ROSE SCHOOL NUMERICAL MODELLING OF SEISMIC BEHAVIOUR OF EARTH-RETAINING WALLS

Transcription

1 Istituto Universitario di Studi Superiori Università degli Studi di Pavia EUROPEAN SCHOOL FOR ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK ROSE SCHOOL NUMERICAL MODELLING OF SEISMIC BEHAVIOUR OF EARTH-RETAINING WALLS A Dissertation Submitted in Partial Fulfilment of the Requirements for the Master Degree in EARTHQUAKE ENGINEERING by RAJEEV PATHMANATHAN Supervisors: Prof.P.E.PINTO Dr.C.G.LAI Dr.P.FRANCHIN June, 006

2 The dissertation entitled Numerical Modelling of Seismic behaviour of Earth-Retaining Walls, by Rajeev, has been approved in partial fulfilment of the requirements for the Master Degree in Earthquake Engineering. Prof.Paolo.E.Pinto Dr.Carlo.G.Lai Dr.Paolo Franchin

3 Abstract ABSTRACT In current engineering practice the design methods for earth retaining walls under seismic conditions are mostly empirical. Dynamic earth pressures are calculated assuming prescribed seismic coefficient acting in the horizontal and vertical directions using the concept of limit equilibrium Mononobe- Okabe method. A research investigation has been undertaken to determine the dynamically-induced lateral earth pressures on a flexible diaphragm wall, a flexible cantilever wall, and a gravity retaining wall with cohesionless and cohesive backfills. Additionally, this report gives information about the point of application of total and incremental dynamic forces, the deformation or displacement of the wall and also the bending moment and shear force envelops for structural design of the wall. A series of non-linear dynamic finite element numerical analyses have been performed using DIANA (DIsplacement ANAlyzer). The analyses consisted of the incremental construction of the wall and placement or excavation of the backfill, followed by dynamic response analyses, wherein the soil was modelled as elasto-plastic. Particular attention has been devoted to ground excitation and determination of the wall and soil model parameters. The results obtained with DIANA have been compared through a series of benchmark tests with those determined using simplified techniques for computing dynamic earth pressure, co-seismic and post-seismic wall displacements. Keywords: Retaining walls, Mononobe-Okabe, Dynamic earth pressure, Finite-element analysis i

4 Acknowledgements ACKNOWLEDGEMENTS I would like to thank Prof.P.E.Pinto, Dr.C.G.Lai and Dr.P.Franchin for their guidance and their patience. I would also like to thank everyone at TNO DIANA B.V. for the opportunity to work at the company s offices in Delft. ii

5 Index TABLE OF CONTENTS ABSTRACT...I ACKNOWLEDGEMENTS... II TABLE OF CONTENTS...III LIST OF FIGURES... VII LIST OF TABLES... XII 1. INTRODUCTION Purpose of the study Motivation of the study...1. LITERATURE REVIEW Dynamic earth pressure computation...5. Limit-state analysis Pseudo-static approaches Mononobe-Okabe (196,199) Arango (1969) Choudhury (00) Ortigosa (005) Pseudo-dynamic approach Steedman-Zeng (1990) Choudhury-Nimbalkar (005) Displacement-based analysis Richards-Elms model Comparison of seismic earth pressure values computed using different approaches Closed form solutions using elastic or viscous elastic behaviour Wood (1973) Veletsos and Younan (1994) Numerical analyses...6 iii

6 Index.4.1 Al-Homoud and Whitman (1999) Green and Ebeling (003) Psarropoulos, Klonaris, and Gazetas (005) TYPE OF RETAINING WALLS ANALYZED Diaphragm wall-soil system Cantilever wall-soil system Gravity wall-soil system Properties of soil Properties of concrete SELECTION AND PROCESSING OF GROUND MOTION Selection criteria List of ground motion Characteristics of ground motion selected Processing of the selected ground motions MODELLING ISSUES AND CHOICES Finite element modeling of soil-structure system Soil constitutive model Mohr-Coulomb Pastor and Zienkiewicz(l986) [P-Z mark III model] HiSS soil model [Hierarchical single surface soil model] Hyperbolic type Osaki model Boundaries Elementary boundaries Local or transmitting boundaries Consistent boundaries Soil-structure interface Size of finite element mesh Overview of DIANA Numerical model Details of diaphragm wall numerical model Details of cantilever wall numerical model Details of gravity wall numerical model Model parameters for soil Model parameters for wall Interface element Dimensions of finite element mesh...61 iv

7 Index Damping DIANA RESULTS AND DISCUSSION Data simplification Determination of forces assuming constant stress distribution Incremental dynamic forces Reaction height of forces Dynamic earth pressure coefficient Presentation and discussion of simplified data Phased analysis stress distribution Dynamic analysis stress distribution Pressure distribution along the diaphragm wall Pressure distribution along the stem of the cantilever wall Pressure distribution along the height of the gravity wall Design lateral earth pressure coefficient and DIANA computed lateral earth pressure coefficient Comparison of lateral earth pressure coefficients for diaphragm wall Comparison of lateral earth pressure coefficients for cantilever wall Comparison of lateral earth pressure coefficients for gravity wall Point of application of total dynamic forces and incremental dynamic forces Point of application of total dynamic forces for diaphragm wall Point of application of total dynamic forces and incremental dynamic forces for cantilever wall Point of application of total dynamic forces and incremental dynamic forces for gravity wall Deformation and displacement of the wall Deformation of diaphragm wall Deformation of cantilever wall Deformation of gravity wall Bending moment and shear force Bending moment and shear force envelop for diaphragm wall SUMMARY AND CONCLUSIONS Diaphragm wall Cantilever wall Gravity wall Problems encountered in DIANA modelling Type of element and material constitutive model v

8 Index 7.4. Interface element and transmitting boundary REFERENCES APPENDIX A STATIC DESIGN OF DIAPHRAGM WALL... A1 9. APPENDIX B STATIC DESIGN OF CANTILEVER RETAINING WALL... B1 10. APPENDIX C DESIGN OF GRAVITY WALL... C1 11. APPENNDIX D ADDITIONAL RESULTS FROM DIANA ANALYSES... D1 1. APPENDIX E DIANA COMMAND FILES...E1 vi

9 Index LIST OF FIGURES Figure 1.1. Damage to a retaining wall due to excessive displacement 004 Niigata-Ken Chuetsu earthquake... Figure 1.. Overturning failure of retaining wall the Shin-Kang Dam due to Chi-Chi earthquake...3 Figure 1.3. Top layers moved away from backfill along the construction joints due to Chi-chi earthquake Taiwan...3 Figure 1.4. The upper section of the retaining wall was uplifted by the thrust fault and the retaining wall was sheared due to Chi-Chi earthquake Taiwan....4 Figure.1. Classification of seismic earth pressure computation methods...7 Figure.. Failure surface and the forces considered by Mononobe-Okabe...9 Figure.3. Analytical model...1 Figure.4. Free body diagram...13 Figure.5. Composite failure surface and forces considered...14 Figure.6. Comparison of passive pressure computation using the method proposed by Choudhury (Choudhury et al 00)...18 Figure.7. System considered by Steedman-Zeng...19 Figure.8. System considered by Choudhury-Nimbalkar...1 Figure.9. Base-excited soil-wall system investigated...4 Figure.10. Mononobe-Okabe active and passive expressions (yielding backfill), Wood expression (nonyielding backfill), and FLAC (Continued) (after Green & Ebeling 003)...8 Figure 3.1. Dimension of diaphragm wall...31 Figure 3.. Dimension of cantilever wall...3 Figure 3.3. Dimension of gravity wall...33 Figure 4.1. Imperial Valley (1940) acceleration time-history...36 vii

10 Index Figure 4.. Imperial Valley (1940) Pseudo-Acceleration spectrum corresponding 5% damping...36 Figure 4.3. Imperial valley (1940) Arias intensity...37 Figure 4.4. Chi-Chi (1999) acceleration time-history...37 Figure 4.5. Chi-Chi (1999) Pseudo-Acceleration spectrum corresponding 5% damping...38 Figure 4.6. Chi-Chi (1999) Arias intensity...38 Figure 4.7. Kobe (1995) acceleration time-history...39 Figure 4.8. Kobe (1995) Pseudo-Acceleration spectrum corresponding 5% damping...39 Figure 4.9. Kobe (1995) Arias intensity...40 Figure 5.1. Shape of yield surfaces in J1-JD space...46 Figure 5.. Non-linear constitutive law for soil...46 Figure 5.3. The dashpot model proposed by Lysmer & Kuhlemeyer...49 Figure 5.4. Compound parabolic callectors...49 Figure 5.5. Lumped-parameter consistent boundary...50 Figure 5.6. CQ16E 8-node -D plane strain element...5 Figure 5.7. SPTR -node translation spring/dashpot...53 Figure 5.8. Finite element mesh for diaphragm wall...53 Figure 5.9. Deformed mesh at the end of the each phased construction (sand), magnification factor Figure Finite element mesh for cantilever wall...55 Figure Deformed mesh after placing of backfill (sand), magnification factor Figure 5.1. Finite element mesh for gravity wall...59 Figure Deformed mesh after constructing and placing backfill (sand)...59 Figure 6.1. Constant stress distribution approximation across the element (from Green & Ebeling 003)...64 Figure 6.. Horizontal acceleration ah, corresponding dimensionless horizontal inertial coefficient kh, of a point in the backfill portion of sliding wedge...66 Figure 6.3. Pressure distribution along the diaphragm wall in sand at the end of the phased analysis...67 Figure 6.4. Pressure distribution along the diaphragm wall in clay at the end of the phased analysis...67 Figure 6.5. Pressure distribution along the cantilever wall in sand at the end of the phased analysis...68 viii

11 Index Figure 6.6. Pressure distribution along the cantilever wall in clay at the end of the phased analysis...69 Figure 6.7. Pressure distribution along the gravity wall in sand at the end of the phased analysis...69 Figure 6.8. Pressure distribution along the gravity wall in clay at the end of the phased analysis...70 Figure 6.9. Pressure distribution along the diaphragm wall in sand at the end of the dynamic analysis (EL Centro)...71 Figure Pressure distribution along the diaphragm wall in sand at the end of the dynamic analysis (Chi-Chi)...71 Figure Pressure distribution along the diaphragm wall in sand at the end of the dynamic analysis (Kobe)...7 Figure 6.1. Pressure distribution along the diaphragm wall in clay at the end of the dynamic analysis (EL Centro)...7 Figure Pressure distribution along the diaphragm wall in clay at the end of the dynamic analysis (Chi-Chi)...73 Figure Pressure distribution along the diaphragm wall in clay at the end of the dynamic analysis (Kobe)...73 Figure Comparison of pressures along the stem of the cantilever wall at the end of the phased and dynamic analysis (sand, EL Centro)...74 Figure Comparison of pressures along the stem of the cantilever wall at the end of the phased and dynamic analysis (sand, Chi-Chi)...75 Figure Comparison of pressures along the stem of the cantilever wall at the end of the phased and dynamic analysis (sand, Kobe)...75 Figure Pressure along the gravity wall at the end of the dynamic analysis (Sand)...76 Figure Pressure along the gravity wall at the end of the dynamic analysis (Clay)...77 Figure 6.0. Comparison of active and passive lateral earth pressure coefficient (KDIANA) back-calculated from DIANA results with values computed using the Mononobe-Okabe expressions (Sand)...79 Figure 6.1. Comparison of active and passive lateral earth pressure coefficient (KDIANA) back-calculated from DIANA results with values computed using the Mononobe-Okabe expressions (Clay)...80 ix

