upper bound theorem of limit analysis, ZHNG et al [6] developed some equations for assessing the seismic sliding stability of gravity retaining wall, in which the effect of embedment depth was considered. In their study, retaining wall, backfill soil Foundation item: Project(44745) supported by the National Natural Science Foundation of China; Project(CQGT-KJ-04049) supported by the Chongqing dministration of Land, Resources and Housing, China; Project(0604CDJZR00009) supported by the Fundamental Research Funds for the Central Universities, China Received date: 04 0; ccepted date: Corresponding author: HUNG Da, PhD, Professor; Tel: ; hdcqy@6.com

2 4084 and earth berm were considered a wall soil system. This work focused on the evaluation of the seismic rotational stability of gravity retaining walls having embedment depth using multi-block upper-bound method. The seismic rotational stability of wall soil system was quantified through critical accelerations (i.e. yield accelerations or threshold accelerations). Theoretical model In order to solve the problem of the seismic rotational stability analysis of gravity retaining wall having embedment depth, the following assumptions were put forward. ) The fill soil is homogeneous, dry and cohesionless. ) The pseudo-static method is used for estimation of seismic inertia forces. 3) The gravity retaining wall is tolerable only for a rotational displacement (i.e. only rotate around its toe without sliding on its bases). 4) The earth berm (namely front cover) is wide enough and the top is horizontal. s shown in Fig., an upright retaining wall was filled with cohesionless soil to its embedment depth H in the front and full height H on the back, respectively. The soil failure zone I I I 3 is behind the wall back face I I ; the soil failure zone J J J 3 is in front of the wall front face J J. Here, β and β are inclination angles of the surface I I 3 and J J 3 of rupture from the horizontal, respectively; γ s and φ are the unit weight and internal friction angle of the fill, respectively; is the friction angle of the wall soil interface; γ w is the unit weight of the wall. Under earthquake, when the ground acceleration exceeds the critical acceleration, the wall will purely J. Cent. South Univ. (05) : rotate with angular velocity ω around its toe (i.e. point J in Fig. ). In the framework of upper bound theorem of limit analysis, the wall and the soil failure zones I I I 3 and J J J 3 are taken as a whole system. s shown in Fig., each soil failure zone is considered a deformation body consisting of an infinite number of rigid soil slices. Every rigid soil slice parallels the rupture surface (I I 3 or J J 3 ) of the corresponding soil failure zone (I I I 3 or J J J 3 ). In Fig., the rigid soil slices were partitioned for the pure rotation movement illustrated in Fig.. Velocity compatibilities between adjacent blocks and velocity hodograph for wall-back fill and wall-front cover system are illustrated in Fig.3 and Fig.4, respectively. s shown in Figs. to 4, P and Q are random points on the back and the front faces of wall, respectively; V s is the velocity of the rigid soil slice adjacent to point P; V P is the velocity of point P; V Ps is the relative velocity between V P and V s and V Ps inclines at angle to the wall back face I I ; is friction angle of wall soil interface I I and J J. ccording to the mechanism to be kinematically admissible (see Fig. 3), the relationship between V P and V s can be found as Vs V P sin( ) cos( ) () where is inclination angle of line J P (0 arctan(h /B), =arctan(h /B) for point P on the top of the back face and =0 for that on the bottom). Notice that when >, the rigid soil slice adjacent to point P moves towards the wall; when <, the slice moves away from the wall; when =, the slice is motionless. Considering assumption 3), point P rotates with angular velocity around the toe J during an incipient rotation, and V P can be written as Fig. Pure rotating failure mechanism of retaining wall having embedment depth

