Seismic Design of Geosynthetic-Reinforced Soil Bridge Abutments with Modular Block Facing (2012)

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CHAPTER 1 LITERATURE REVIEW INTRODUCTION Few methods for the seismic design of geosynthetic-reinforced soil (GRS) bridge abutments exist in the literature. However, methods proposed for seismic design of geosynthetic-reinforced soil retaining walls and reinforced earth bridge abutments have been investigated and can be readily adopted for the use in design of GRS bridge abutments. This literature review presents analytical and numerical methods for the seismic design of reinforced soil retaining walls and reinforced soil bridge abutments constructed over competent foundations for which settlement and collapse of the foundation materials are not a concern. These methods can be divided into three categories: 1) Pseudo-Static methods that are based on the original Mononobe-Okabe approach; 2) Pseudo-Dynamic methods; and 3) Displacement methods that originate from the Newmark sliding block models. PSEUDO-STATIC METHOD The pseudo-static method is an extension of conventional limit equilibrium method for analysis of earth structures that include destabilizing body forces related to horizontal and vertical components of ground accelerations. The method uses the Mononobe-Okabe approach to calculate dynamic earth forces acting on reinforced soil retaining walls. Mononobe-Okabe Approach Calculation of Dynamic Earth Force The Mononobe-Okabe approach, an extension of the classical coulomb wedge analysis, is used to calculate the dynamic active earth forces acting on a planar surface inclined at an angle ψ, into an unsaturated, homogeneous, cohesionless soil mass. In Figure 1.1, W refers to the static weight of the active wedge of soil acting behind the wall and Ww refers to the static weight of the facing column. Quantities kh and kv are the horizontal and vertical seismic coefficients, respectively, expressed in terms of the gravitational constant, g.

6 Figure 1.1: Forces and Geometry used in Pseudo-Static Seismic Analysis of Segmental Retaining Walls (Bathurst and Cai, 1995) The positive sign convention for the horizontal seismic coefficient, +kh, is to be consistent with active earth pressure conditions in which the horizontal inertial forces are assumed to act outward. The sign convention for the positive vertical seismic coefficient, +kv, corresponds to a seismic inertial force that acts downward. The total dynamic active earth force, PAE, transmitted by the backfill soil is calculated as: 2)1( 2 1 HKkP AEvAE γ±= (1) Where: γ = Unit weight of the soil H = Height of wall The dynamic earth pressure coefficient, KAE, as given by Bathurst and Cai (1995) is calculated as: ( ) ( ) ( )( ) ( ) 2 2 2 coscos sinsin 1coscoscos )(cos       ++− −−+ ++− −+ = βψθψδ θβφδφθψδψθ θψφ AEK (2)

7 Where: φ = Peak soil friction angle ψ = Total wall inclination (positive in a clockwise direction from the vertical) δ = Mobilized interface friction angle assumed to act at the back of the wall β = Backslope angle (from horizontal) θ = Seismic inertia angle given by:       ± = − v h k k 1 tan 1θ (3) kh and kv = Horizontal and vertical seismic coefficients, respectively The seismic inertia angle represents the angle through which the resultant of the gravity force and the inertial forces, both horizontal and vertical, is rotated from vertical. Equations 1 through 3 are an exact analytical solution to the classical Coulomb wedge problem which has been modified to include the inertial forces khW and kvW. From Equation 2, it can be shown that solutions are only possible for βφθ −≤ . Given this limitation, the maximum horizontal seismic coefficient in Equation 2 is restricted to ( ) ( )βφ −±≤ tan1 vh kk . Equations 1 and 2 can be modified to include additional surcharge loads acting behind the wall (Okabe 1924; Motta 1994). A closed-form solution for the calculation of dynamic earth force for φ−c soils in retaining wall design for the special case of β = 0 and kv = 0, is reported by Prakash (1981). Seed and Whitman (1970) decomposed the total dynamic active earth force, PAE, calculated according to Equations 1 and 2 into two components representing the static earth force component, PA, and the incremental dynamic earth force due to inertial seismic effects, ΔPdyn. Hence: PAE = PA + ΔPdyn (4) or (1 ± kv) KAE = KA + ΔKdyn (5)

8 Where: KA = Static active earth pressure coefficient ΔKdyn = Incremental dynamic active earth pressure coefficient Distribution of Dynamic Lateral Earth Pressures and Point of Application The position of the dynamic earth force, PAE, acting against gravity retaining walls is variable and depends on the magnitude of ground acceleration. The application point of the incremental dynamic earth force increment, ΔPdyn, is located at ηH above the toe of the wall where η is assumed to be 0.6 for segmental retaining wall structures (Seed and Whitman, 1970). The application point of the dynamic earth force, PAE, is given by mH, where m is limited to a range of 0.33 ≤ m ≤ 0.60. The distribution of static and dynamic active earth pressures is illustrated in Figure 1.2. To simplify calculations, only the horizontal component of PAE is used in stability calculations, i.e. PAE cos(δ-ψ). This assumption ignores the benefit of the stabilizing vertical component of PAE and is therefore conservative. Figure 1.2: Calculation of Dynamic Earth Pressure Distribution due to Soil Self-Weight (Bathurst and Cai, 1995)