12 Index Figure 6.. Comparison of active lateral earth pressure coefficient (KDIANA) backcalculated from DIANA results with values computed using the Mononobe-Okabe expressions (Sand)...8 Figure 6.3. Comparison of active lateral earth pressure coefficient (KDIANA) backcalculated from DIANA results with values computed using the Mononobe-Okabe expressions (Clay)...83 Figure 6.4. Comparison of active lateral earth pressure coefficient (KDIANA) backcalculated from DIANA results with values computed using the Mononobe-Okabe expressions (Sand)...84 Figure 6.5. Comparison of active lateral earth pressure coefficient (KDIANA) backcalculated from DIANA results with values computed using the Mononobe-Okabe expressions (Clay)...85 Figure 6.6. Point of application of total active dynamic force for diaphragm wall...86 Figure 6.7. Point of application of total passive dynamic force for diaphragm wall in sand.87 Figure 6.8. Point of application of total passive dynamic force for diaphragm wall in clay..87 Figure 6.9. Point of application of total dynamic force at stem and heel section of cantilever wall...88 Figure Point of application of incremental dynamic force at stem and heel section of cantilever wall...89 Figure Point of application of total dynamic force of gravity wall...90 Figure 6.3. Point of application of incremental dynamic force for gravity wall...91 Figure Deformed shape of the diaphragm wall in sand at different time step (EL Centro)...9 Figure Deformed shape of the diaphragm wall in sand at different time step (Chi-Chi) 9 Figure Deformed shape of the diaphragm wall in clay at different time step (EL Centro)...93 Figure Deformed shape of the diaphragm wall in clay at different time step (Chi-Chi).93 Figure Annotated deform mesh from EL Centro analysis...94 Figure Relative permanent displacement time-history of base of the cantilever wall in sand (El Centro)...95 Figure Bending moment envelop for diaphragm wall in sand (EL Centro)...96 Figure Shear force envelop for diaphragm wall in sand (EL Centro)...97 Figure Bending moment envelop for diaphragm wall in sand (Chi-Chi)...97 x

13 Index Figure 6.4. Shear force envelop for diaphragm wall in sand (Chi-Chi)...98 Figure Bending moment envelop for diaphragm wall in clay (EL Centro)...98 Figure Shear force envelop for diaphragm wall in clay (EL Centro)...99 Figure Bending moment envelop for diaphragm wall in clay (Chi-Chi)...99 Figure Shear force envelop for diaphragm wall in clay (Chi-Chi) Figure Bending moment envelop for cantilever wall in sand Figure Bending moment envelop for cantilever wall in clay Figure Shear force envelop for cantilever wall in sand Figure Shear force envelop for cantilever wall in clay...10 Figure Maximum pressure envelop along the height of the gravity wall in sand...10 Figure 6.5. Maximum pressure envelop along the height of the gravity wall in clay xi

14 Index LIST OF TABLES Table.1 Comparison of K pγd values obtained by Choudhury s method and available theories in seismic case for β=0º, i=0º, φ=30º...17 Table. Comparison of seismic earth pressure computation using different approach...3 Table 3.1 Properties of sand...33 Table 3. Properties of clay...34 Table 3.3 Properties of reinforced concrete...34 Table 4.1 Ground motion...35 Table 5.1 Typical parameters for P-Z model...45 Table 5. DIANA input properties of sand...60 Table 5.3 DIANA input properties of clay...60 Table 5.4 DIANA input properties of concrete...61 Table 6.1 Pressure values along the height of the cantilever wall...68 Table 6. Relative permanent displacement of base of the cantilever wall...95 Table 6.3 Relative permanent displacement of base and permanent tilt of gravity wall...95 xii

15 Chapter 1 Introduction 1. INTRODUCTION 1.1 Purpose of the study Understanding the behaviour of earth retaining structures in seismic events is one of the oldest problems in geotechnical engineering. The devastating effects of earthquakes make the problem more important. Despite the multitude of studies that have been carried out over the years, the dynamic response of earth retaining structures is far from being well understood. As a result, current engineering practice lacks conclusive information that may be used in design. The most commonly used methods to design retaining structures under seismic conditions are force equilibrium based pseudo-static analysis (e.g. Mononobe-Okabe 196, 199), pseudodynamic analysis (Steedman and Zeng 1990), and displacement based sliding block method (e.g. Richards and Elms 1979). Even under static conditions, prediction of actual retaining wall forces and deformations is a complicated soil-structure interaction problem. The dynamic response of even the simplest type of retaining wall is quite complex. The dynamic response depends on the mass and stiffness of the wall, the backfill and the underlying ground, the interaction among them and the nature of the input motions. The purpose of this study is to develop finite element numerical models to understand the dynamic behavior of retaining structures, and, in particular, to find the magnitude and distribution of dynamic lateral earth pressure on the wall, as well as the displacement and forces induced by horizontal ground shaking. Retaining structures considered include a flexible diaphragm wall, a cantilever wall and a gravity wall. In all the analyses, the soil is assumed to act as a homogeneous, elasto-plastic medium with Mohr-Coulomb failure criterion and the walls are assumed to act as linear elastic. The numerical models for the three walls have been developed using DIANA, a commercially available finite element program. The numerical analyses encompass the incremental construction of the wall and the placement of the backfill or excavation of soil, followed by the seismic response analysis. Particular attention is given to how the ground motions are selected, processed and prescribed as an external loading to the numerical model. The results obtained with DIANA model are compared with results from simplified analysis techniques for computing dynamic earth pressure. 1. Motivation of the study Several types of structures, such as various gravity walls and cantilever sheet pile walls, are used to retain soils in two different levels such as slope and abutments of highway bridges. In order to evaluate the stability of earth-retaining structures during earthquakes the seismic earth pressures and their point of application must be estimated. Many authors have reported [4, 0] numerous cases of damage or failure of bridges induced by excessive abutment displacement or failure during recent earthquakes. Observed failures of 1

16 Chapter 1 Introduction retaining walls were due to sliding, overturning and loss of the bearing capacity of soil underlying the wall. In 1989, the Loma Prieta Earthquake, a severe 7.1 Richter magnitude event, shook the San Francisco area, causing serious damage to bridges and buildings. In 1994, the Northridge earthquake, a severe 6.7 Richter magnitude event, shook the densely populated San Fernando Valley, 0 miles northwest of Los Angeles. Severe damage occurred to buildings, bridges and freeways. These bridge related damages were mostly due to the failure of retaining walls. During the October 3, 004 Chuetsu earthquake, several residential developments constructed on reclaimed land in Nagaoka city, Niigata Prefecture, have experienced damages to houses and roads due to seismically-induced failure of artificial fill slopes. Post-earthquake field reconnaissance surveys revealed that many fill slope failures were caused by the excessive seismic displacements of the gravity retaining walls supporting the fill material. Figure (1.1) shows a retaining wall failure due to the excessive displacement, during the Niigata-Ken Chuetsu earthquake 004. Figure 1.1. Damage to a retaining wall due to excessive displacement 004 Niigata-Ken Chuetsu earthquake Figure (1.) shows retaining walls failed due to the overturning, during the Chi-Chi earthquake (1999).

17 Chapter 1 Introduction Figure 1.. Overturning failure of retaining wall the Shin-Kang Dam due to Chi-Chi earthquake Figure (1.3) shows the top layers moved way from the backfill along the construction joints due to the inadequate frictional resistance at the top portion of the retaining wall. Figure 1.3. Top layers moved away from backfill along the construction joints due to Chi-chi earthquake Taiwan Figure (1.4) shows the wall uplifted by thrust fault, vertical and horizontal displacement of.0m and 1.3m respectively. 3

18 Chapter 1 Introduction Figure 1.4. The upper section of the retaining wall was uplifted by the thrust fault and the retaining wall was sheared due to Chi-Chi earthquake Taiwan. The 1995 Kobe earthquake provides many opportunities for documenting the behavior of retaining structures in waterfront areas, along transportation facilities, and throughout various public and private developments. Many walls failed catastrophically, but some survived virtually intact. This should provide an opportunity to understand better some important issues concerning the design of these structures. Compiling information on what happened is essential so that researchers can try to understand why the structures behaved as they did. There are at least few factors that may have contributed to the movements or failure of earth retaining structures such as: (1) inertial forces on the wall themselves, () dynamic lateral pressures from the backfill in the absence of liquefaction, (3) static and dynamic pressures associated with the liquefaction of the backfill, and (4) reduced resistance to sliding because of liquefaction of soils surrounding the base of the wall in waterfront areas. There are no data that identify directly the relative contribution of these factors, thus requiring studies using numerical and possibly physical models to clarify their interactions. The problem is complex, and developing an understanding of it is so complex that it will require refined finite-element or finite-different model analyses. 4

19 Chapter Literature review. LITERATURE REVIEW.1 Dynamic earth pressure computation The methods that are used to compute the dynamic earth pressure on the retaining walls nowadays can be classified into three main groups: (1) Limit state analyses, in which a considerable relative movement occurs between the wall and soil to mobilize the shear strength of the soil () Elastic analyses, in which the relative movement in between the soil and wall is limited, therefore the soil behaves within its linear elastic range. The soil can be considered as a linear elastic material. (3) Numerical analyses, in which the soil is modelled with actual non-linear hysteretic behaviour. The limit-state analyses were developed by Mononobe and Okabe (Mononobe and Matuo 199; Okabe 194). The Mononobe-Okabe approach has several variants (Kapila 196, Arango 1969, Seed and Whitman 1970; Richards and Elms 1979; Nadim and Whitman 1983, Richards et al 1999, Choudhury 00). A wedge of soil bounded by the wall is assumed to move as a rigid block, with prescribe a horizontal and a vertical acceleration. This method was basically developed to calculate the active and passive earth pressure for dry cohesionless materials by Mononobe-Okabe. The use of a graphical construction, such as Coulomb or Melbye construction procedure, has been described by Kabila (196). Arango (1969) has developed a simple procedure for obtaining the value of the dynamic lateral earth pressure coefficient for active conditions from standard charts for static lateral earth pressure coefficient for active condition using Coulomb method. The contributions for the elastic analyses came from the works done by Matuo and Ohara (1960), Wood (1973), Scott (1973), Veletsos and Younan (1994a, 1994b, 1997, 000), Li (1999), and Ortigosa and Musante (1991). In particular, Wood (1973) analyzed the dynamic response of homogeneous linear elastic soil trapped in between two rigid walls connected to a rigid base, providing an analytical exact solution. An approximate model proposed by Scott (1973) represents the retaining action of the soil by a set of massless, linear horizontal springs. The stiffness of the springs is defined as subgrade modulus. Veletsos and Younan (1994a, 1994b, 1997, and 000) improved the Scott s model, by using a semi-infinite, elastically supported, horizontal bars with distributed mass, to include the radiational damping of the soil and using horizontal springs with constant stiffness, to model the shearing action of the stratum. Li (1999) included the foundation flexibility and damping into the Veletsos and Younan analyses. In this study, the rigid wall with viscoelastic backfill is considered to rest on viscoelastic half-space foundation. Ortigosa and Musante (1991) proposed a simplified kinematic method, in which the wall is supported in several locations. The possible wall movement is the flexural deformation. The free-field shear modulus is used to calculate the subgrade modulus. 5

20 Chapter Literature review Detailed accounts of pervious analytical and experimental studies on the limit state and elastic analysis matter have been presented by Nazarian and Hadjian (1979), Prakash (1981), Whitman (1991), and Veletsos and Younan (1995). In elastic analyses in which the wall is considered to be fixed against both deflection and rotation at the base, usually the wall pressure and associated forces computed are.5 to 3 times larger than those determined by the Mononobe-Okabe approach, hence elastic solutions are generally believed to be excessively conservative and inappropriate for use in design applications. Conclusions from some recent exploratory studies (Finn et al.1989; Siller et al. 1991; Sun and Lin 1995) suggest that the existing elastic solutions are limited to nondeflecting rigid walls and do not provide for the important effect of wall flexibility. A recent study by Veletsos and Younan (1994b) concluded that for walls that are rigid but elastically constrained against rotation at their base, both the magnitude and distribution of the dynamic wall pressures and forces are quite sensitive to the flexibility of the base constraint. For realistic base flexibilities these effects may be significantly lower than those computed for nondeflecting rigid walls. Moreover, Li (1999) showed that, if foundation compliance is taken into account, the computed based shear may be of the same order of that estimated with Mononobe-Okabe, even for a rigid gravity wall. Therefore, after these studies, the initial limitations to the elastic approach seem to be overcome and this method might be considered as a valuable tool for the seismic design of non-yielding walls. The third group involves nonlinear numerical analysis to find earthquake-induced deformations of retaining walls. Numerical analyses should be capable of accounting for nonlinear, inelastic behaviour of the soil and of the interfaces between the soil and wall. Among the relatively few examples of numerical analyses that are finite element and/or finite difference methods are those reported by Alampalli and Elgamel (1990), Finn et al (199), Iai and Kameoke (1993), Al-Homoud and Whitman (1999), Green and Ebeling (003) Psarropoulos, Klonaris and Gazetas (005) for different type and configurations of retaining walls. The flowchart below (Figure.1) summarizes the theories and method those are used to design the retaining walls in dynamic conditions. 6

