The Influence of Strain Localization and Phase on the Estimation of Earth Pressures by Pseudo-Dynamic Method IGC 2009, Guntur, INDIA
THE INFLUENCE OF STRAIN LOCALIZATION AND PHASE ON THE ESTIMATION OF EARTH PRESSURES BY PSEUDO-DYNAMIC METHOD
B. Munwar Basha Research Associate, Department of Civil Engineering, Indian Institute of Science, Bangalore–560012, India. E-mail: [email protected]. G.L. Sivakumar Babu Associate Professor, Department of Civil Engineering, Indian Institute of Science, Bangalore–560012, India. E-mail: [email protected]
ABSTRACT: The paper focuses on the determination of seismic active earth pressure coefficients when subjected to high earthquake loading (ah = 0.4-1.0g). Planar failure surface has been used in conjunction with the modified pseudo-static approach to compute the seismic active earth pressures on the retaining wall. The proposed modified Mononobe-Okabe method, considers the effects of strain localization in the backfill soil and associated post-peak reduction in the shear resistance from peak to residual values along a previously formed failure plane, phase difference in shear waves and soil amplification along with the horizontal seismic accelerations. The influence of various design parameters on the magnitude of seismic active earth pressure is presented.
1. INTRODUCTION soil structures respectively using modified pseudo-static method considering the amplification of vibrations. However A retaining wall should be designed so as to withstand the the above studies did not take into account the effect of strain damage from the lateral earth pressure and earthquake localization in the shear zone and the effect of the resultant loading. In the following sections the modified pseudo-static strain softening as discussed in the following section. method for the estimation of seismic active earth pressure coefficients is discussed. 1.2 Modified Pseudo-Static Procedure Considering the 1.1 Modified Pseudo-Static Procedure Considering the Effects of Strain Localization Influence of Phase Change The M-O method further assumes that the shear strength of Pseudo-static method, known as the Mononobe-Okabe (M- backfill is isotropic and constant. In reality, the internal O) method (Kramer, 2003) is a popular method and assumes friction angle of soil along a failure surface decreases from that the soil block is rigid and therefore acceleration is the peak strength to residual strength. For example, Bolton & constant throughout the block. However, it does not represent Steedman (1985) conducted shake table tests on cantilever the actual oscillatory nature of earthquake loading. retaining wall and observed that the shear resistance angle o o Additionally, this method does not account for certain mobilized along a failure plane reduces from 50 to 33 . aspects such as earthquake duration, frequency of motion, Koseki et al. (1998) proposed a modified M-O procedure that effects of phase difference in the waveform between the base takes into consideration the effect of strain localization in the and crest of the abutment and the amplification of shear zone and the effect of the resultant strain softening as accelerations within the backfill. To addresses the above illustrated below: aspects, Steedman & Zeng (1990) modified the pseudo-static For the calculation of the angle of failure plane, the peak method. Choudhury & Nimbalkar (2006) computed the 1. active earth pressures considering both shear and primary internal friction angle of backfill (peak) should be waves propagating through the backfill with variation in time employed. In addition, it assumes that the first failure by considering sinusoidal horizontal and vertical seismic plane forms at kh = 0.0 and the same failure plane accelerations for rigid retaining walls. Basha & Babu (2008) continues to be activated up to certain level of kh value. and Basha (2009) proposed an approach for computing However, as the seismic coefficient, kh is increased to a seismic passive earth pressure coefficients using composite certain level (i.e. 0.6 – 0.8), another failure plane will be failure surface (log-spiral and planar) based on the modified activated. pseudo-static method which takes into account the influence 2. For the computation of active earth pressure, the residual of phase change in the shear and primary waves. internal friction angle (res) should be used, since the post peak reduction of internal friction angle evolves Further, Basha & Babu (2009a,b) presented a formulation for after the shear band formation. the calculation of sliding component of response of bridge abutments and seismic reliability assessment of reinforced