9 Using the decomposed equations for the total dynamic earth force proposed by Seed and Whitman (1970), it has been recommended by The Reinforced Earth Company (1995) that half the incremental dynamic earth force, 0.5ΔPdyn, acting at 0.6H above the base in addition to the static earth force, PA, be used in stability calculations for reinforced earth bridge abutments. According to The Reinforced Earth Company (1995), applying half the incremental dynamic earth force accounts for the particle acceleration not reaching maximum everywhere at the same time, either in the reinforced fill or in the retained earth, and that some small horizontal displacement leading to stress release is acceptable. Orientation of Active Failure Plane Closed-form solutions for the orientation of the critical planar surface from the horizontal, αAE, reported by Okabe (1924) and Zarrabi (1979) are as follows:       +− +−= − AE AEAE AE E DA1tanθφα (6) Where: ( )βθφ −−= tanAEA (7) ( )( )1++= AEAEAEAEAEAE CBBAAD (8) ( )[ ]AEAEAEAE BACE ++= 1 (9) ( )ψθφ +−= tan/1AEB (10) ( )ψθδ −+= tanAEC (11) The orientation of the assumed active failure plane within the reinforced soil mass and in the retained soil can be calculated using Equation 6. Selection of Parameter Values Soil and Interface Friction Angles. For cohesionless backfill soils, the friction angle, φ , of the soil is assumed to be the peak value determined from conventional laboratory practice and its magnitude is assumed not to change under seismic excitations (Bathurst and Cai, 1995).

10 The interface friction angle, δ, is assumed to be equal to 3/2φ for internal stability analysis (facing column-reinforced soil interface) and equal to φ for external stability analysis (reinforced soil-retained soil interface). (Bathurst and Cai, 1995) Seismic Coefficients. The selection of seismic coefficients greatly affects the design of reinforced soil structures. Multiple relationships for kh to the peak ground acceleration of a site have been reported although a general agreement to this relationship has not been established. The average peak horizontal acceleration in the soil behind the wall can differ from the sites peak ground acceleration due to the influence that a reinforced soil structure can have on a site's ground acceleration. Equation 11.10.7.1-1 in AASHTO LRFD Bridge Design Specifications (2007) relates peak ground acceleration, A, to average maximum horizontal acceleration Am, for A < 0.45 using: Am = (1.45-A)A (12) Where: kh = Am in the reinforced earth volume In the use of the Mononobe-Okabe method, the choice of a positive or negative kv values influence the magnitude of dynamic earth forces calculated using Equations 1 and 2. The selection of a non-zero value of kv implies that peak horizontal and vertical accelerations are time coincident. While significant vertical accelerations may occur at sites located near the epicenter, both positive and negative values of kv must be evaluated in order to ensure the most critical case has been accounted for. In the study performed by Bathurst and Cai (1995), kh and kv are assumed to be uniform and constant throughout the facing column, the reinforced and retained soil mass. Bathurst and Cai (1995) also state that this assumption may not be true for walls higher than 7 m, or walls with complex geometries, surface loadings and/or structures with special foundation conditions. External Stability Based on the recommendations of The Reinforced Earth Company (1995), the verification of external stability is done in two parts: First, the stability of the sill with respect to forward

11 sliding, bearing and overturning; Second, verification of the stability of the overall reinforced earth abutment. Sill Stability The Reinforced Earth Company (1995) recommends that the free field acceleration, A, to be used in the stability check of the sill itself given that little is known on the actual accelerations reaching the top of the structure. Loads Transmitted From the Bridge Deck. For the calculation of the safety factor with respect to sliding and overturning of the sill, the live load transmitted from the bridge is excluded. The live load is excluded here because it would have tendency to increase the factor of safety for sliding and the negligible effect it has on overturning. Using dead load, Qd, of the bridge superstructure, the horizontal inertia of the dead load, Fd, acting at the location of bearing, is calculated as: AQF dd = (13) For the bearing capacity check of the sill and surcharge effect for internal stability, the dead load, Qd, plus 50% of the live load, 0.5Qll, are applied vertically while the seismic force of the dead and live loads are applied horizontally. The seismic force of the dead load plus live load, Fd+l, is calculated as: AQQF lldld )5.0( +=+ (14) Although AASHTO LRFD Bridge Design Specifications (2007) allows omission of live loads for seismic stability analysis, it is likely that traffic loads exist during a seismic event. Therefore, 50% of the maximum live load applied for seismic analysis should conservatively represent the conditions associated with rush hour automobile traffic. (The Reinforced Earth Company, 1995) Forces Developed From the Sill. The sill has a total weight, Ws, which includes its backwall and the soil over its heel. The inertial force of the sill weight is: AWP sis = (15)