21 Chapter Literature review Methods Used to Analyze the Retaining Walls in Seismic Conditions Limite State Analysis Closed Form Solution Active pressure computation without cohession Mononobe - Okabe (196, 199) Arango (1969) [modification for Mononobe-Okabe] Seed - Whitman (1970) [modification for Mononobe-Okabe] Pseudo-Static Approach Passive pressure computation without cohession Mononobe - Okabe (196, 199) Morisson - Ebeling(1995) Soubra (000) Kumar (001) Choudhury (00) Forced Based Analysis Displacement Based Analysis Earth pressure computation with cohession Prakash (1981) Richards - Shi (1999) [interaction model] Choudhury (00) Ortigasa (005) Pseudo-Dynamic Approach Steedman - Zeng (1990) [active pressure computation - only horizontal seismic acceleration] Choudhury - Nimbalkar (005) [active & passive pressure computation - vertical & horizontal seismic acceleration] Richards-Elms model (1979) (Newmark's Method) Nadim - Whitman (1983) [stochastic nonlinear model] Matuo - Ohara (1960) Wood (1973) Scott (1973) Ortigosa - Musante (1991) Veletsos & Younan (1994[a,b], 1996, 000) Li (1999) Numerical Analysis Alampalli - Elgamll (1990) Finn (199) Iai - Kameoke (1993) Al - Homoud & Whitman (1999) Green & Ebeling (003) Psarropoulos, Klonaris,&Gazetas (005) Figure.1. Classification of seismic earth pressure computation methods 7

22 Chapter Literature review. Limit-state analysis..1 Pseudo-static approaches The seismic stability of earth retaining structures is usually analyzed by the pseudo-static approach in which the effects of earthquake action are expressed by constant horizontal and vertical acceleration attached to the mass. The common form of pseudo-static analysis considers the effects of earthquake shaking by pseudo-static accelerations that produce inertia forces, F h and F v, which act through the centroid of the failure mass in the horizontal and vertical directions respectively. The magnitudes of the pseudo-static forces are; where, ah W (.1) Fh = = kh W g av W (.) Fv = = kv W g a h and a v - horizontal and vertical pseudo-static accelerations k h and k v - coefficients of horizontal and vertical pseudo-static accelerations W - weight of the failure wedge. A pseudo-static analysis is relatively simple and very straightforward. Representation of the complex, transient, dynamic effects of earthquake shaking by a single constant unidirectional pseudo-static acceleration is obviously quite crude. Experiences have shown that pseudostatic analysis can be unreliable for soils that build up large pore pressures or show more than about 15% degradation of strength due to earthquake shaking. [see Kramer, 1996]..1.1 Mononobe-Okabe (196,199) Okabe (196) [34], Mononobe and Matsuo (199) [30] were the early pioneers to obtain the active and passive earth pressure coefficients under seismic conditions. It was an extension of Coulomb s method in the static case for determining the earth pressures by considering the equilibrium of a triangular failure wedge. The method is now commonly known as Mononobe - Okabe method. For active and passive cases, planar rupture surfaces were assumed in the analysis. Figure (.) shows the failure surfaces and the forces considered in the analysis. The Mononobe-Okabe approach is valuable in providing a good assessment of the magnitude of the peak dynamic force acting on a retaining wall. However, the method is based on three fundamental assumptions, 1. The wall has already deformed outwards sufficiently to generate the minimum (active) earth pressure. 8

23 . Chapter Literature review. A soil wedge, with a planar sliding surface running through the base of the wall, is on the point of failure with a maximum shear strength mobilized along the length the surface. 3. The soil behind the wall behaves as a rigid body so that acceleration can be assumed to be uniform throughout the backfill at the instant of failure. i. kvw khw plane failure surface ah=khg av=kvg Active. AE P a.. kvw P p. PE khw. i Passive plane failure surface Figure.. Failure surface and the forces considered by Mononobe-Okabe The expression for computing the seismic active and passive earth force, P ae,pe, is given by 1 (.3) P ae, pe = γ H (1 kv) Kae, pe And cos ( φ m β θ ) (.4) K ae, pe = 0.5 sin( φ + δ )sin( φ m i θ cosθ cos β cos( δ ± β + θ ) 1 ± cos( )cos( ) δ ± β + θ i β Where γ - unit weight of soil H - vertical height of the wall K ae,pe - seismic active and passive earth pressure coefficient φ - soil friction angle 9

24 Chapter Literature review δ - wall friction angle β - wall inclination with respect to vertical i - ground inclination with respect to horizontal k h - seismic acceleration coefficient in the horizontal direction k v - seismic acceleration coefficient in the vertical direction k h (.5) θ = tan 1 1 kv Seed and Whitman (1970) [43] gave convenient solutions for practical purposes for the incremental dynamic force in equation (.3) for the active pressure condition, and gave an approximate solution for the case of zero vertical acceleration, a vertical wall, horizontal backfill, and effective friction angle approximately 35º. Their approximation can be expressed as ΔPae 3 (.6) k h γh = 8 in which ΔP ae active wall force increment due to horizontal earthquake load The approximation is in close agreement with the more exact solution for k h <0.35. In Mononobe-Okabe analysis the point of application of the total seismic earth force is considered to be at H/3 from base of the wall, but experimental results (Jacobse 1939, Matsuo 1941) [3] show it is slightly above H/3 from base of the wall for seismic active case. Prakash and Basavanna (1969) [39] have made an analysis to determine the height of the resultant force in the Mononobe-Okabe analysis. Seed and Whitman (1970) [43] recommended that the dynamic component to be taken at 0.6H. Mononobe-Okabe analyses show that k v, when taken as one-half to two-thirds the value of k h, affects total active or passive pressure by less than 10%. Seed and Whitman (1970) concluded that vertical accelerations can be ignored when the Mononobe-Okabe method is used to estimate the total pressure for typical wall designs. The Mononobe-Okabe method is very simple and straightforward, has been used by designers, because experimental and theoretical studies have shown that it gives satisfactory results in cases where the backfill deforms plastically and the wall movement is large and irreversible (Whitman 1990 [56]). However, there are many practical cases, such as massive gravity walls or basement walls braced at top and bottom, where the wall movement is not sufficient to induce a limit state in the soil...1. Arango (1969) Arango has developed a simple procedure for obtaining the value of K ae from standard charts for K a as determined by Coulomb method for static condition, thus providing a convenient 10

25 Chapter Literature review general solution for any inclination of wall and backfill slope, any angle of wall friction and any value of angle of friction of backfill material and earthquake acceleration. The classical Coulomb solution for static earth pressure can be expressed as 1 (.7) Pa = γ H Ka Where, (.8) K a = cos β cos ( δ β ) cos 1 + P a may thus be expressed in the following form (.9) Pa = γh Ka = γh A c cos β Where A c = K a cos β ( φ β ) sin( δ + φ) sin( φ i) cos( δ + β ) cos( β i) On the other hand Mononobe-Okabe value for K ae can be written as follows: P 1 = H 1 k K 1 = γh 1 k 1 cosθ cos γ (.10) ae ( v ) ae ( v ) m Where A m = K ae cosθcos β A β A comparison of the expression for A m with that for A c shows that A m can be determine from the solution for A c by redefining the slope of the back of the wall and as β * and the inclination of the backfill as i *, where β * = β+θ and i * = i+θ Thus And Where A = A β, i = K β, i cos β (.11) ( ) ( ) m c (.1) P = P ( β, i )( 1 k )F ae a ( β ) cos ( β θ ) cos + F = = cosθ cos β cosθ cos β a v Values for F corresponding to different values for β and θ have been computed by Arango. Since charts are available for determining P a for a wide variety of combination of φ, δ, i, and 11

26 Chapter Literature review β, the corresponding dynamic pressure determined by the Mononobe-Okabe analysis can be readily obtained Choudhury (00) Terzaghi (1943) [50] showed that active earth pressures determined assuming a planar rupture surface almost match the exact or experimental values of earth pressures, while for the passive case, when wall friction angle, δ, exceeds one-third of soil friction angle, φ, the assumption of planar failure surface seriously overestimates the passive earth pressures. To correct the error in the Mononobe-Okabe method for the passive case, Morisson and Ebeling (1995), Soubra (000) and Kumar (001) considered curved rupture surfaces in their analysis of the passive case. However, all of these analyses were performed only for sands. Initially Choudhury, Subba Rao and Ghosh gave a complete solution for the distribution of seismic passive pressure behind rigid retaining walls using the method of horizontal slices by considering seismic forces in a pseudo-static manner. Only planar rupture surfaces have been considered and hence wall friction angle has been restricted up to one-third the soil friction angle. This approach results is the same seismic passive earth pressure coefficients as those obtained by Mononobe-Okabe approach, besides giving additional information about the distribution of earth pressures. It has been found that in the seismic case, passive resistance acts at a point other than at 1/3 rd from the base of the wall. Under seismic conditions, the extension of the failure zone is more than that under static conditions. Figure (.3) shows the system considered by Choudhury, Subba Rao and Ghosh, a rigid retaining wall of vertical height H, supporting dry, homogeneous cohesionless backfill material with horizontal ground. The seismic forces are considered as pseudo-static forces along with other static forces. The equilibrium of each elemental slice is considered. It is assumed that the occurrence of earthquake does not affect the basic soil parameters such as soil friction angle φand soil unit weight γ. A displacement C y H a d c b dy B PE Figure.3. Analytical model 1

27 Chapter Literature review py a b Pxtan dy dw kh Rtan Px (1-kv)dW R d c PE py+ dpy Figure.4. Free body diagram In figure (.4), the freebody diagram of an elemental slice shows the action of different forces. The thickness of the slice is dy, at a depth of y from the top ground surface. The vertical pressure p y is acting on the top of the element and (p y + dp y ) on the bottom of the element. The reaction p x normal to the wall and the shear force p x tanδ are acting on the interface between the retaining wall and the backfill material. The normal force R and the shear force Rtanφ act on the sliding surface. The other forces are, the weight dw of the element, the seismic forces dwk h in the horizontal direction and dwk v in the vertical direction. The critical directions of these seismic forces are as shown in figure (.4). The horizontal planes are assumed as principal planes. Resolving all the forces in the vertical and horizontal directions, from the boundary condition p y = 0 at y = 0, and ignoring higher order differential terms and upon simplifying, the expression for the seismic passive earth pressure at any depth y is obtained as: Where (.13) p γ n K + a K ( + a K ) X = 1 H ( ) ( ) ( ) H y + a K H y p X - seismic passive pressure at any depth y from top acting normal to the wall K - lateral earth pressure coefficient used for the static analysis n - (1-k v -k h.b) b - cot ( α + φ) a - PE ( tanφ cotα PE )( 1+ tan β tanδ ) ( tan β + cotα )( 1+ tanφ cotα ) PE PE + ( tanδ tan β ) ( tan β + cotα ) δ - wall friction angle α PE - angle of inclination of the failure plane with horizontal PE 13

28 Chapter Literature review Integrating over the height of the wall, the total passive resistance P X is given by, (.14) P X γ n H = a 1000 a K K γ n H 1 ( ) ( ) + a K ( + a K ) The equivalent seismic passive earth pressure coefficient with respect to normal to the wall is found out as, PX (.15) K pe = γ H The critical value of α PE is obtained by minimizing P X with respect to α PE keeping all other parameters constant and it is found to exactly match with the Coulombic values for the static case. Recently Choudhury (00) [6] has given the complete solution for passive earth pressure coefficients for rigid retaining walls under seismic conditions for variations in parameters, such as wall inclination, ground inclination, wall friction angle, soil friction angle, wall adhesion to soil cohesion ratio, and the horizontal and vertical seismic accelerations. The limit equilibrium method based on a pseudostatic approach was adopted for determining individually the seismic passive earth pressure coefficients corresponding to own weight, surcharge and cohesion components. In the determination of each of these components, composite (logarithmic spiral and planar) failure surfaces were considered. It was considered that the occurrence of an earthquake does not affect the basic soil parameters: unit cohesion c, friction angle φ, and unit weight γ. Uniform seismic accelerations are assumed in the domain under consideration. qgekv F? θ A q i qagkh qag(1-kv) G Wkv qgekh E H H/ Ppcd+Ppqd Pp d Ca (1-kv)W1 H/3 W1 kh C+Ntan N D Wkh PpcR+PpqR i Pp R i y/3 y/ B Figure.5. Composite failure surface and forces considered FB=r 0 FD=r f FA=L DG=y W 1 =weight ABDGA W =weight DGE q=uniform surcharge 14