3 J. Cent. South Univ. (05) : Fig. Wall block and two groups of parallel rigid soil slices V P B () cos where B is the width of the wall. Hence, substituting Eq. () in Eq. (), V s can be found as V s Bsin( ) cos cos( ) (3) Similarly, the relationship between V Q and V s can be written as (see Fig. 4) V s cos cos( ) (4) where V s is the velocity of the rigid soil slice adjacent to point Q; V Q is the velocity of point Q; V Qs is the relative velocity between V Q and V s ; V Qs inclines at angle to the wall front face J J (see Figs. and 4). Fig. 3 Velocity compatibility between adjacent blocks and velocity hodograph for wall-back fill system: (a) Profile of vectors; (b) Relationship between vectors Fig. 4 Velocity compatibility between adjacent blocks and velocity hodograph for wall-front cover system: (a) Profile of vectors; (b) Relationship between vectors Similarly to Eq. (), the velocity of point Q can be written as VQ x (5) where x is the distance between point Q and the base of wall. Substituting Eq. (5) in Eq. (4), it can be found as x cos Vs (6) cos( ) The area d of one rigid soil slice inside zone I I I 3 can be calculated from the following equation: d cot ( H Btan )d( Btan ) Bcot sec ( H B tan ) d (7) The area d of one rigid soil slice inside zone J J J 3 can be written as d cos ( H x)dx (8) In the framework of limit analysis, the external forces of wall soil system include gravities and seismic inertia forces. The balance equation of the work rate in the mechanism of pure rotation is obtained as W W W W W W D (9) gw ew es es gs gs where W gw, W gs and W gs are the rates of work done by gravities of the wall, zone I I I 3 and zone J J J 3, respectively; W, ew W es and W es are the rates of work done by seismic inertia forces of the wall, zone I I I 3 and zone J J J 3, respectively; D is the rate of energy dissipation of the overall wall soil system. When the yielding state is first reached, the rate of work done by external forces and energy dissipation can be calculated by the following equations.

4 4086 ) The rate of work done by the wall weight is: W gw wb H (0) ) The rate of work done by seismic inertia force of the wall is: W ew khwbh () where k h is horizontal seismic coefficient. 3) The rate of work done by the gravity of soil zone I I I 3 is: W V sin( )d () gs s s Substituting Eqs. (3) and (7) in to Eq. (), W gs can be obtained as W V sin( )d gs s s where sv 0 ssin( ) Bcot sec ( H B tan ) d sin( )cot f cos( ) s 3/ H (3) f B [9H cos(3 ) 48 B 5Bsin(3 ) 9H cos( ) 3Bsin( ) 3Hcos( ) 3Hcos(3 ) 9Bsin( ) Bsin(3 )] (4) arctan H B (5) 4) The rate of work done by seismic inertia force of soil zone I I I 3 is: W k V cos( )d (6) es h s s Substituting Eqs. (3) and (7) in Eq. (6), the can be obtained as W k V cos( ) d es h s s kh 0 svscos( ) Bcot sec ( H B tan )d cos( )cot k f cos( ) h s W es (7) 5) The rate of work done by gravity of soil zone J J J 3 is: W V cos(90 ) d (8) gs s s J. Cent. South Univ. (05) : Substituting Eqs. (6) and (8) in to Eq. (8), W gs can be written as: W V cos(90 ) d gs s s H x cos 0 s cos( ) cos(90 ) cot ( H x)dx ssin( ) f (9) where f 3 H cos cot 6sin( ) (0) 6) The rate of work done by inertia force of soil zone J J J 3 is: W k V cos( )d () es h s s Substituting Eqs. (6) and (8) into Eq. (), can be found as W k V cos( )d es h s s H x cos k 0 h s cos( ) cos( ) W es cot ( H x)dx k cos( ) f () h 7) The rate of energy dissipation Since there is no cohesion, the rate of energy dissipation along the rupture surfaces is zero[7]. D 0 (3) The wall soil system should be in limit state if acceleration coefficient k h reaches threshold value k cr. Substituting Eqs. (0), (), (3), (7), (9), () and (3) in to Eq. (9), the acceleration coefficient k h can be found as sin( )cot kh wb Hs f cos( ) sin( )cos s f wbh cos( ) cos( ) cot cos( ) cos s f s f cos( ) cos( ) (4) The critical acceleration coefficient k cr is obtained by minimizing k h with respect to the variable angles β and β. The corresponding values of β and β are denoted by β cr and β cr (i.e. rupture angles), respectively. 3 Comparison with limit equilibrium method To verify the validity of the present methodology for design, it is instructive to compare the value of k cr