12 Force Transmitted From Backfill. Stability checks of the sill also include the static and dynamic pressure exerted directly behind the seat and its backwall from the backfill overlying the reinforced earth mass. The dynamic force is calculated using the acceleration, A. The forces acting against the sill include: the static earth pressure, P2; the static pressure due to the reduced surcharge, P2q, and the pseudo-static pressure, PAES. The pseudo-static pressure, PAES, as given by The Reinforced Earth Company, 1995, is calculated as: ( )AAEsAES KKHP −= 22 1 γ (16) Where KAE is calculated from Equation 2 using: A1tan −=θ (17) The reduced traffic surcharge is also incorporated into the total dynamic earth pressure. The total dynamic earth pressure applied at 0.6Hs above the base of the sill is:       + 2 21 P P P qAES (18) The free body diagram of the forces acting on the sill can be seen in Figure 1.3. Figure 1.3: External Stability of Sill (The Reinforced Earth Company, 1995)

13 External Stability of the Reinforced Earth Mass Potential external failure modes of the reinforced mass include translational sliding along the base, overturning about the toe of the reinforced mass and bearing capacity of the foundation as shown in Figure 1.4. It is assumed that the foundation provides a competent base such that excessive settlement and bearing capacity failure is not of concern. Figure 1.4: Potential External Modes of Failure: a) Base Sliding, b) Overturning, c) Bearing Capacity (Bathurst and Cai, 1995) Forces Transmitted From the Deck. The only forces considered from the bridge superstructure for external stability calculations are the dead load, Qd, and the inertia of the dead load, Fd. The live load would have tendency to increase the safety factor for sliding and has little or no effect on overturning and is therefore excluded. The inertia of the dead load, Fd, is calculated using the free field acceleration, A, as shown in Equation 13. Forces Transmitted From the Sill. Verification of the overall stability of the reinforced earth abutment considers the sill, including its backwall and the backfill over the heel, as an integral part of the abutment. As a result, the inertia of the sill is calculated using the average maximum acceleration, Am. The inertia of the sill for the calculation of the stability of the reinforced earth mass is: smshis WAWkP == (19) Inertial Forces of the Reinforced Earth Mass. The effective inertial force, Pir, is a horizontal load acting at the center of gravity of the effective mass. The total weight of the effective mass, W, is defined by The Reinforced Earth Company (1995), as the weight of the reinforced mass which extends 0.5H in from the face of the wall as shown in Figure 1.5. The inertial force due to

14 the effective weight of the overlying fill, Pi2, is also assumed to act at the center of gravity of its weight. The total weight of the overlying fill, W2, is defined as the weight of the overlying fill that extends 0.5H in from the face of the wall. These inertial forces are calculated by: mmhir AHWAWkP 25.0 γ=== (20) and 222 WAWkP mhi == (21) Forces Transmitted to the Structure from the Backfill. As shown in Figure 1.5, the forces transmitted to the structure from the retained backfill include the static earth pressure, P, assumed to act at H/3 above the base, and half the dynamic earth pressure, 0.5PAE, which is assumed to act at 0.6H above the base. The dynamic earth pressure is calculated as: dynAE KHP ∆= 2 2 1 γ (22) Where: ( ) AAEvdyn KKkK −−=∆ 1 (23) Figure 1.5: External Stability of Abutment (The Reinforced Earth Company, 1995) External Factors of Safety. For external stability use the following factors:

15 External Stability Static Seismic F.S. with respect to base sliding 1.5 1.1 F.S. with respect to overturning 2.0 1.5 F.S. with respect to bearing capacity 2.0 Note 1 Note 1: A factor of safety of 2.0 with respect to foundation bearing capacity is considered acceptable for static conditions. Eccentricity of the structure and applied bearing pressure are not determined during a seismic event due to the temporary and transient nature of the loading condition. Bearing pressure at the toe of the structure during a seismic event should not vary appreciably from the static case. However, this commentary shall serve as a reminder that it may be necessary to check that an earthquake will not alter the inherent strength characteristics of the foundation soils. (The Reinforced Earth Company 1995) External Stability Calculations The safety factors with respect to sliding and overturning are verified using calculations similar to those that apply for the static condition. The eccentricity and bearing pressure under the reinforced earth mass is not calculated because a seismic event is a temporary and transient loading condition on a very flexible system. The bearing pressures at the foundation level are assumed not to increase significantly during a seismic event. Bathurst and Cai (1995) provide the following external stability calculations for reinforced soil retaining walls. Base Sliding. The dynamic factor of safety against base sliding for purely frictional soils is: ( ) ( ) ( )       + − +−± ±      + − = H L a H LL kakK k H L a H LL FS ww hvAE v ww bsl 2 2 1 2 cos1 2 1 tan1 λψδ φ (24) Where: βtan11 H LL a w − += (25)