29 Chapter Literature review The seismic passive force P pd was divided into three components as: 1. Unit weight component P pγd (γ 0,c=q=0). Surcharge component P pqd (q 0,γ=c=0), and 3. Cohesion component P pcd (c 0,γ=q=0) Where γ, c, and q are the unit weight of the soil, unit cohesion, and surcharge pressure respectively. The principle of superposition was assumed to be valid and the minimum of each component was added to get the minimum seismic passive force. Hence, (.16) P pd = Pp γ d + Ppqd + Ppcd The failure surfaces for each of these components that are P pγd, P pqd and P pcd will be different. A particular single failure surface can be found for which the total seismic passive force is minimum. For this failure surface, the corresponding components are not necessarily the minimum values. It was shown that the error magnitude between the method of superposition considering minimum of each component and finding the minimum of total earth force was very small, less than 3%. In figure (.5), the failure surface includes portion BD, which is a logarithmic spiral and a planar portion DE, which is similar to the Rankine passive planar failure surface including the pseudostatic seismic forces. F is the focus of logarithmic spiral and is located at a distance L from A. The initial radius r 0 and the final radius r f of the logarithmic spiral are given by distances FB and FD, respectively. For the force system shown in figure (.5), after satisfying equilibrium equations and Mohr Coulomb criterion of failure, the exit angle ξ at point E on the ground surface becomes π φ (.17) ξ = tan 1 kh 1 k v + i 1 sin 1 sin tan 1 kh 1 k sinφ v i For k h =k v =0, Equation. (.10) yields the same value as given by Rankine and the same value as given by Kumar (001) for k v =0. As given in the figure (.5), the point of application of P pγd is assumed at a height of H/3 from the base of the wall (Chen and Liu 1990), whereas P pqd and P pcd are assumed to act at a height of H/. A uniform surcharge pressure of q is assumed along AE weight of the failure wedge ABDGA is W 1. Cohesive force C is assumed to act on the failure surface BD along with normal force N and frictional force N tan φ. Adhesive force C a is acting on the retaining wall- 15

30 Chapter Literature review soil interface AB. Rankine passive forces P pcr, P pqr, and P pγr are assumed to act on the surface DG. Pseudostatic forces due to seismic weight components for zone DGE are W k h and W k v in the horizontal and vertical directions, respectively. Pseudostatic forces due to seismic weight component for zone ABDGA are W 1 k h and W 1 k v in the horizontal and vertical directions, respectively. Pseudostatic forces due to q.ag.k h and q.ag.k v in the horizontal and vertical directions, respectively, are assumed to act on AG. Similarly pseudostatic forces q.ge.kh and q.ge.kv in the horizontal and vertical directions, respectively, are assumed to act on GE. Considering the moment equilibrium of all forces about the focus F, 1. The seismic passive earth pressure coefficient for the unit weight component K pγd corresponds to the minimum value of the seismic passive earth force P pγd. The minimum value of Ppγd is obtained by considering different logspirals by varying the distance L. The coefficient Kpγd in normal direction to the wall is then obtained as: Pp γd cosδ (.18) K pγ d = γh. The seismic passive earth pressure coefficient for the surcharge component K pqd corresponds to the minimum value of P pqd. The minimum value of P pqd is obtained by varying L. The coefficient K pqd is then obtained as: (.19) K pqd = P pqd cosδ qh 3. The seismic passive earth pressure coefficient for the cohesion component K pcd corresponds to the minimum value of P pcd. By assumption, the cohesion component will not be affected by the seismic accelerations, and hence the static and seismic cohesion values remain the same. The minimum value of P pcd is obtained by varying L. The coefficient K pcd is then obtained as: Ppcd cosδ (.0) K pcd = ch The total seismic passive force P pd on the retaining wall of height H becomes 1 1 (.1) P pd = chk pcd + qhk pqd + γh K pγd cosδ The force P pd acts at an angle δ with the normal to the retaining wall. The points of applications for passive force under seismic condition were determined by Choudhury (00) using the method of horizontal slices. The results showed that the points of applications values ranges from 0.8H to 0.4H from base of the wall with vertical height H, for different seismic conditions and wall inclinations. 16

31 Chapter Literature review Table.1 Comparison of K pγd values obtained by Choudhury s method and available theories in seismic case for β=0º, i=0º, φ=30º δ/φ k h k v Mononobe- Okabe Morrison & Ebeling Chen & Liu Soubra Kumar Choudhury Table.1 shows a comparison of K pγd values obtained from different analyses. For δ/φ=0.5, it is seen that the Choudhury s method results in the least values of the coefficients. However for δ/φ =1.0, it is not necessarily the least in cases for higher k h values. But the difference is very marginal. 17

32 Chapter Literature review Figure.6. Comparison of passive pressure computation using the method proposed by Choudhury (Choudhury et al 00)..1.4 Ortigosa (005) The Mononobe and Okabe expression has the limitation of not introducing soil cohesion, which was later included by Prakash (1981) [38], also employing the Coulomb s wedge method. Based on Prakash s expressions, Ortigosa (005) [31] proposed resolving the problem uncoupling the static and seismic thrust in the following manner: 1. Determining the resultant static thrust, P c, including soil cohesion, c, with tension cracks.. Determining the resultant static plus seismic thrust, P ae, with the Mononobe and Okabe expression, which implicates considering c = Determining the resultant static thrust, P 0, by making c = Determining the seismic thrust component as: 18

33 Chapter Literature review (.) Pe = Pae P0 In this manner, the resultant of the static plus seismic thrust is obtained as: (.3) P ec = Pc + Pe It is important to point out that the thrust uncoupling is valid if P c > 0, which equates to consuming all soil cohesion in the static thrust component. If the cohesion is such that the critical height of the soil is equal to that of the wall, the uncoupling gives P c = 0 and if it is greater gives P c = 0 and an overvalued seismic component. More recently, Richards and Shi (1994) [4] utilize an interaction model between the retaining element and the free field seismic movement of the soil in which they incorporate cohesion... Pseudo-dynamic approach In this approach, the advancement over the previous approach is that the dynamic nature of the earthquake loading is considered in an approximate and simple manner. The phase difference and the amplification effects within the soil mass are considered along with the accelerations to the inertia....1 Steedman-Zeng (1990) Steedman and Zeng (1990) [45] considered the harmonic horizontal acceleration of amplitude a h at base of the wall. For a typical fixed base cantilever wall (figure.7) and assuming i = β = k v = 0 for simplicity. A C Z Qh w Z H h Pae F Vs B ah(t) Figure.7. System considered by Steedman-Zeng Then at depth z below the ground surface the acceleration can be expressed as: 19

34 Chapter Literature review Where, H z (.4) a h( z, t) = ah sin ω t Vs ω - angular frequency t - time elapsed V s - shear wave velocity The planar rupture surface, inclined at an assumed angle α to the horizontal, is considered in the analysis along with the seismic force and weight of failure block. The total seismic active force on the wall is given by, And (.5) Qh ( t)cos( α φ) + W sin( α φ) Pae ( t) = cos( δ + φ α) Where, λγa 4π g tanα h (.6) Q ( t) = [ πh cosωζ + λ(sinωζ sinωt) ] πvs (.7) λ = ω H (.8) ζ = 1 V s h The point of application of the total seismic active force is h d from the base of the wall and given by, (.9) h d π H = H cosωζ + πλh sinωζ λ (cosωζ cosωt) π H cos ωζ + πλ (sin ωζ sin ω t) This point of application of the seismic force for very low frequency motions (small H/λ, so the backfill moves essentially in phase) is at h d = H/3. For higher frequency motions, h d moves upwards from base of the wall. This solution accounts for non uniformity of acceleration within the soil mass but disregards dynamic amplification.... Choudhury-Nimbalkar (005) Steedman & Zeng (1990) did not consider the effect of vertical seismic acceleration on the active earth pressure, which was corrected by Choudhury & Nimbalkar (005) [7]. Also, Choudhury & Nimbalkar (005) considered using pseudo-dynamic method to determine the seismic passive resistance behind a rigid retaining wall. 0

35 Chapter Literature review The effect of variation of different parameters such as wall friction angle δ, period of lateral shaking T, soil friction angle φ, horizontal and vertical seismic coefficients k h, k v, shear wave velocity V s and primary wave velocity V p are considered in the present analysis. A planar rupture surface BC, inclined at an angle α to the horizontal, is assumed for the analysis to avoid further complication of the problem. A C H Ppe Z Qv w Qh F Z h Vs, Vp B ah(t) Figure.8. System considered by Choudhury-Nimbalkar The base is subjected to harmonic horizontal and vertical accelerations of amplitudes a h and a v, the horizontal accelerations at depth z below the top of the wall is give in equation (.15), and the vertical acceleration can be given as: (.30) H z a v = av sin ω t Vp The total horizontal inertia (Q h (t)) force acting on the wall is given in equation (.17). The total vertical inertia force acting on the wall is expressed as: Where, ηγa 4π g tanα v (.31) Q () t = [ πh cosωψ + λ( sinωψ sinωt) ] (.3) η = π Vp ω (.33) v H ψ = t V p The total (static and dynamic) passive resistance can be obtained by resolving forces on the wedge: that is, 1

36 Chapter Literature review (.34) P pe W sin = ( α + φ) Qh cos( α + φ) Qv sin( α + φ) cos( α + δ + φ) Typical results show the highly non-linear nature of the seismic passive earth pressure distribution by this pseudo-dynamic method compared with the existing linear seismic passive earth pressure distribution using a pseudo-static approach. Comparisons of the present method with the available pseudo-static methods are shown, leading to the minimum seismic passive resistance by pseudo-dynamic method...3 Displacement-based analysis A retaining structure subjected to earthquake motion will vibrate with the backfill soil and the wall can easily move from the original position due to an earthquake. The methods available for displacement based analysis of retaining structures during seismic conditions are based on the early work of Newmark (1965) [see Kramer, 1996]. The basic procedure was developed for evaluating the deformation of an embankment dam shaken by earthquake based on the analogy with a sliding block-on-a-plane Richards-Elms model The model proposed by Richards and Elms (1979) [41] is based on the basic Newmark s model, developed originally for evaluation of seismic slope stability, modified for the design of gravity retaining walls. Richards and Elms recommended that the dynamic active earth force calculated using Mononobe-Okabe method is given by, (.35) P 0.5γH (1 ± k v )cos ( φ β θ ) cosθ cos β cos( δ + β + θ ) 1+ ae = The permanent block displacement is given by, Where, Vmaxa (.36) d perm = a V max - ground velocity a max - ground acceleration y 3 max a y - yield acceleration for the backfill-wall system Pae cos( δ + β ) Pae sin( δ + β ) (.37) a y = tanφ b g W 1 sin( φ + δ )sin( φ i θ ) cos( i β )cos( δ + β + θ )

37 Chapter Literature review..4 Comparison of seismic earth pressure values computed using different approaches It is batter to compare the earth pressure values computed using different approaches. The table (.) shows the comparison. For the computation following data have been used: H=6m, c=0, φ=34º, δ=17º, γ=17.3kn/m, k h =0.3, and k v =0.3. A : Forced-based analysis, B : Displacement-based analysis Table. Comparison of seismic earth pressure computation using different approach Method Mononobe- Okabe Choudhury Steedman- Zeng Richards & Elms EC8 Seismic active earth pressure, P ad (kn/m) p.o.a of P ad from base (m) Seismic passive earth pressure, P pd (kn/m) p.o.a of P pd from base Displacement (mm) Remarks A A A B B.3 Closed form solutions using elastic or viscous elastic behaviour.3.1 Wood (1973) The massive gravity walls founded on rock or basement walls braced at both top and bottom, do not move sufficiently to mobilize the shear strength of the backfill soil. Wood (1973) analyzed the response of a rigid nonyielding wall retaining a homogeneous linear elastic soil and connected to a rigid base. For such conditions, Wood established that the dynamic amplification was insignificant for relatively low-frequency ground motions (that is, motions at less than half of the natural frequency of the unconstrained backfill), which would include many or most earthquake problems. For uniform, constant k h applied throughout the elastic backfill, Wood (1973) developed the dynamic thrust, ΔP E, acting on smooth rigid nonyielding walls as: Where, (.38) Δ P E = F k γh P h F P - Dimensionless thrust factor for various geometry and soil Poisson s ratio values 3