11 The Influence of Strain Localization and Phase on the Estimation of Earth Pressures by Pseudo-Dynamic Method As pointed out by Koseki et al. (1998), the above proposed hypothesis has a few advantages: 1). It provides active earth pressure coefficients even at high seismic loads (0.6 to 0.8 g are considered in the seismic design against the level 2 earthquakes, and 2). It provides more realistic values of seismic active earth pressure coefficients and size of active failure zone than the conventional methods. For the practical considerations, Shirato et al. (2002, 2006) further modified the above stated procedure which was introduced in Japanese Specifications for Highway Bridges (JRA, 2002) as follows. It is suggested to evaluate the angles of the initial failure plane at kh = 0.0 and second failure plane at kh = 0.42, but consider only the second failure plane for the computation of Fig. 1: Schematic Diagram of Gravity Wall under Active seismic active earth pressure coefficients for all values of kh. Case with Planar Failure Surface Accordingly, JRA (2002) has adopted the following two equations to evaluate the seismic active earth pressure coefficients (Kae) in the revised seismic design specifications 2.1. Phase Angle and Amplification Effect on Sinusoidal of highway bridges: Accelerations
Kae=0.22 + 0.81 k h for dense sand and gravel (1) The present method considers finite shear wave and primary velocity within the backfill behind the abutment. Similar to K=0.26 + 0.97 k for dense sandy soils (2) ae h Steedman & Zeng (1990), Choudhury & Nimbalkar (2006), Basha (2009) and Basha & Babu (2009a,b), it is assumed that 1.3 Seismic Stability of Retaining Walls the horizontal seismic acceleration in soil vary linearly from The review of the above literature indicates that the seismic the input seismic acceleration at the base to the higher value stability of gravity retaining walls should consider the at the top of the wall (Fig. 2). revisions suggested by Steedman & Zeng (1990) in M-O 2 method, like the influence of phase change in the horizontal Period of lateral shaking can be defined as T (3) acceleration in the backfill behind a retaining wall as shear where w = angular frequency of the base shaking. The waves propagate from the base of wall towards ground horizontal sinusoidal acceleration at any depth, z below the surface, and amplification of soil vibration. In addition, it top of the wall and time ( t ) with soil amplification factor (f) should consider the revised seismic design specifications can be expressed as follows: suggested in JRA (2002) like strain localization in the backfill soil and associated post-peak reduction in the 骣 H- z骣 H - z ah( z , t )=琪 1 + ( f - 1) k h g sinw 琪 t - (4) internal friction angle from peak to residual values along a 桫 H桫 vs previously formed failure plane. Hence, a proper design where v is the shear wave velocity propagating through the procedure is presented in this paper to address these issues. s backfill. In the study, it is assumed that the base of the wall is subjected to horizontal sinusoidal acceleration with 2. SEISMIC ACTIVE EARTH PRESSURE amplitude of accelerations, k,g, where ‘g’ is the acceleration During earthquakes, the peak values of horizontal inertial due to gravity. The details of the calculation of horizontal force, vertical inertial force and active earth pressure may not inertial force due to triangular wedge CGD (Qh+CGD) are be activated simultaneously and the phase differences among reported in Choudhury & Nimbalkar (2006). The forces these three components should be largely different among acting on the triangular wedge CGD are shown in Figure 2. different earthquakes. In addition, Seed & Whitman (1970) The estimated values of Qh+CGD are useful in the derivation of reported that for most earthquakes the horizontal acceleration seismic active earth pressure (Pae(t)) as explained below. The components are considerably greater than the vertical weight of the triangular wedge CGD can be written as, 1 2 components and concluded that vertical acceleration can be WCGD = g H cot a (5) neglected for practical purposes. Accordingly, following the 2 The seismic active earth pressure (P (t)) can be obtained by suggestions of Seed & Whitman (1970), Koseki et al. (1998), ae resolving the forces on the wedge, CGD horizontally and Shirato et al. (2002, 2006) and Japan Road Association JRA vertically as follows: By considering the horizontal (2002), the vertical ground acceleration is neglected in the equilibrium condition ( �H 0 ), we get: present study. In the following sections, the seismic active Q+ Rsina - f = P t cos d earth pressure due to weight and inertia of the wedge ‘ CGD ’ h_ CGD( ) ae ( ) (6) (Fig. 1) behind the retaining wall is presented using modified Rsin(a- f) = Pae( t) cos d - Q h_ CGD (7) pseudo-static procedure as explained in the above sections.