16 βtan 2 12 H LL a w − += (26) L = Minimum width of the gravity mass Lw = Width of the facing column λ = An empirical constant used to artificially reduce the internal force of the gravity mass used under the assumption that the inertial forces in the gravity mass and the retained soil will not peak simultaneously during an earthquake. A value of λ = 0.6 has been used for design purposes. a1 & a2 = Geometric constants that account for the effect of the backslope angle on the calculation of the mass of the reinforced soil zone. Figure 1.6 shows the static factor of safety against base sliding to give a minimum dynamic factor of safety of 1.125 against base sliding for a range of seismic coefficients, kh and kv, and backslope angle, β. Figure 1.6: Static Factor of Safety Against Base Sliding to give a Minimum Dynamic Factor of Safety of 1.125 Against Base Sliding for a Range of Seismic Coefficients, kh and kv, and Backslope Angle, β (Notes: WR = Weight of reinforced zone plus weight of facing column; and R = Base sliding resistance.) (Bathurst and Cai, 1995)

17 Overturning. The dynamic force moment arm, Ydyn, normalized with respect to wall height, H, is given by m as shown in equation 27: ( )[ ] ( )vAE AvAEA dyn kK KkKK H Y m ± −±+ == 1 1 3 1 η (27) Where: m = Normalized moment arm η = Normalized dynamic force increment location The relationship between normalized moment arm, m, and horizontal seismic coefficient, kh, is shown in Figure 1.7. Figure 1.7: Influence of Seismic Coefficient, kh, Normalized Dynamic Force Increment, η, Wall Inclination Angle, ψ, and Wall-Soil Interface Friction Angle, δ, on Location of Normalized Dynamic Moment Arm, m (Bathurst and Cai, 1995)

18 The dynamic factor of safety against overturning about the toe of the free body comprising of the reinforced soil mass and the facing column given by Bathurst and Cai (1995) is: ( ) ( ) ( )       + − +−± ±              +      − +      − = H L b H LL kakmK k H L a LL L b H LL FS ww hvAE v w w ww bot 1 3 1 2 22 2 cos1 1 2 λψδ (28) Where: ( )2111 13 1 −+= aab (29) ( )212 13 2 1 −+= ab (30) a1 & a2 are defined by Equations 25 and 26 Figure 1.8 shows the static factor of safety, FSbot (static), required to satisfy, FSbot (dynamic) 5.1275.0 =×= . The vertical component of seismic force has been taken as upward (- kv) in order to calculate results for the most critical orientation.

19 Figure 1.8: Minimum Static Factor of Safety Against Overturning Required to give a Factor of Safety of 1.5 Against Dynamic Overturning for a Range of Seismic Coefficients, kh and kv, Backslope Angle, β, and Length to Height Ratio, L/H (Bathurst and Cai, 1995) Internal Stability Seismic loading increases the magnitude of the horizontal force carried by the geosynthetic reinforcement as well as the percentage of total lateral force to be carried by the reinforcing elements in the upper portions of the wall. Also, the influence of ground acceleration on the volume of the internal potential failure wedge leads to an increase in required length of the reinforcement layers. Potential internal modes of failure are shown in Figure 1.9 As recommended by The Reinforced Earth Company (1995), internal stability calculations are performed in two parts. First, tensile forces resulting from static loads alone are calculated. Second, tensile forces from an overall internal dynamic load, Pi, connected with both the

20 reinforced mass itself and the concentrated load transmitted by the sill are calculated. The load, Pi, is distributed among the reinforcement layers in proportion to their resistant area and added to the tensile load calculated in the static case. Figure 1.9: Potential Internal Modes of Failure: d) Tensile Over-Stress, e) Pullout, f) Internal Sliding (Bathurst and Cai, 1995) Loads Considered in the Calculation of Pi. The dynamic force, Pi, is directly connected with the “active zone” of the reinforced mass, through its own weight and the weight it carries. Based on the recommendations of The Reinforced Earth Company (1995), the weight of the idealized (bilinear) active zone is multiplied by a factor of 0.67 to simulate the correct weight of the active zone, Wa. The three main configurations of the active zone envelope are shown in Figure 1.10. The applied load from the sill is then added to the active zone weight to obtain the total vertical load. The total vertical load is multiplied by the acceleration, Am, to obtain the dynamic force, Pi, which is distributed amongst the reinforcement layers.

21 Figure 1.10: Calculating Internal Dynamic Force, Pi, in Different Cases (The Reinforced Earth Company, 1995) The applied load from the sill consists of the dead load, Qd, 50% of the live load, 0.5Qll, and the weight of the sill, Ws, which includes the backwall and soil above the heel. As shown in Figure 1.10, the active zone may include part of the fill behind the sill, in which case a reduced surcharge, Q1’, acting on the roadway surface, over the width of concern, shall be taken into account. The dynamic force, Pi, is calculated as: ( ) mslldai AWQQQWP +++′+= 5.067.0 1 (31) The dynamic load, Pi, is added to the maximum tensile forces, Tm, created by static forces. The dynamic loads, 0.5PAE and Pir, are not taken into account in the static calculation of the maximum tensile force, Tm. Internal Stability Calculations In the study performed by Bathurst and Cai (1995) numerous internal modes of failure were examined for GRS walls. Factors of safety relating to these failure modes are shown in the following.