38 Chapter Literature review The value of F P is approximately equal to unity (Whitman, 1991) leading to the following approximate formulation for a rigid nonyielding wall on a rigid base: (.39) Δ P E = k γh h As for yielding walls, the point of application of the dynamic thrust is taken typically at a height of 0.6H above the base of the wall. Wood s simplified procedures do not account for: (1) vertical accelerations, () the typical increase of modulus with depth in the backfill, (3) the influence of structures or other loads on the surface of the backfill, (4) the phased response at any given time for the accelerations and the dynamic earth pressures with elevation along the back of the wall, and (5) the effect of the reduced soil stiffness with the level of shaking induced in both the soil backfill and soil foundation. Depending on the dynamic properties of the backfill as well as the frequency characteristics of the input ground motion, a range of dynamic earth pressure solutions would be obtained for which the Mononobe-Okabe solution and the Wood (1973) solution represent a lower and an upper bound, respectively..3. Veletsos and Younan (1994) The system examined by Veletsos and Younan is shown in figure (.9). It consists of a semiinfinite, uniform layer of linear viscoelastic material of height h that is free at its upper surface, is bounded to a rigid base, and is retained along one of its vertical boundaries by a rigid wall. The wall may be either fixed or elastically constrained against rotation at its base, by a spring of stiffness R θ. Both the base of the layer and the wall are subjected to a spaceinvariant, harmonic, uniform horizontal motion, a(t), at any time t., characteristic by a frequency ω and a maximum amplitude A. Material damping for the medium is considered to be of the constant hysteretic type, frequency-independent, and the same for both shearing and axial deformations. y Rigid Wall h R? θ x a(t) Figure.9. Base-excited soil-wall system investigated 4

39 Chapter Literature review The properties of the soil stratum are defined by its mass density (ρ), shear modulus of elasticity (G), Poisson s ratio (υ), and the material damping factor (δ). Veletsos and Younana realized that the Scott s model (1973) fails to provide for the capacity of the medium between the wall and the far field to transfer forces vertically by horizontal shearing. In addition to the horizontal normal stresses and inertia forces, a horizontal element of the medium is acted upon along its upper and lower faces by horizontal shearing stress, the difference of which Δτ is given by: (.40) τ Δτ = y xy 1 τ xy = h η With η being the dimensionless distance given by η=y/h. On the assumption that the horizontal variation of the vertical displacements is negligible, τ xy can be expressed as: (.41) τ = G u h η G Δτ = h u η xy and Where u is the relative horizontal displacement of the medium with respect to the moving base. If u is expressed by the method of separation of variables as a linear combination of modal terms, through mathematical manipulation (Veletsos and Younana, 1994b), the n th component of Δτ, denoted as (Δτ) n, may be recognized to be: Δ τ = ρω u (.4) ( ) n n n Where u n is the n th component of the displacement u and ω n is the circular frequency of the stratum, considered to respond as a cantilever shear-beam, given by: ( n 1) πvs (.43) ωn = h Where n refers to the mode being considered and V s is the shear wave velocity of the stratum. The equation (.33) represents a force per unit of length that is identical to the force induced by a massless linear spring of stiffness k n given by: (.44) k n = ρω = n ( n 1) π G h This corresponds to modeling the shearing action of the medium, for each modal component, with a set of horizontal linear spring of constant stiffness k n, connected at their lower ends to the common base, subjected to the prescribed ground acceleration a(t). The other end of the spring connected to the medium that may be modeled by a series of semiinfinitely long, elastically supported horizontal bars with distributed mass (For more detail see []). In 1996, Veletsos and Younan analyzed the response of flexible cantilever retaining walls that are elastically constrained against rotation at their base for horizontal ground shaking. The 5

40 Chapter Literature review retaining medium is idealized as a uniform, linear, viscoelastic stratum of constant thickness and semi-infinite extent in the horizontal direction. The parameters varied including the flexibilities of the wall and its base, the properties of the retained medium, and the characteristics of the ground motion. They discovered that the dynamic pressures depend profoundly on both the wall flexibility and the foundation rotational compliance, and that for realistic values of these factors the dynamic pressures are substantially lower than the pressures for a rigid, fixed-based wall. In fact, they found out that the dynamic pressures may reduce to the level of the Mononobe Okabe solution if either the wall or the base flexibility is substantial. More recently (000), Veletsos and Younan [60] have proposed a solution technique for the dynamic analysis of flexible both cantilever and top-supported walls. However, these analytical solutions are based on the assumption of homogeneous retained soil, and there are reasons for someone to believe that the potential soil inhomogeneity may lead to significant changes in the magnitude and distribution of the dynamic earth pressures. Furthermore, as the presence of the foundation soil layers under the retained system is only crudely modelled through a rotational spring, these solutions do not account for the potential horizontal translation at the wall base, which in general may have both an elastic and an inelastic (sliding) component..4 Numerical analyses Earthquake-induced pressures on retaining walls can also be evaluated using dynamic response-analyses. A number of computer programs are available for such analyses. Linear or equivalent linear or non-linear analysis can be used to estimate wall pressures. Non-linear analyses are capable of predicting permanent deformations as well as wall pressure..4.1 Al-Homoud and Whitman (1999) A finite element numerical model has been developed for gravity walls founded on dry sand by Al-Homoud and Whitman (1999), using Weidlingger Associates two-dimensional (D) finite element computer code, FLEX. Dynamic analyses in FLEX are performed using an explicit time integration technique. The suggested model for studying the dynamic response of rigid gravity wall can be summarized as follows: 1. The soil (dry sand in this study) is modelled by a D finite element grid. This includes the backfill material and the foundation soil.. The gravity retaining wall is modeled as a rigid substructure. 3. The strength and deformation of the soil material are modeled using the viscous cap constitutive model. This model consists of a failure surface and a hardening cap together with an associated flow rule. The cap surface is activated only for the soil under the wall to represent compaction during wall rocking. In addition, viscoelastic 6

41 Chapter Literature review behavior is provided for the state of stress within the region bounded by these surfaces, so as to provide for hysteretic-like damping of soil during dynamic loading. 4. Interface (continuum approximation) elements are used between the soil and the wall (at the back face of the wall and under its base), allowing for sliding and for the opening and closing of the gaps (i.e. debonding and bonding). 5. The finite element grid is truncated by using an absorbing boundary approximation developed by Lysmer and Kuhlemeyer. This proposed model was verified by comparing its prediction to results from three dynamic centrifuge tests conducted by Andersen et al (1987). Some of the conclusions which were made from above study were summarized below. For more details see Al-Homoud and Whitman [5]. 1. The results from the analysis shows the outward tilt of rigid abutments is the dominant mode of response during dynamic shaking and that these walls end up with a permanent outward tilt at the end of shaking.. The results from the current study showed that the Seed Whitman [8] simplified equation is conservative while the location of the maximum dynamic earth force is higher than 0.6H above the base, which is the value suggested by Seed and Whitman [8]..4. Green and Ebeling (003) A research investigation was undertaken to determine the dynamically induced lateral earth pressure on the stem portion of a concrete cantilever earth retaining wall with dry medium dense sand by Green and Ebeling (003). The numerical model has been developed using FLAC finite difference code. The results obtained from above numerical model were compared with the results from simplified techniques for estimating the permanent wall displacement and the dynamic earth pressures. In this investigation, highly non-linear model has been developed to cover almost all the aspects, such as non-linear behaviour of soil and interface in between the wall and soil. The model was numerically constructed similarly to the way an actual wall would be construed. From the analyses, the lateral earth pressure coefficients have been computed and checked with those values computed using Mononobe-Okabe equation. 7

42 Chapter Literature review Figure.10. Mononobe-Okabe active and passive expressions (yielding backfill), Wood expression (nonyielding backfill), and FLAC (Continued) (after Green & Ebeling 003) The repot also presents the total resultant force, incremental dynamic force and their point of application. Permanent relative displacement of the wall has been computed using Newmark sliding block-type analysis.4.3 Psarropoulos, Klonaris, and Gazetas (005) A study has been carried out by Psarropoulos, Klonaris, and Gazetas to validate the assumptions of Veletsos and Younan analytical solution and to define the range if its applicability. The numerical models were developed using the commercial finite-element package ABAQUS. Pre summing plane-strain conditions, the numerical model was twodimensional. The versatility of the finite-element method permits the treatment of some more realistic situations that are not amenable to analytical solution. So the modelling was extended to account for: (a) soil inhomogeneity of the retained soil, and (b) translational flexibility of the wall foundation. They studies three different type of soil retaining system, 1. coincides with the single-layer case in where the retained soil is characterized by homogeneity.. models the same single-layer case, but the retained soil is inhomogeneous, with the shear modulus vanishing at the soil surface. 8

43 Chapter Literature review 3. refers to a rigid wall founded on a soil stratum. The results show that the inhomogeneity of the retained soil leads to reduced earth pressures near the top of the wall, especially in the case of very flexible walls, while the compliance of the foundation may not easily be modelled by a single rotational spring, due to wave propagation phenomena. In case of homogeneous soil wall system, the factors examined are the characteristics of the ground motion, the properties of the soil stratum, and the flexibilities of the wall and the rotational constraint at its base. Emphasis is given on the long-period effectively static harmonic excitations. The response for a dynamically excited system is then given as the product of the corresponding static response with an appropriate amplification (or deamplification) factor. The whole approach is based on the following simplifying assumptions: (1) no de-bonding or relative slip is allowed to occur at the wall-soil interface, () no vertical normal stresses develop anywhere in the medium, i.e. σ y = 0, under the considered horizontal excitation, (3) the horizontal variation of the vertical displacement are negligible, and (4) the wall is considered to be massless. Quasi-static response and harmonic response with resonance and high frequency has been analyzed in the work. In reality, the soil shear modulus is likely to increase with depth. Such inhormogeneity reflects in very simple way, not only the unavoidably-reduced stiffness under the small confining pressures prevailing near the top, but two more strong shaking effects: a. the softening of the soil due to the larger shearing deformations, and b. the non-linear wall-soil interface behaviour, including separation and slippage. This feature is too complicated to be incorporated in analytical formulations. Veletsos and Younan have examined analytically the inhormogeneity with depth, regarding a rigid wall elastically constrained against rotation at its base. To simplify, the equations of motion a specific parabolic variation of the shear modulus was used. But the greatest advantage of the finite element method allows including the flexibility of the wall and/or additional variations of the shear modulus to extend the analytical solution. The resultant force and the corresponding overturning moment values for inhomogeneous soil are substantially lower compared to those of homogeneous soil. The third category was the two-layers system (gravity wall founded on soil) has been numerically analyzed. In the aforementioned single-layer models the rotational stiffness of the wall foundation is simulated by a rotational elastic constraint at the base of the wall. It is evident that in this way, while the potential rotation of the wall is taken into account, horizontal translation is not allowed, thus reducing by one the degrees of freedom of the system. That simplification is expected to have a substantial effect on the response of the retaining structure. In this part of the study, in order to assess this effect, a more realistic model is examined, in which wall and the retained soil overlie a linearly visco-elastic soil layer. As the aim in this section is to evaluate the role of the wall foundation, only rigid gravity walls are examined. 9

44 Chapter Literature review From the static response analyses, it is observed that in general the increase in the degrees of freedom of the system leads to a decrease of the induced wall pressures. As it was already stated, the replacement of the Veletsos and Younan rotational spring at the base of the wall by an actual elastic soil layer introduces an additional degree of freedom to the system: the horizontal (transverse) elastic displacement of the wall. Consequently, the wall soil system becomes more flexible, which, as anticipated, leads to a decrease in the wall pressures. Furthermore, the decrease in pressures is more noticeable when the base width-height (B/H) ratio attains relatively high values. This observation may be easily explained through the following example: consider two systems with identical soil profile geometry, and wall widths B 1 =B and B =B. But the rotational stiffness, K r, is approximately proportional to the square of the wall width (B). This fact implies that for a given (constant) value of K r, the stiffness of the soil supporting the wider wall has to be about a quarter of that supporting the narrower wall for the rotational stiffness to remain the same. In turn, the horizontal stiffness, K h, is proportional to the stiffness of the underlying soil. Thus, the horizontal stiffness of the wider wall will be substantially lower than that of the narrower wall. For the particular cases examined herein, the horizontal stiffness of the wall with width 0.8H is about 30% of that of the wall with width 0.4H. So, although the two systems have identical rotational stiffness (d θ ), the overall flexibility of the wall soil system is higher in the case of B=0.8H, which is reflected on the resulting pressure distributions. Harmonic response analyses have been also carried out to examine the influence of the underlying soil layer to the dynamic characteristics of the system. Generally, the remarks made for the statically excited systems apply for the case of resonance as well. Furthermore, the dynamic texture of the excitation amplifies the discrepancies observed in the static case. It is of great interest to examine the shear-base and the overturning-moment maximum dynamic amplification factors. According to the spring model, the more flexible the wall soil system is, the higher the dynamic amplification factors are. The consideration of a more realistic model, as the one adopted in this study, leads to the opposite conclusion. The discrepancy between the two approaches can be justified as follows: In the spring model the stiffness of the rotational constraint is real-valued, and therefore, the damping capacity of the wall itself cannot be taken into consideration. As a consequence, the impinging waves on the wall cannot be dissipated, while the rotational oscillation of the wall increases the wave amplitude. So, in the spring model, the increase in the wall base flexibility leads to higher values of the dynamic amplification factors. On the contrary, at the present approach the rotation of the wall is governed by not only the rotational stiffness, but the damping characteristics (radiation and material damping) of the foundation layer, as well. In this way, the wave energy can be dissipated by the boundaries of both the retained and the underlying soil. Additionally, higher values of impedance contrast cause larger wave dissipation, and consequently, smaller dynamic amplification. 30