5 J. Cent. South Univ. (05) : obtained from Eq. (4) with the results obtained by traditional limit equilibrium method (i.e. tilting moment-balance method), in which earthquake dynamic earth thrust acting on the wall was obtained by Mononobe-Okabe approach [8 9]. Figure 5 shows the purely rotating failure mechanism of gravity retaining wall having embedment depth using the limit equilibrium method [7 8]. base of the wall: PPE sh KPE (30) K PE is coefficient of passive earth pressure. KPE cos ( ) / cos cos( ) sin( )sin( ) (3) cos( ) The critical acceleration coefficient k cr were obtained by soling the sophisticated Eq. (5). Figure 6 shows the gravity retaining wall in the present work. s an illustration,for H =8 m, B=3.6 m, γ w =4 kn/m 3, γ s =0 kn/m 3, =35, the critical acceleration coefficients obtained from two methods are presented in Table. Fig. 5 Forces and acceleration on gravity retaining wall in limit equilibrium method [7 8] In Fig. 5, when the ground acceleration threshold for pure rotation is exceeded, the limit equilibrium method can be given by kcrw H PE cos H W B PEBsin 3 PE cos H P (5) where W is the weight of the wall: W BH (6) w P E is the dynamic active thrust due to the backfill; the point of application of P E is at H /3 above the base of the wall. PE sh KE (7) K E is coefficient of active earth pressure: KE cos ( ) / cos cos( ) sin( )sin( ) (8) cos( ) ψ is seismic angle: arctan k cr (9) P PE is the dynamic passive thrust due to the earth berm; the point of application of P PE is at H /3 above the Fig. 6 Retaining wall considered in analysis Table Comparison of results with limit equilibrium method k H /m /() cr for proposal k cr for limit method equilibrium method ctually, the all critical acceleration coefficients (k cr ) from Eq. (5) equal those obtained by limit equilibrium

6 4088 method for the gravity retaining wall with a purely rotating failure mechanism under earthquake. Though the same results were obtained using the two methods above, they are based on different theory backgrounds: the proposed method is based on the balance of energy in the wall soil system and does not require estimation of the magnitude, distribution or point of application of the seismic earth pressure; and the balance of moments is adopted in the limit equilibrium method, for which the moments acting on the wall (including the magnitude and the direction of moments, particularly for magnitude) are complicated to be determined. 4 Parameters analysis The effect of variations in wall soil friction angle and embedment depth H on the critical acceleration coefficient k cr was discussed below. 4. Variation of k cr with The effect of the wall soil friction angle on the critical acceleration coefficient k cr is shown in Fig. 7 (for γ w =4 kn/m 3, γ s =0 kn/m 3, H =5 m, H = m, B=3 m). s can be seen from Fig. 7, the critical acceleration coefficient k cr increases almost linearly with the increased ratio of wall-soil friction angle to internal friction angle of soil. J. Cent. South Univ. (05) : Fig. 8 Variation of critical acceleration coefficient k cr with embedment depth H important parameter for estimating cr and cr. The effects of this parameter on the cr and cr are shown in Figs. 9 and 0 (for γ w =4 kn/m 3, γ s =0 kn/m 3, H =5m, H = m B=3 m). s can be found, the both rupture angles ( cr and cr ) decrease nearly-linearly with the increased ratio of /. Fig. 9 Variation of cr with wall soil friction angle Fig. 7 Variation of critical acceleration coefficient k cr with wall soil friction angle 4. Variation of k cr with H Non-linear relations between the ratio of the embedment depth to the total height of the wall (i.e. H /H ) and the k cr are illustrated in Fig. 8 (for γ w =4 kn/m 3, γ s =0 kn/m 3, H =8 m, B=3 m, =0.5). The curves of H /H k cr show an initial slow increase of k cr when H /H is from 0 to 0.5 and a drastic increase when H /H is greater than Variation of cr and cr with The wall soil friction angle is also a very Fig. 0 Variation of cr with wall soil friction angle