22 Over-Stressing of Reinforcement. For the geometry shown in Figure 1.11, the dynamic factor of safety against over-stressing, FSos, of a reinforcement layer at depth z below the crest of the wall is given by: ( ) ( ) ( ) vwhdynAdyn allow dyn allow os HS H L k H z KKK T F T FS γψδψδ     +−∆−+−∆ == cos6.0cos8.0 (32) Where: Tallow = The allowable tensile load for the reinforcement under seismic loading Sv = Contributory area of each reinforcing layer Fdyn = Dynamic tensile force Figure 1.11: Calculation of Tensile Load, Fdyn, in a Reinforcement Layer due to Dynamic Earth Pressure and Wall Inertia (Bathurst and Cai, 1995) Figure 1.12 shows the influence of seismic coefficient values and normalized depth below the crest of wall, z/H, on dynamic reinforcement force amplification factor, rF, (the ratio of dynamic tensile force to static tensile force). The results show the largest increase in the reinforcement force occurs in the shallowest layer of the reinforced soil wall. These results imply that the number of reinforcement layers at the upper portions of the wall may need to be increased to keep tensile loads within allowable limits. Also, in Figure 1.12 it can be seen that rF is reasonably independent of the magnitude of kv for kh ≤ 0.35 such that solutions using kv = 0 are

23 sufficiently accurate for designs within this range. Figure 1.12: Influence of Seismic Coefficients, kh and kv, and Normalized Depth Below Crest of Wall, z/H, on Dynamic Reinforcement Force Amplification Factor, rF (Bathurst and Cai, 1995) Reinforcement Anchorage. The dynamic tension load in the reinforcement is resisted by the length of reinforcement that is anchored. This anchorage length is located between the internal active failure plane and the reinforcement free end. As shown in Figure 1.13, seismic loading results in a larger active wedge due to the internal failure plane angle, αAE, decreasing as kh increases. The length of reinforcement may need to be increased in order to capture the larger active zone.

24 Figure 1.13: Influence of Seismic Coefficients, kh and kv, and Soil Friction Angle,φ , on Ratio of Minimum Reinforcement Lengths, statdyn LL / , to Capture the Inertial Failure Wedge in Pseudo- Static Coulomb Wedge Analyses (Bathurst and Cai, 1995) Internal Sliding. Internal sliding includes sliding along horizontal planes which pass along the reinforcement-soil interface as well as sliding through the facing column between facing units. The dynamic shear resistance, Vu, available at a horizontal interface in the facing column, can be described as: ( ) uvwuu kWaV λtan1±+= (33) Where: au = Minimum interface shear capacity λu = Equivalent interface friction angle

25 The dynamic factor of safety against internal sliding along a horizontal surface at depth z below the crest of the wall is: ( ) ( ) ( )       +      −+−± ±      −+ = z L a z LL kakK ka z LL z V FS ww hvAE dsv wu isl 2 2 1 22 cos1 2 1 tan1 λψδ φ γ (34) Where: dsφ = Direct sliding interface friction angle between the geosynthetic reinforcement and the cohesionless reinforced soil a1 & a2 are described by Equations 25 and 26 with H = z Facing Stability Facing stability analysis of segmental retaining walls include: interface shear failure, connection failure and local overturning (toppling) as shown in Figure 1.14. Figure 1.14: Potential Facing Modes of Failure: g) Shear Failure, h) Connection Failure, i) Local Overturning (Bathurst and Cai, 1995) Interface Shear. As shown in Figure 1.15, the out of balance force to be carried through shear at the bottom of facing unit j is the sum of the incremental column inertial force, kh ΔWw j, plus the force due to the corresponding contributory area of CDEF. The factor of safety against dynamic interface shear failure at a reinforcement layer is: dyn u sc S V FS =

26 ( ) ( ) ( ) 24 cos6.0cos8.0 vwh v dynAdyn u HS H L k H S H z KKK V γ ψδψδ       +      −−∆−+−∆ = (35) Where: Sdyn = Interface shear force Vu = Shear capacity Figure 1.16 shows the ratio of dynamic factor of safety to static factor of safety against shear failure for a range of seismic coefficients. Data shows the potential for shear interface failure increases towards the crest of the wall for seismic loading. Figure 1.15: Calculation of Dynamic Interface Shear Force Acting at a Reinforcement Elevation (Note: N = Total number of reinforcement layers; and M = Total number of facing units.) (Bathurst and Cai, 1995)