45 Chapter 3 Type of retaining walls analyzed 3. TYPE OF RETAINING WALLS ANALYZED The previous chapter presents an overview of earth pressure analysis on earth retaining structures and on past works that support or find drawback of the methods. The following chapter presents the three case-studies analyzed in this thesis. 3.1 Diaphragm wall-soil system The dimension of diaphragm wall analyzed in this investigation is illustrated in figure (3.1). The wall is designed for static forces using Rankine theory pressure distribution [See Appendix - A]. Soil is assumed homogeneous with the properties given in the table (3.1 &3.). The water table and bed rock are well below the domain considered for modelling. 0.5 m Retaining W all 6 m 5 m Figure 3.1. Dimension of diaphragm wall The primary parameters governing the dynamic response of the system are the relative flexibility of the wall and retained medium and relative flexibility of the rotational point constrain given by retained soil. The characteristics of the base motion also affect the response. The seismic loads acts with static loads on the wall when it is subjected to dynamic analysis. Seismically induced shear and bending moments must be considered in design, but are not important in global stability checks. A dry site (i.e, no water table) will be analyzed in this first of a series of analyses of diaphragms using DIANA (DIsplacement ANAlyzer). This gives a better understanding of the dynamic behavior of diaphragm wall retaining dry backfill before adding additional complexities associated with submerged or partially submerged backfills. This report summarizes the results of detailed numerical analyses performed on a diaphragm wall. The detailed numerical analyses were performed using the commercially available computer program DIANA. 31

46 Chapter 3 Type of retaining walls analyzed 3. Cantilever wall-soil system The dimension of diaphragm wall analyzed in the investigation is illustrated in figure (3.). The wall was designed for static forces using Rankine theory pressure distribution [See Appendix - B]. Soil is assumed homogeneous with following properties given in the table (3.1 & 3.). The water table and bed rock are well below from the domain used for modelling. 0.9 m.4 m Cantilever Wall Back Fill 6 m Toe 4 m Heel Figure 3.. Dimension of cantilever wall A dry site (i.e, no water table) will be analyzed in this first of a series of analyses of cantilevers using DIANA (DIsplacement ANAlyzer). This gives a better understanding of the dynamic behavior of cantilever wall retaining dry backfill before adding additional complexities associated with submerged or partially submerged backfills. 3.3 Gravity wall-soil system The dimension of diaphragm wall analyzed in the investigation is illustrated in figure (3.4). The wall was designed using the traditional approach to seismic design for an earthquake with a peak acceleration of 0.g [See Appendix - C]. Soil is assumed homogeneous with following properties given in the table (3.1 & 3.). The water table is well below from the domain used for modelling. The bedrock is fond at 1m below from the free surface. 3

47 Chapter 3 Type of retaining walls analyzed 0.8 m Gravity Wall Back Fill 8.0 m 3.0 m Figure 3.3. Dimension of gravity wall A dry site (i.e, no water table) will be analyzed in this first of a series of analyses of gravity walls using DIANA (DIsplacement ANAlyzer). This gives a better understanding of the dynamic behavior of gravity wall retaining dry backfill before adding additional complexities associated with submerged or partially submerged backfills. 3.4 Properties of soil The non-linear dynamic analyses were carried out for all the walls mentioned above with dense sand and clay soils separately. The properties of those soils are given in the following tables. Table 3.1 Properties of sand Parameters Value Relative density (%) 75 Total unit weight (kn/m 3 ) 19.6 Peak effective angle of internal friction 40º Residual effective angle of internal friction 35º Constant volume friction angle 30º OCR 1.00 Porosity (%) 45 Permeability (cm/s) 10-3 Cyclic undrained shear strength (kpa) 34 33

48 Chapter 3 Type of retaining walls analyzed Table 3. Properties of clay Parameters Value Total unite weight (kn/m 3 ) 18 Peak effective angle of internal friction 8 Drained cohesion (kn/m ) 10 OCR 1.15 Elastic modulus (kn/m ) 30,000 Permeability (cm/s) Properties of concrete The retaining walls were constructed using reinforced concrete. The properties of the concrete mixture and steel are given in the table (3.3). Table 3.3 Properties of reinforced concrete Parameters Value Unite weight (kn/m 3 ) 3.6 Compressive strength of concrete (MPa) 30 Poisson ratio 0.0 Yield strength of reinforcement (MPa)

49 Chapter 4 Selection and processing of ground motion 4. SELECTION AND PROCESSING OF GROUND MOTION 4.1 Selection criteria The selection criteria of time-history for dynamic analyses of a numerical system are summarized below: (1) A real earthquake motion was desired, not a synthetic motion. () The earthquake magnitude and site-to-source distance corresponding to the motion should be representative of characteristic scenarios at the site. (3) The motion should have been recorded on rock or stiff soil. The criteria described above were used to assemble a list of candidate acceleration timehistories. By applying additional conditions to these candidate time-histories, a certain number of suitable time-histories can be selected for dynamic analyses. The response of nonlinear dynamic soil-structure system may be strongly affected by the time-domain character of time-histories even if the spectra of different time-histories are nearly identical. More time-histories are required for nonlinear dynamic analyses than for linear analyses. The dynamic response of nonlinear system may be influenced by frequency content, shape, and number of pulses of time-history, in addition to the response spectrum characteristic. However, for this research investigation, only three real acceleration timehistories were selected for use in the dynamic analyses. 4. List of ground motion The acceleration time histories used in dynamic analyses of numerical model are listed below. Table 4.1 Ground motion Earthquake Station PGA (g) Significant duration (s) Imperial Valley (1940) 117 EL Centro Array Chi-Chi (1999) CHY 006N Chi-Chi Kobe (1995) KJMA Kobe These records were obtained by searching the Strong Motion Database maintained by the Pacific Earthquake Engineering Research (PEER) center. ( 35

50 Chapter 4 Selection and processing of ground motion 4.3 Characteristics of ground motion selected As stated above, three acceleration time-histories were selected and include, Imperial Valley (1940), Chi-Chi (1999), and Kobe (1995), corresponding to low, medium, high PGA (g) respectively and were used to analyze the numerical models. Figures below are the acceleration time-history, pseudo-acceleration response spectrum corresponding 5 percentage damping, and Arias intensity of acceleration time-histories describe above Acceleration (g) Time (s) Figure 4.1. Imperial Valley (1940) acceleration time-history Pseudo-Acceleration (g) Period (s) Figure 4.. Imperial Valley (1940) Pseudo-Acceleration spectrum corresponding 5% damping 36

51 Chapter 4 Selection and processing of ground motion Arias intensity (%) D 5, 95 =0.34s Time (s) Figure 4.3. Imperial valley (1940) Arias intensity Acceleration (g) Time (s) Figure 4.4. Chi-Chi (1999) acceleration time-history 37

52 Chapter 4 Selection and processing of ground motion Pseudo-Acceleration (g) Period (s) Figure 4.5. Chi-Chi (1999) Pseudo-Acceleration spectrum corresponding 5% damping Arias intensity (%) D 5, 95 =6.03s Time (s) Figure 4.6. Chi-Chi (1999) Arias intensity 38

53 Chapter 4 Selection and processing of ground motion Acceleration (g) Time (s) Figure 4.7. Kobe (1995) acceleration time-history 3.5 Pseudo-Acceleration (g) Period (s) Figure 4.8. Kobe (1995) Pseudo-Acceleration spectrum corresponding 5% damping 39

54 Chapter 4 Selection and processing of ground motion Arias intensity (%) D 5, 95 =8.36s Time (s) Figure 4.9. Kobe (1995) Arias intensity 4.4 Processing of the selected ground motions Several stages of processing are required to selected acceleration time-histories before using that as an external loading to a numerical model. In some cases a record is simply scaled upward by a factor in between two to three without distorting the realistic characteristic of the ground motion. This type of upward scaling may be desired even though the record has a high PGA, and induced largest permanent relative displacement. But, still that displacement is not enough to ensure active earth pressure. The second processing stage involves filtering high frequencies and computing the ground motion at the base of the numerical model; both operations are required for finite element computation. In finite element formulation, the mesh size perpendicular to the wave propagation direction and wave propagation velocity of the material limit the maximum frequency that can be transferred through the finite element mesh. Generally the frequencies above 15Hz are not significant for soil-structure system. Because of these reason removal of frequencies above 15Hz from ground motion is typical, however if a soil-structure system has higher natural frequency above 15Hz then the cut off frequency is increased well beyond the system natural frequency. The filtering operation was done using Seismosignal, for that Bessel filter type and low-pass filter configuration had been selected. The outcrop ground motion was simply divided by two and applied at the base of the numerical model. This idea came from one-dimensional wave propagation theory. For the purpose of illustrating the features of the boundary formulation, the vertical propagation of shear waves is considered. The equation of motion may be expressed as: (4.1) ρ u, tt = Gu, xx Where a comma is used to indicate partial differentiation ρ mass density 40

55 Chapter 4 Selection and processing of ground motion G - shear modulus u - horizontal displacement t - time x - depth coordinate, with the x -coordinate assumed oriented upwards positively. The fundamental solution of equation (4.1) can be expressed as: x x = c c (4.) u ( x, t) I t + R t + G where c =, ρ and I and R are two arbitrary functions of their arguments: I (t-x/c) represents a wave motion propagating upwards in the positive x -direction with the velocity c, and is referred to as the incident motion; R (t+x/c) presents a wave motion propagating downwards in the negative x - direction with the velocity c, and is referred to as the reflected motion. The boundary at x=h is free, setting τ (h,t)=0 leads to: x h x h (4.3) R t = I t c c Resulting in the total wave motion: x h c (4.4) u ( x, t) = I t + I t + x h c Therefore, at a free boundary, the incident wave is reflected back with the same shape and the same sign. The motion is doubled at the free surface. This is the reason that the out crop motion is simply divide by two applied at the base, if the soil is homogeneous. The figure (4.10) shows the wave motion through a semi-infinite layered soil. 41

56 Chapter 5. Modelling issues and choices 5. MODELLING ISSUES AND CHOICES This chapter provides the general issues and choices such as soil constitutive model, boundaries and interface element in finite-element modeling of soil structure system. It also gives the information about advanced soil constitutive models and absorbing boundaries that are used nowadays in commercial finite-element codes. Subsequent part of this chapter specially dedicates for DIANA modelling issues and choices. 5.1 Finite element modeling of soil-structure system Two important characteristics that distinguish the dynamic soil-structure system from other general dynamic structural systems are the unbounded nature and the nonlinearity of the soil medium. Generally, when establishing numerical dynamic soil-structure models, the following problems should be taken into account: 1. Radiation of dynamic energy into the unbounded soil. The hysteretic nature of soil damping 3. Separation of soil from the structure 4. Possibility of soil liquefaction under seismic loads 5. Other inherent nonlinearities of the soil and the structure However, due to the complexity of dynamic soil-structure behaviour, numerical modeling of this phenomenon still remains a challenge. There still exist many difficulties to cover in one model all the aspects listed above. Current models usually stress one or several of these problems Soil constitutive model A constitutive model is a mathematical model that describes the material behaviour, and exhibits a wide range of complexity in engineering. To describe the material behavior, one needs to consider both the properties of the material and nature of external excitation. It is difficult to develop a general model that covers all aspect of material behavior. The same material may exhibit very different patterns when subjected to different external loadings. Consequently, it is usually necessary to focus on a specific material and a specific external loading of interest. The behaviour of soil under earthquake loading is complex. It is essential that the constitutive model used is able to capture the important features of the soil behaviour under cyclic loading such as permanent deformation, dilatancy, hysterisis and damping, etc. Constitutive models based on plasticity formulations (e.g., Iwan, 1967; Dafalias and Herman, 198; Pastor et al., 1985; Wathugala and Desai, Bardet, 1995) have contributed significantly to the development of analytical procedures (e.g., Zienkiewicz et al., 1984 and 1990). However, several important aspects of dynamic soil behavior are not yet incorporated into these models, including cyclic mobility, post-liquefaction behavior, large deformation potential of sandy soils, cyclic 4