7 J. Cent. South Univ. (05) : Conclusions ) new methodology with regard to the estimation of the critical acceleration coefficient is proposed by adopting the pseudo-static method. This work aimes at analyzing the seismic rotation stability of gravity retaining walls having embedment depth based on upper bound theorem of limit analysis. The wall and the soil failure zones in front cover and back fill are taken as a whole wall soil system. ) The critical acceleration coefficients k cr obtained by the limit equilibrium method (Mononobe-Okabe approach) and the present method are identical. nd using the limit equilibrium method the earthquake dynamic earth thrust must be analyzed, while only energy dissipation of wall soil system is discussed by the present upper-bound solution. 3) Both of increases of wall soil friction angle and decrease of the embedment depth H of wall can improve the critical acceleration coefficient k cr of the wall soil system. The results for the critical acceleration coefficient and the rupture angles obtained show important light on the seismic rotational stability of gravity retaining walls. References [] ZENG X, STEEDMN R S. Rotating block method for seismic displacement of gravity walls [J]. Journal of Geotechnical and Geoenvironmental Engineering, 000, 6(8): [] CHOUDHURY D, HMD S M. Stability of waterfront retaining wall subjected to pseudo-static earthquake forces [J]. Ocean Engineering, 007, 34(4): [3] HMD S M, CHOUDHURY D. Seismic rotational stability of waterfront retaining wall using pseudodynamic method [J]. International Journal of Geomechanics, 00, 0(): [4] BSH B M, BBU G L. Seismic rotational displacements of gravity walls by pseudodynamic method with curved rupture surface [J]. International Journal of Geomechanics, 009, 0(3): [5] BSH B M, BBU G L. Optimum design of bridge abutments under seismic conditions: Reliability-based approach [J]. Journal of Bridge Engineering, 00, 5(): [6] SIDDHRTHN R, R S, NORRIS G M. Simple rigid plastic model for seismic tilting of rigid walls [J]. Journal of Structural Engineering, 99, 8(): [7] NOURI H, FKHER, JONES C. Development of horizontal slice method for seismic stability analysis of reinforced slopes and walls [J]. Geotextiles and Geomembranes, 006, 4(3): [8] POWRIE W. Limit equilibrium analysis of embedded retaining walls [J]. Geotechnique, 996, 46(4): [9] DIKOUMI M, POWRIE W. Mobilisable strength design for flexible embedded retaining walls [J]. Geotechnique, 03, 63(): [0] L L, SITR N. Seismic earth pressures on cantilever retaining structures [J]. Journal of Geotechnical and Geoenvironmental Engineering, 00, 36(0): [] ULBCH B, ZIEGLER M, SCHÜTTRUMPF H. Design aid for the verification of resistance to failure by hydraulic heave [J]. Procedia Engineering, 03, 57: 3 9. [] SU Q, ZHNG X. Kinematic approach for seismic stability analysis of cantilever retaining wall [C]// CICTP 04@Safe, Smart, and Sustainable Multimodal Transportation Systems. SCE, 04: [3] LI X, WU Y, HE S. Seismic stability analysis of gravity retaining walls [J]. Soil Dynamics and Earthquake Engineering, 00, 30(0): [4] HE S, OUYNG C, LOU Y. Seismic stability analysis of soil nail reinforced slope using kinematic approach of limit analysis [J]. Environmental Earth Sciences, 0, 66(): [5] ZHNG D, SUN Z, ZHU C. Reliability analysis of retaining walls with multiple failure modes [J]. Journal of Central South University, 03, 0: [6] ZHNG X, HE S, SU Q, JING W. Seismic stability analysis of pre-stressed rope of anti-slide retaining wall [J]. Geotechnical and Geological Engineering, 03, 3(4): [7] CHEN W F. Limit analysis and soil plasticity [M]. msterdam: Elsevier, 03. [8] MIKOL R G, SITR N. Seismic earth pressures on retaining structures in cohesionless soils [R]. Berkeley: California Department of Transportation (Caltrans), 03. [9] KRMER S L. Geotechnical earthquake engineering [M]. New Jersey: Prentice-Hall, 996. (Edited by DENG Lü-xiang)

Objectives. In this section you will learn the following. Rankine s theory. Coulomb s theory. Method of horizontal slices given by Wang (2000)

Objectives. In this section you will learn the following. Rankine s theory. Coulomb s theory. Method of horizontal slices given by Wang (2000) Objectives In this section you will learn the following Rankine s theory Coulomb s theory Method of horizontal slices given by Wang (2000) Distribution of the earth pressure Height of application of the