27 Figure 1.16: Influence of Seismic Coefficients, kh and kv, and Normalized Depth Below Crest of Wall, z/H, on the Ratio of Dynamic to Static Interface Shear Factor of Safety (Bathurst and Cai, 1995) Connections. The peak connection load envelope is: ( ) (max)tan1 ccsvwcsc FkWaF ≤±+= λ (36) Where: acs = Minimum connection capacity λcs = Slope of the connection strength envelope The dynamic factor of safety for connection failure is: ( ) ( ) ( ) vwhdynAdyn c dyn c cn HS H L k H z KKK F F F FS γψδψδ     +−∆−+−∆ == cos6.0cos8.0 (37)

28 Toppling. Internal moments that cause a net outward moment at the toe of a facing unit provide a possible failure mechanism and must be evaluated. The factor of safety for local overturning at a reinforcement layer i under dynamic loading conditions is: ( ) ( ) ( ) 2 1 2 1 cos1.04.0cos 6 1 1 Hz H L kK H z H z K YFkM FS w hdyna i c N i i cvR lot γψδψδ       +−∆      −+− +± = ∑ + (38) Where: MR = Resistance to static overturning due to facing column self-weight above the toe of the target facing unit N = Number of reinforcement layers ∑ + N i i c i c YF 1 = Resisting moment due to the connection capacity at each of reinforcement layer, icF , and their corresponding moment arm, icY , from the point of rotation As shown in Figure 1.17, the uppermost interface layers require a higher static factor of safety against overturning to maintain a dynamic factor of safety equal to or greater than one. In order to minimize potential toppling at the top of the wall, reinforcement layers should be placed close to the crest and have adequate facing connection capacity.

29 Figure 1.17: Influence of Seismic Coefficients, kh and kv, and Normalized Depth Below Crest of Wall, z/H, on the Ratio of Dynamic to Static Local Toppling Factor of Safety (Bathurst and Cai, 1995) PSEUDO-DYNAMIC METHOD Steedman and Zeng (1990) proposed a pseudo-dynamic earth pressure theory to account for dynamic amplification which considers the effect of phase difference over the height of a vertical retaining wall. The method recognizes that a base acceleration input will propagate up through the retained soil at a speed corresponding to the shear velocity of the soil. However, this model only considers the effect of horizontal seismic acceleration due to vertically propagating shear waves through the backfill behind the retaining wall. The inclusion of vertical seismic effects due to vertically propagating primary waves through the backfill soil was proposed by Choudhury and Nimbalkar (2006).

30 Considering a typical fixed base cantilever wall as shown in Figure 1.18, when the base is subjected to a harmonic horizontal seismic acceleration, ah ( = khg), and harmonic vertical seismic acceleration, av ( = kvg), the accelerations at depth z and time t are given by: ( )       − −= S hh V zH tatzA ωsin, (39) ( )       − −= P vv V zH tatzA ωsin, (40) Where: ρ GVS = = Shear wave velocity ρ GK VP 3 4+ = = Primary wave velocity K = Soil bulk modulus G = Soil shear modulus ρ = Soil density T πω 2= = Angular frequency T = Period of lateral shaking It is initially assumed that the soil shear modulus, G, is constant with depth through the backfill and only the phase and not the magnitude of acceleration varies.

31 Figure 1.18: Model Retaining Wall Considered for Computation of Pseudo-Dynamic Active Earth Pressure (Choudhury et al. 2006) Considering the mass of a horizontal element in the wedge at depth z, the total horizontal inertia force, Qh, acting on the wall is: ( ) ( )[ ]tHkdztzAzHtQ hh H h ωωξλωξπαπ λγ α ρ sinsincos2 tan4 , tan )( 2 0 −+=      −= ∫ (41) And the total vertical inertia force acting on wall is: ( ) ( )[ ]∫ −+=     −= H v vv tH k dztzA zH tQ 0 2 sinsincos2 tan4 , tan )( ωωψλωψπ απ ηγ α ρ (42) Where: =γ Unit weight of soil STV=λ = Wavelength of shear wave PTV=η = Wavelength of primary wave SVHt /−=ξ PVHt /−=ψ α Direction of wall H h PAE δ z dz φ F Qv W Qh z VS, VP