57 Chapter 5. Modelling issues and choices modulus degradation of cohesive soils, low-mean effective stress behavior of soils, and strainrate effects (Ishihara, 1993). Normally, Mohr-Coulomb, modified Mohr-Coulomb, egg cam-clay or Durcker-Prager can be used for the finite element analyses. Some of most advanced constitutive laws for soil that are used in finite element modeling are listed below. All of these models include plasticity and work hardening Mohr-Coulomb The yield condition of Mohr Coulomb is an extension of the Tresca yield condition to a pressure dependent behavior. The formulation of the yield function can be expressed in the principal stress space (σ 1 > σ > σ 3 ) as: 1 1 (5.1) f ( σ, k) = ( σ1 σ 3) + ( σ1 + σ 3) sinφ( k) c( k) cosφ0 with c(k) the cohesion as a function of the internal state variable k, and φ the angle of internal friction which is also a function of the internal state variable. The initial angle of internal friction is given by φ 0. The flow rule is given by a general non-associated flow rule g not equal to f, but with the plastic potential given by, 1 1 ( which results for the plastic strain rate vector (5.) g σ, k) = ( σ σ ) + ( σ + σ ) sinψ ( ) (5.3) ε = p λ 1 1 where ψ is dilatancy angle. 1 k ( 1+ sinψ ) 0 ( 1 sinψ ) The relation between the internal state variable k and the plastic process is given by the hardening hypothesis. For the Mohr Coulomb yield condition it considers only the strain hardening hypothesis. In the case of strain hardening the relation is given in the principal space by, (5.4) = p p + p p + p p k 3 ε 1 ε1 ε ε ε3 ε3 which can be elaborated to k = λ (5.5) 1+ sin ψ 43

58 Chapter 5. Modelling issues and choices Pastor and Zienkiewicz(l986) [P-Z mark III model] The P-Z mark III model is a generalised plasticity-bounding surface model with non associated flow rule. The model is described by means of potential surfaces given by: (5.6) G( p', q, p ) g q M 1 p' α g p' p g α = g where p is the mean confining stress, q is the deviatoric shear stress, M g, is slope of the Critical State line, α g, is a constant and p g is a size parameter. The shapes of yielding surfaces and potential surfaces follow the same family of curves given by above equation. For the analysis the parameter M g, is obtained from the effective angle of friction φ of the soil and Lode s angle θ by Mohr-Coulomb relation; (5.7) M g 6sinφ'sin3θ = 3 sinφ'sin3θ M g is determined by assuming that sinφ is constant and by considering M g =M gc when θ = π/6. M gc is obtained from the triaxial compression tests. The dilatancy of sands is approximated as suggested by Nova and Wood (198); (5.8) d ( 1+ α )( M η) = g g Where η is the stress ratio (q/p ). The direction of plastic flow is defined by means of a unit normal n g given by, (5.9) { n } { d, s} T g 1 = for loading 1+ d { s} T (5.10) { n } = abs( d ), g 1 1+ d for unloading Where s=+l during compression and s=-1 during extension. The typical parameters that are required for this constitutive model are presented in table (5.1) with typical values for medium dense sand. 44

59 Chapter 5. Modelling issues and choices Table 5.1 Typical parameters for P-Z model HiSS soil model [Hierarchical single surface soil model] A nonlinear soil model HiSS has been used to introduce the effect of plasticity. There is a series of these models, as mentioned in Wathugala and Desai (1993). Both plasticity and work hardening of the soil are considered in the model, which is based on an incremental stress strain relationship and assumes associative plasticity. Further, this version assumes the constitutive relationship for nonvirgin loading (i.e., loading or unloading) to be elastic. A simplified formulation used for virgin loading in HiSS is described here. Further details can be found in Wathugala and Desai. In this model, a material parameter β is used to define the shape of the yield surface in the octahedral plane. Assuming β = 0, the dimensionless yield surface F can be simplified as (5.11) J F = p D a J + α ps p 1 a η J γ p 1 a Where J 1 is the first invariant of the stress tensor σ ij ; J D is the second invariant of the deviatoric stress tensor; p a is the atmospheric pressure; α ps is the hardening function; and γ and η are material parameters that influence the shape of F in J1-(J D ) 0.5 space. The parameter η is related to the phase-change point, which is defined as the point where material changes from contractive to dilative behaviour [Figure (5.1)]. The hardening function, α ps, is defined in terms of plastic strain trajectory, ξ ν, as h1 (5.1) α ps = h ξ v 45

60 Chapter 5. Modelling issues and choices Where h 1 and h are material parameters and ξ ν denotes the trajectory of the volumetric plastic strain. Typical yield surfaces for this model are shown in figure (5.1). Figure 5.1. Shape of yield surfaces in J 1 -J D space Hyperbolic type Osaki model The stress-strain relationship of soil can be divided into the volumetric and deviatoric components. For the deviatoric component, the hyperbolic type Osaki model can be adopted, as shown in figure (5.). In this model, loading, unloading and reloading paths are prescribed by the relationship of second invariant of deviatoric stress and strain, as formulated in equation (5.8), defined by the initial shear stiffness and shear strength (Ohsaki 1980). Figure 5.. Non-linear constitutive law for soil Where, (5.13) ' J G0 = 1 + M G0M 100S u J SuM J 1 B J - second invariants of deviatoric stress J ' - second invariants of deviatoric strain 46

61 Chapter 5. Modelling issues and choices G 0 - initial shear modulus (N/mm) S u - shear strength at 1% shear strain (N/mm) B - material parameter (sand: 1.6, clay: 1.4) M - hysteretic parameter (loading: 1.0, unloading/reloading:.0). Initial shear modulus can be calculated with equation (5.9), formulated for Gifu sand used in the experiment (Ishida et al. 1981), and shear strength is evaluated by Coulomb's friction theory, described by equation (5.10) Where, (.17 e) (5.14) G0 = 630 σ c 1+ e (5.15) S u = ccosφ + σ c sinφ e - void ratio σ c confining pressure c cohesion φ internal friction angle For sandy soil, variations in relative density (or void ratio) and confining stress strongly affect the static and dynamic behavior of the soil. Additionally, positive and negative dilatancy arises due to repeated shear deformation that results in nonlinear volumetric behavior. This model had already been verified for to 1% shear strain amplitude in the stiffness degradation and the variation of material hysterics damping (Ohsaki 1980) Boundaries For computational efficiency it is desirable to minimize the number of elements in a finiteelement analysis. Minimizing the number of elements usually becomes a matter of minimizing the size of the discretized region. As the size of the discretized region decreases, the influence of boundary conditions becomes more significant. For many dynamic response and soil-structure interaction problems, rigid or near-rigid boundaries such as bedrock are located at considerable distances, particularly in the horizontal direction, from the region of interest. As a result, wave energy that travels away from the region of interest may effectively be permanently removed from that region. In a dynamic finite-element analysis, it is important to simulate this type of radiation damping behaviour. The most commonly used boundaries for finite-element analyses can be divided into three groups, 47

62 Chapter 5. Modelling issues and choices 1. Elementary boundaries. Local or transmitting boundaries 3. Consistent boundaries Elementary boundaries Conditions of zero displacement or zero stress are specified at elementary boundaries. Elementary boundaries can be used to model the ground surface accurately as a free (zero stress) boundary. For lateral or lower boundaries, however, the perfect reflection characteristics of elementary boundaries can trap energy in the mesh that in reality would radiate past the boundaries and away from the region of interest. The resulting box effect can produce serious errors in a ground response or soil-structure interaction analysis. If elementary boundaries are placed far enough from the region of interest, reflected wave may be damped sufficiently to negate their influence Local or transmitting boundaries The simulation of waves by finite-difference or finite-element methods in unbounded domains requires a specific treatment for the boundaries of the necessarily truncated computational domain. Two solutions have been proposed: absorbing boundary conditions and absorbing layers. The most common analytical soil-structure models are based on the assumptions that the soil domain may be represented by an elastic half space and that dashpots may be used to represent the absorbing boundary conditions (Wolf 1985). These boundary conditions are required to model both radiation damping of the waves propagating outward into the infinite domain and to prevent reflections back into the system from any artificially introduced finite domain of the half-space. In 1969, Lysmer &Kuhlemeyer developed absorbing boundaries only with dashpots. This type of absorbing boundary is used in most of the finite-element and finite different codes, such as DYNOFLOW, DIANA, FLAC and ABAQUS. Lysmer & Kuhlemeyer have investigated different possibilities for expressing this boundary condition analytically and have found that the most promising way is to express it by the conditions (5.16) σ (5.17) τ xx xy = aρv p u t u y = bρvs t x In which σ xx and τ xy are the normal and shear stress, respectively; u x and u y are the normal and tangential displacement respectively; ρ is the mass density; V s and V p are the velocities of S- waves and P-waves, respectively; and a and b are dimensionless parameters, usually a=b=1. 48

63 Chapter 5. Modelling issues and choices Figure 5.3. The dashpot model proposed by Lysmer & Kuhlemeyer The absorption cannot be made perfect over the whole range of incident angles by any choice of a and b. It has been shown that the nearly perfect absorption is obtained in the range incident wave angle grater than 30º for a=b=1. While carrying out the finite-element analysis the boundaries are rendered non-reflective by various schemes such as Smith-Cundall boundary or Lysmer-Kuhlemeyer boundary. An elegant scheme based on compound parabolic collectors was developed by Madabhushi, (1993). In this scheme the boundary is modelled with a compound parabolic shape which will transmit stress waves arriving at any angle less than the angle of receptance into a region where they undergo multiple reflections. This scheme is particularly suitable for transient analysis. These types of boundaries are used in SWANDYNE code. Figure 5.4. Compound parabolic callectors Absorbing layers model are an alternative to absorbing boundary condition. The idea is to surround the domain of interest by some artificial absorbing layers in which wave are trapped and attenuated. For elastic waves, several models have been proposed. For instance, Sochacki et al suggest adding inside the layers some attenuation term, proportional to the first time derivative of the displacement to the elastodynamic equations. This technique is inspired by Physics and revealed to be quite delicate in practice. The main difficulty is that, when entering the layers, the waves see the change in impedance of the medium and then is reflected artificially into the domain of interest. The use of smooth and not too high attenuation profiles allows user to weaken the difficulty but requiring the use of thick layers. 49

64 Chapter 5. Modelling issues and choices Consistent boundaries Boundaries that can absorb all types of body waves and surface waves at all angles of incidence and all frequencies are called consistent boundaries. Consistent boundaries can be represented by frequency-dependent boundary stiffness matrices obtained from boundary integral equations or boundary element method. Wolf (1991), for example, developed a lumped-parameter model consisting of an assemblage of discrete springs, mass and dashpots which can approximate the behaviour of a consistent boundary Figure 5.5. Lumped-parameter consistent boundary Soil-structure interface Based on SSI analyses of four hypothetical earth retaining structures, Ebeling, Duncan, and Clough (1990) concluded that the interface shear stiffness has a significant influence on the distribution of forces on the structure. They performed two different analyses of the same structure using the expected maximum and minimum values of shear stiffness of the backfillto-structure interface. A difference of 1.5 percent was found between the values of friction angle mobilized at the base of the structure for the two analyses. Filz (199) and Filz and Duncan (1997) showed that the distribution of the backfill-tostructure interface shear stresses is not uniform along the height of the wall. As the backfill in contact with the wall rises, the shear stresses at the interface decrease. The magnitude of the vertical shear forces acting on the back of the wall may have a significant impact on the stability of the structure. These vertical shear forces have a stabilizing effect that could produce economies if accounted for in the design of the structure. Reliable calculation of these forces requires an adequate constitutive model for the interface response. All the commercial finite-element or finite-difference codes have interface elements. User wants to select a constitutive model for the interface elements. Most used constitutive model for interface element, especially for retaining wall modeling, is hyperbolic interface model proposed by Clough and Duncan (1971). 50