12 The Influence of Strain Localization and Phase on the Estimation of Earth Pressures by Pseudo-Dynamic Method
H at any z = 0 vs
Qh_CGD
Pae H z at any z H vs WCGD
Pae h
0 at z = H
Fig. 2: Forces Acting on Failure Wedge CGD where f = friction angle of the backfill, d = soil-abutment This study includes the influence of the strength parameters interface friction angle, a = angle of the failure plane with of backfill and characteristics of earthquake ground motions. horizontal, and R = resultant force acting on the triangular The parameters values involved in the study are the unit 3 o o wedge ‘CGD ’. By considering the vertical equilibrium weight (g ) = 18 kN/m , peak = 50 , res = 35 , / =0.5, kh = condition ( �V 0 ) for the wedgeCGD , we have 0.0-1.0, f = 1.0, vs = 65 m/s, vsw = 1500 m/s, (), H = 6 m, T Rcos(a- f) + P( t) sin d = W ae CGD (8) = 0.3, H/ = H/(Tvs) = 0.2 – 0.5 and H/ = H/(Tvsw) = 0.3.
Rcos(a- f) = WCGD - P ae ( t) sin d (9) 3.1 Comparison of M-O Method, Modified M-O Method solving the Eqs. (7) and (9), Pae ( t) is given by the following by Shirato et al. (2006) and Present Method equation:
tan(a- f )WCGD + Q h_ CGD The seismic active earth pressure coefficients ( K ) P( t ) = (10) ae ae cosd+ sin d tan ( a - f ) calculated using M-O method (refer the appendix), modified M-O method by Shirato et al. (2006) and present method are plotted in Figure 3 for the horizontal seismic coefficient ( k ) 2.2 Location of the Critical Failure Surface by h Optimization = 0.0 to 1.0. It is assumed that the internal friction angles of o o dense sand and gravel are fpeak = 50 and fres = 35 . It can be In this section, the state at which Pae(t) achieves a maximum observed that for constant value of = res, the magnitudes of value and the failure surface corresponding to this state i.e. K k critical failure surface are determined. The geometry of the ae increases gradually with increase in h value until –1 planar failure surface (GD ) is governed by the angle (a ) as res > tan (kh), beyond which Kae value cannot be evaluated –1 shown in Figure 2. Following the suggestions of JRA (2002) when the term (res > tan kh) in the square root becomes and Shirato et al. (2006), the condition of the initial active negative. It can be noted that Shirato et al. (2006) approach failure plane for kh = 0.0 and second active failure plane for predicts the critical angle (acri ) for second active failure f × kh = 0.42 (note here that the influence of amplification o o d= f / 2 factor is also taken into consideration) in the backfill should plane as 49.7 for fpeak = 50 , peak and kh = 0.42, be evaluated for = , = /2 to obtain the critical o peak peak (Fig. 3) and the following linear function using fres = 35 , angles (acri ) and t/ T ratios of the initial and second failure d= fres / 2 was proposed to estimate the magnitude of Kae : planes subjected to bound constraints such as 0 < < 90 K=0.22 + 0.81 k and 0 骣Qh_ CGD incorporates the effect of phase change in Shirato et al. tan(acri- f res) cot a cri + 琪 2 桫0.5g H (11) (2006) approach and the following differences have been K = noted. It can be observed from Figure 3 that the present study ae cosd+ sin d tan a - f ( cri res ) predicts the critical values for second active failure plane as o o Refer the appendix for the derivation of Qh_ CGD and Kae . acri = 52 and t/ T = 0.45 for fpeak = 50 , d= fpeak / 2 , kh = 3. RESULTS AND DISCUSSION 0.42, H / l = 0.3 and f = 1.0. Substituting the values of acri o o = 52 and t/ T = 0.45 for fres = 35 , d= fres / 2 , H / l = 0.3 13 The Influence of Strain Localization and Phase on the Estimation of Earth Pressures by Pseudo-Dynamic Method and f = 1.0 in Eq. (12), the following linear function has backfill soil. Hence, addressing these issues, a new linear function has been proposed to evaluate more realistic values been proposed to estimate the value of Kae : of K (= 0.22 + 0.68 k ) with respect to the horizontal K=0.22 + 0.68 k (13) ae h ae h seismic acceleration coefficient. REFERENCES Basha, B.M. (2009). “Optimum Design of Retaining Structures under Static and Seismic Loading: A Reliability Based