32 The total (static +seismic) earth pressure on the wall is computed as: ( ) ( ) ( ) ( )αφδ φαφαφα −+ −+−+− = cos sincossin vh AE QQW P (43) Where: W = Weight of active failure wedge α = Angle of active failure surface The seismic earth pressure can be separated into a static component, PA, and a dynamic component, ΔPdyn as shown in Equation 44: ( ) ( ) ( ) ( )             − −+ − + −+ − =+= S h dynAAE V z t zkz PPP ω αφδ φα α γ αφδ φα α γ∆ sin cos cos tancos sin tan ( ) ( )            − −+ − + P v V z t zk ω αφδ φα α γ sin cos sin tan (44) The force ΔPdyn acts at a height h above the base, which is given as: ( ) ( )tH tHH Hh ωωξπλωξπ ωωξλωξπλωξπ sinsincos2 coscossin2cos2 2 222 −+ −−+ −= (45) The acting point of the dynamic force increment is seen to be independent of soil friction angle,φ , and the slope angle, ψ , but a function of shear wave velocity and the period, T, of the assumed harmonic horizontal acceleration function. DISPLACEMENT METHODS The pseudo-static approach, like all limit equilibrium methods of analysis, does not consider wall deformations. Since the performance of a geosynthetic reinforced soil wall after an earthquake can be controlled by unacceptable deformations without structural collapse, methods of analysis that predict the permanent displacements of a GRS wall have been investigated. Richards and Elms (1979) developed a method for seismic design of gravity retaining walls based on the concept of an allowable permanent displacement. The approach is similar to the method suggested by Newmark (1965) to evaluate the amount of slip occurring in dams and

33 embankments during earthquakes. Cai and Bathurst (1996) adopted the Newmark sliding block theory to examine cumulative displacements of geosynthetic reinforced segmental retaining walls associated with three sliding mechanisms: 1) external sliding along the base of the total wall structure; 2) internal sliding along a reinforcement layer and through the facing column; and 3) block interface shear between facing column units. Permanent displacements are assumed to accumulate each time the critical acceleration, ac (ac = kcg, where kc is the critical horizontal acceleration coefficient), of a sliding mechanism is exceeded by the horizontal ground acceleration a(t). Calculation of Critical Accelerations External Sliding Along Base Horizontal sliding of the entire reinforced soil mass is assumed to occur through the soil at the base of the reinforced soil mass. The destabilizing forces are the dynamic active earth force, PAE, and the seismic inertial force, PI. The resisting force is the frictional resistance, R, acting along the base of the reinforced soil mass. The free body diagram for base sliding can be seen in Figure 1.19. The dynamic factor of safety against base sliding is shown by Equation 46. The critical horizontal acceleration coefficient, kc, for base sliding corresponds to a value of kh which gives FSdyn = 1.0 in Equation 46. ( ) ( ) ( ) ( )       + − +−± ±      + − = +− = H L a H LL kakK k H L a H LL PP R FS ww hvAE v ww IAE dyn 2 2 1 2 cos1 2 1 tan1 cos λψδ φ ψδ (46) Where: βtan11 H LL a w − += (47) βtan 2 12 H LL a w − += (48) L = Minimum width of the gravity mass Lw = Width of the facing column

34 λ = An empirical constant used to artificially reduce the internal force of the gravity mass used under the assumption that the inertial forces in the gravity mass and the retained soil will not peak simultaneously during an earthquake. A value of λ = 0.6 has been used for design purposes. a1 & a2 = Geometric constants that account for the effect of the backslope angle on the calculation of the mass of the reinforced soil zone. Figure 1.19: Free Body Diagram of Composite Gravity Mass Comprising of Facing Column and Reinforced Soil Zone for Base-Sliding Analysis (Cai and Bathurst, 1996) Internal Sliding Along Soil-Geosynthetic Interface Internal sliding at the soil-geosynthetic interface refers to a portion of the GRS wall sliding along the soil-geosynthetic interface at a depth z below the crest of the wall. The free body diagram of this sliding mechanism can be seen in Figure 1.20. The destabilizing forces are the dynamic active earth force, PAE, and the seismic inertial force, PI. Here PI = khλWz where Wz = Ws + Ww is the total weight of the sliding mass, Ws being the weight of the reinforced soil and Ww the weight of the facing column. The resisting force is composed of two parts: First being the frictional resistance of the soil-geosynthetic interface, Rs, given as: ( ) dsvss kWR φtan1−= (49)

35 Where: =dsφ Soil-geosynthetic interface friction angle The second component is the shear resistance of the geosynthetic-block interface at the same depth given by: ( ) uvwuu kWaV λtan1−+= (50) Where: au = Minimum available shear capacity λu = Equivalent interface friction angle The critical horizontal acceleration coefficient, kc, for internal sliding is given by the value of kh when FSdyn = 1.0 in Equation 51. ( ) ( ) ( ) ( )       + − +−− −      + − + = +− + = z L c z LL kckK k z L c z LL z a PP RV FS ww hvAE vu w ds wu IAE su dyn 2 2 1 22 cos1 2 1 1tantan cos λψδ λφ γ ψδ (51) Where: βtan11 z LL c w − += (52) βtan 2 12 z LL c w − += (53)

36 Figure 1.20: Free Body Diagram of Sliding Mass along a Soil-Geosynthetic Interface and through the Facing Column at Depth z below Crest of Wall (Cai and Bathurst, 1996) Block Interface Shear between Facing Column Units Sliding at the block-block or block-geosynthetic interface may occur when shear capacities of these interfaces are exceeded. The analysis of interface shear transmission on facing column stability is treated as a beam in which the integrated lateral earth pressures equal the sum of the reactions. The calculation of dynamic interface shear force acting at a reinforcement elevation can be seen in Figure 1.21. The out-of-balance force at interface j is equal to the sum of the incremental column inertial force, jwh Wk ∆ , plus the force due to area CDEF in Figure 1.21. The critical horizontal acceleration coefficient corresponds to kh when Fdyn = 1.0 in Equation 54. ( ) ( ) ( ) ( ) 24 cos6.0cos8.0 tan1 vw h v dynAdyn uvwu dyn HS H L k H S H z KKK kWa FS γ ψδψδ λ       +      −−∆−+−∆ −+ = (54) Where: Sv = Height of contributory area for the considered reinforcement layer at depth z

37 Figure 1.21: Calculation of Dynamic Interface Shear Force Acting at a Reinforcement Elevation. Fdyn, Dynamic Force in Reinforcement Layer; Sdyn, Dynamic Interface Shear Force; N, Total Number of Reinforcement Layers; M, Total Number of Facing Units (Cai and Bathurst, 1996) Calculation of Permanent Displacements The permanent displacement of a GRS wall resulting from sliding or shear mechanisms can be calculated using one of two methods depending on the input acceleration. Newmark’s double integration method for a sliding block may be used to find permanent displacements when the acceleration time history is given. When only the peak ground acceleration and peak ground velocity are given, the permanent displacement can be estimated using empirical displacement methods. Newmark Sliding Block Analysis Permanent displacement of a mass occurs whenever the seismic forces acting on the soil mass, plus the existing static force, exceed the resistance available at the potential sliding surface. The acceleration corresponding to this seismic force is the critical acceleration. The permanent displacement accumulated is calculated by integrating the portions of the acceleration time

38 amax = Ag aT = Ng BN = bN WBT φ tantan == history that are above and below the critical acceleration until the relative velocity between the sliding mass and sliding base become zero. Consider the rigid block shown in Figure 1.22. FI = M aT = W/g aT W (a) (b) Figure 1.22: Notation and Forces for Sliding Block on a Plane; (a): Notation for Block and Plane Accelerations; (b): Free Body Diagram of Block Where: amax = Maximum plane acceleration A = Maximum ground acceleration aT = Acceleration transmitted to block through friction N = Transmittable block acceleration W = Weight of the block tan bφ = Coefficient of friction between block and plane FI = Inertia force BN = Base normal force T = Shear force

39 Figure 1.23: Acceleration and Velocity Profiles of Block and Plane Subjected to a Rectangular Pulse Excitation Suppose that a plane is subjected to a rectangular earthquake impulse of magnitude Ag and the maximum acceleration transmitted to the block through friction forces is aT = Ng. The acceleration and the resulting velocity profiles of block and plane are shown in Figure 1.23. The plane’s velocity increases linearly at slope of Ag and levels off at time to, the end of the rectangular impulse. The block’s velocity increases at a slope of Ng until its velocity reaches the velocity of plane at time tm. The resulting relative displacement between the block and the plane can be calculated as the difference between the integrals of plane and block velocities over time which is simply the shaded area shown in Figure 1.23. These basic concepts are applicable to more complex earthquake acceleration time histories. Acceleration Time Plane Acceleration Block Acceleration to tm Ng Ag Velocity Time Plane Velocity V=A.g.t0 Block Velocity Vibe=N.g.t to tm V (a) (b)

40 For evaluation of retaining wall displacements according to Newmark’s sliding block theory, additional vertical and horizontal earth pressure forces should be considered as shown in Figure 1.24: Figure 1.24: Idealization of the Retaining Wall Problem by Richards-Elms (1979); (a): Wall and Backfill Accelerations; (b): Free Body Diagram of Wall (Richards and Elms, 1979) Richards-Elms Design Procedure Richards-Elms proposed the following equation for calculating block displacement, dR, in the medium to low range of N/A (Transmittable block acceleration / Maximum ground acceleration): 42 087.0 −      = A N Ag V d R (55) Where: N = Transmittable block acceleration = aT / g V = Maximum ground velocity If tolerable permanent displacements of the structure are specified, the wall can be designed according to the Richards-Elms design method. After choosing a maximum acceptable displacement, N can be calculated using Equation 55. Next, PAE should be obtained using the M- O method as shown in Equation 56. The required weight of the wall to meet the specified displacement can be calculated using Equation 57: ( ) AEVAE KNHP −= 12/1 2γ (56) amax = Ag (a) aT = Ng aT T (b) FI WW (PAE)V (PAE)H B

41 ( ) ( ) N PP W b bVAEHAE W − − = φ φ tan tan (57) Finally, a factor of safety of 1.5 should be applied to the wall weight, WW. The conservative safety factor of 1.5 compared to the usual values of 1.0 to 1.2, takes into account the deformability of the backfill or possible tilting and the statistical variability of earthquake ground motions.