Approach”, Ph.D. thesis, Civil Engineering Dept., Indian Institute of Science Bangalore, Karnataka, India. Basha, B. M. and Babu, G.L.S. (2008). “Seismic Passive Earth Pressure Coefficients by Pseudo-dynamic Method using Composite Failure Mechanism”, In Geo- sustainability and Geohazard Mitigation, ASCE Geotechnical Special Publication, 178, 343 –350. Basha, B.M. and Babu, G.L.S. (2009a). “Computation of Sliding Displacements of Bridge Abutments by Pseudo- dynamic Method”, Soil Dynamics and Earthquake Engineering, 29(1), 103–120. Fig. 3: Comparison of Kae values using Mononobe-Okabe Basha, B.M. and Babu, G.L.S. (2009b). “Seismic Reliability (M-O) method, Modified M-O method by Shirato et al. Assessment of External Stability of Reinforced Soil Walls using Pseudo-dynamic Method”, Geosynthetics (2006) and present method for peak = 50, res = 35 and / = 0.5 International, 16(3), 197–215. Bolton, M.D. and Steedman, R.S. (1985). “Modeling the Seismic Resistance of Retaining Structures,” Proc. of It can also be noted from Figure 3 that the values of Kae 11th International Conference on Soil Mechanics and predicted by the present method are lower than the modified Foundation Engineering, 4, 1845–1848. M-O method by Shirato et al. (2006) approach (which was Choudhury, D. and Nimbalkar, S.S. (2006). “Pseudo- adopted in JRA, 2002). As for an illustration from Figure 3, dynamic Approach of Seismic Active Earth Pressure for the case of kh = 0.6, the estimated values of Kae are 1.05, Behind Retaining Wall”, Geotechnical and Geological 0.70 and 0.63 with M-O method, Modified M-O method of Engineering, 24(5), 1103–1113. Shirato et al. (2006) and present method respectively. This JRA (2002). Specification for Highway Bridges, Part V, difference occurs due to the limitations of pseudo-static “Seismic Design”, Japanese Road Association. analysis such as non-consideration of the effect of time and Koseki, J., Tatsuoka, F., Munaf, Y., Tateyama, M. and phase difference due to finite shear wave velocity. Kojima, K. (1998). “A Modified Procedure to Evaluate Active Earth Pressure at High Seismic Loads (Special 4. CONCLUSIONS Issue)”, Soils and Foundations, 209–216. Kramer, S.L. (2003). “Geotechnical Earthquake Engineering”, A formulation is outlined in this paper for the computation of Englewood Cliffs, NJ: Prentice-Hall. seismic active earth pressure considering the influence of Seed, H.B., and Whitman, R.V. (1970). “Design of Earth oscillatory nature of earthquake loading (like time, soil Retaining Structures for Dynamic Loads,” State-of-the-art amplification and phase difference in the shear wave paper. In Proceedings of the 1970 ASCE Specialty propagating through the backfill behind the abutment), strain Conference on Lateral Stresses in the Ground and Design localization in the shear zone and the effect of the resultant of Earth Retaining Structures, Ithaca, N.Y. American strain softening in the framework of modified pseudo-static Society of Civil Engineers, New York. pp. 103–147. method. Further, the procedure for estimating the point of Shirato, M., Fukui, J. and Koseki, J. (2002). “Ductility application of total thrust is presented. The methodology Design of Foundations of Highway Bridge Abutments”, presented in the paper provides a rational and systematic 18th US-Japan Bridge Engineering Workshop 1–15. procedure for the determination of active earth pressure Shirato, M., Fukui, J. and Koseki, J. (2006). “Current Status acting on retaining walls during earthquake conditions. The of Ductility Design of Abutment Foundations Against main findings of the present investigation are as follows: It is Large Earthquakes”, Soils and Foundations, 46(3), 377– shown that the seismic active earth pressure coefficients ( Kae 396. ) predicted by the present method are lower than the Steedman, R.S., and Zeng, X. (1990). “The Influence of modified Mononobe-Okabe earth pressure theory by Shirato Phase on the Calculation of Pseudo-static Earth Pressure et al. (2006) which does not take into account the effect of on a Retaining Wall”, Geotechnique, 40(1), 103–112. phase difference in body wave propagating through the 14