65 Chapter 5. Modelling issues and choices Size of finite element mesh A proper dimensioning of the finite-element mesh or finite-difference zones is required to avoid numerical distortion of propagating ground motions, in addition to accurate computation of model response. The response of both equivalent linear and nonlinear finiteelement models can be influenced by discretization. In particular, the use of coarse finiteelement meshes can result in the filtering of high-frequency components whose short wavelengths cannot by widely spaced nodal points. Kuhlemeyer and Lysmer (1973) recommended that the length of the element Δl be smaller than one-tenth to one-eighth of the wavelength (λ) associated with the highest frequency (f max ) component of the input motion. Lysmer (1975) recommended that Δl be smaller than one-fifth the λ associated with f max. Wave length (λ) is related to the shear wave velocity of the soil V s and the frequency f of the propagating wave by the following relation: (5.18) V λ = s f The finite-element mesh size or finite-difference zone size can be selected according to the following expression: (5.19) V Δ l Γ s f max Γ takes different values according to the recommendation given by different code. Γ=10 for FLAC analysis Γ=5 for FLUSH and DIANA analysis (5.0) f max V Γf s max As may be observed from these expressions (5.15), the finite-element mesh and finitedifference zone with the lowest V s and a given Δl will limit the highest frequency that can pass through the zone without numerical distortion. 5. Overview of DIANA DIANA is a multi-purpose finite element program, based on displacement method. It has been under development at TNO the Netherlands since 197. As stated in chapter1, the numerical analyses of earth retaining structures were performed using DIANA. Dynamic analysis can be performed with DIANA using the optional dynamic calculation module, wherein user can specify acceleration or velocity or displacement time-history to a model. Incremental-Iterative solution procedures may be selected for nonlinear system with different 51

66 Chapter 5. Modelling issues and choices iterative procedures such as regular Newton-Raphson, modified Newton-Raphson, quasi Newton- Raphson and line search etc. DIANA offers several convergence norm such as force norm, displacement norm, energy norm, and residual norm to stop the iteration process, if the results are satisfactory. User can specify more than one convergence criterion simultaneously. DIANA terminates the iterative process when all the specified criteria are satisfied simultaneously. User defined mass matrix can be computed as lumped or consistent or including rotational terms in the matrix by DIANA. In a same way, damping matrix can also be computed as lumped or consistent or Rayleigh damping matrix. DIANA allows user to select different type of time integration method such as Euler backward, Newmark s method, Hilber-Hughes-Taylor, Wilson-θ method, and Runge-Kutta time integration method for transient analyses. DIANA has eight built-in isotropic plasticity models such as Mohr-Coulomb, Drucker-Prager, Egg Cam-Clay, and Modified Mohr-Coulomb etc and allows user-defined model to be incorporated. [See DIANA User s manual 9.0] 5.3 Numerical model In this investigation, the soil and wall are modeled using eight-node quadrilateral isoparametric -D plane strain elements as shown in figure (5.7). These elements are based on quadratic interpolation and Gauss integration. Each node has two translation degrees of freedom along the X and Y coordinate directions. The polynomial for the displacements u x and u y can be expressed as: u i ξ, η = a + a ξ + a η + a ξη + a ξ + a η + a ξ η + a ξη (5.1) ( ) Figure 5.6. CQ16E 8-node -D plane strain element Typically, this polynomial yields a strain ε xx which varies linearly in x direction and quadratically in y direction. The strain ε yy varies linearly in y direction and quadratically in x direction. The shear strain γ xy varies quadratically in both directions (DIANA User s manual 9.0). These behaviors are enough to capture the bending behavior of the wall. To simulate an infinite soil medium, transmitting boundary elements consisting of two nodes translational spring-dashpot that are shown in figure (5.8) were attached at all the bottom node along the finite element mesh boundaries. The value for the spring has been set to zero and the dashpot coefficient has been calculated using following equation, 5

67 Chapter 5. Modelling issues and choices (5.) c = ρvs where ρ is mass density of soil Figure 5.7. SPTR -node translation spring/dashpot For the vertical boundary, horizontal active soil pressure has to be applied to the corresponding nodes manually. This was done by recording the reaction forces in the model with fixed boundaries and applying them with opposite sign to the model with horizontal rollers. The horizontal displacements after applying self weight should be very small Details of diaphragm wall numerical model DIANA created finite element mesh is shown in figure (5.9). As shown in the figure, the model contains only the top15m of soil profile. Figure 5.8. Finite element mesh for diaphragm wall The small strain natural frequency of the DIANA model of the retaining wall-soil system is estimated to be 4.8 Hz ( 5.0 Hz). At higher stains, it is expected that the natural frequency of the system will be less than 5 Hz. The cutoff frequency for dynamic analysis was set at 15 Hz. 53

68 Chapter 5. Modelling issues and choices This value was selected based on the natural frequency of the wall-soil system. The dimensioning of the finite element mesh selected such as to ensure proper transfer of frequencies up to 15 Hz is discussed in section 5.6. All along the model, the size of the elements varied from 0.5 to 1.0 m in both directions that was less than one eighth of the shortest wave length that corresponds to the highest frequency of 15 Hz considered in the transient analysis (Kuhlemeyer and Lysmer, 1973). The total of 806 elements was used in the model. The retaining wall model was "numerically constructed" in DIANA similar to the way an actual wall would be constructed. The excavation at the left hand side of the wall was done in 3m cuts, with the model being brought to static equilibrium after the each cut. This allowed realistic earth pressures to develop as the wall deformed and moved due to the excavation of soil. Figure (5.10) shows the deformed grid, magnified 50 times, after the construction of the wall and excavation of 6 m. Stage 1 Stage 54

69 Chapter 5. Modelling issues and choices Stage 3 Figure 5.9. Deformed mesh at the end of the each phased construction (sand), magnification factor Details of cantilever wall numerical model The cantilever wall-soil system numerical model that was created by DIANA is illustrated in figure (5.11). The finite element model contains the top 9.1m of soil profile. The small strain natural frequency of the cantilever wall-soil system of DIANA model has been estimated is 9. Hz ( 9.0 Hz). At higher strain, it is expected that the natural frequency of the system will be less than 9.0 Hz. The size of the elements in the numerical model is around 0.6 m in both directions that was less than one eighth of smaller wave length corresponding to higher frequency of 15 Hz that was set for dynamic analysis. The total of 688 elements was used in the model. Figure Finite element mesh for cantilever wall The retaining wall model was "numerically constructed" in DIANA similar to the way an actual wall would be constructed. Backfill has been lifted as 0.5m thickness layer in 1 steps. After 55

70 Chapter 5. Modelling issues and choices placing of each layer the numerical model was brought to static equilibrium. This allowed realistic earth pressures to develop as the wall deformed and moved due to the excavation of soil. Figure (5.1) shows the deformed grid, magnified 150 times, after the placement of the backfill of 6 m. Stage1 Stage Stage 3 Stage 4 56

71 Chapter 5. Modelling issues and choices Stage 5 Stage 6 Stage 7 Stage 8 Stage 9 57

72 Chapter 5. Modelling issues and choices Stage 10 Stage 11 Stage 1 Stage 13 Figure Deformed mesh after placing of backfill (sand), magnification factor Details of gravity wall numerical model The numerical model contains top 1m of soil profile. The finite element model created in DIANA is illustrated in figure (5.13). 58

73 Chapter 5. Modelling issues and choices Figure 5.1. Finite element mesh for gravity wall The total of 384 elements was used in the model. The small strain natural frequency of the gravity wall-soil system of DIANA model has been estimated is 4.4 Hz ( 4.5 Hz). At higher strain, it is expected that the natural frequency of the system will be less than 4.5 Hz. Figure (5.14) shows the deform mesh, magnified 150 times, after the construction and placement of soil. Figure Deformed mesh after constructing and placing backfill (sand) Model parameters for soil The stress-strain behaviour of soil was modelled using the Mohr-Coulomb constitutive model. The parameters that are required for Mohr-Coulomb model are effective internal friction angle (φ ), effective cohesion (c ), angle of dilation (ψ) and mass density (ρ). The mass density is the total unit weight of the soil (γ t ) divided by the acceleration due to gravity (g), i.e. ρ = γ t /g. 59

74 Chapter 5. Modelling issues and choices For the plane strain finite element, additional parameters that are Young s modulus (E) and Poisson s ratio (ν) are required. By using theory of elasticity: (5.3) E = (1 + ν ) G Several correlations exist that relate G that is the shear modulus of the soil to other soil parameters. However, the most direct relation is between G and shear wave velocity (V s ): (5.4) G = ρ V s Consequently the value for the bulk modulus K can be calculated using following relation: (5.5) K = G (1 + ν ) 3 (1 ν ) ν may be estimated using the following expression: ' 1 sinφ (5.6) ν = sinφ' Which was also derived from the theory of elasticity (e.g. Terzaghi 1943), in conjunction with the correction relating K o and φ proposed by Jaky (1944): (5.7) K 0 = sin ' 1 φ Table 5. DIANA input properties of sand Parameters Value Poisson s ratio 0.6 At-rest pressure coefficient 0.36 Young s modulus (MPa) Effective friction angle 40º Density (kg/m 3 ) 000 Table 5.3 DIANA input properties of clay Parameters Value Poisson s ratio 0.34 At-rest pressure coefficient 0.53 Drained cohesion (kn/m ) 10 Effective friction angle 8º Elastic modulus (MPa) Density (kg/m 3 )

75 Chapter 5. Modelling issues and choices Model parameters for wall The concrete diaphragm wall was assigned to act as linear elastic material for whole analysis. The wall was also modeled using -D 8-node plane strain elements. Young s modulus (E) and Poisson ratio (ν) were input as additional parameters. Where, (5.8) ' E = 5000 fc ' f c - compressive strength of concrete Table 5.4 DIANA input properties of concrete Parameters Value Elastic modulus of concrete (MPa) 30,000 Yield strength of steel (MPa) Young s modulus of steel (GPa) 00 Density (kg/m 3 ) Interface element In this study no special interface elements have been used in between the soil and the wall. In the case of the diaphragm and cantilever walls, the horizontal displacement of the wall element nodes and the soil element nodes are tied together and the vertical displacement are independent of each other. Both the vertical and horizontal displacements of the soil and wall elements were tied together for the gravity wall Dimensions of finite element mesh As mentioned previously, proper dimensioning of finite element mesh is required to avoid numerical distortion of propagating ground motions, in addition to accurate computation of model response. Kuhlemeyer and Lysmer (1973) recommend that the length of the element Δl be smaller than one-tenth to one-eight of the wavelength λ associated with the highest frequency f max component of the input motion. Lysmer (1975) recommended that Δl be smaller than one-fifth the λ associated with f max. The shear wave velocity of both soils was taken as 180 m/s. The maximum frequency that was allowed for the transient analysis was 15 Hz, corresponding minimum wave length (λ) is 1m.The maximum allowable mesh size is in the range of 1.m to.4m (correspond to 5-10 times less than the wave length). One-fifth the λ was used as a the criterion for the modelling works Damping An elastroplastic Mohr-Coulomb constitutive model was used for soil in DIANA numerical model. Inherent in this model is the characteristic that once the induced dynamic shear 61

76 Chapter 5. Modelling issues and choices stresses exceed the shear strength of the soil, the plastic deformation of the soil introduced considerable hysteretic damping. However, for dynamic stresses less than the shear strength, the soil behaves elastically (without damping). DIANA allows mass proportional, stiffness proportional and Rayleigh damping to be specified, where the later provides relatively constant level of damping over the restricted range of frequencies. For the DIANA analysis performed, Rayleigh damping was specified, for which the critical damping ratio ξ may be determined by the following relation: Where, 1 α (5.9) ξ = + β ω ω α - the mass-proportional damping constant β - the stiffness-proportional damping constant ω - angular frequency associated with ξ For Rayleigh damping, the damping ratio and the corresponding central frequency need to be specified. A lower bound damping ratio one percentage was set to soil at first natural frequency of the system and the predominant frequency of the excitation. 6

77 Chapter 6. DIANA results and discussion 6. DIANA RESULTS AND DISCUSSION In the previous chapter, an overview was given of the numerical models used to analyze the different type of retaining walls. In this chapter an overview is given on how the DIANA data was simplified, followed by a presentation and discussion of the simplified data. Each numerical model was subjected to two types of analyses: phased analysis (static) including the construction sequences and then dynamic analysis. 6.1 Data simplification Time-histories for the lateral stresses acting on the elements composing the wall and the soil were computed by DIANA at Gauss integration points within the elements, as well as acceleration and displacement time-histories. From the DIANA computed stresses, the resultant forces and the points of applications were computed for the wall sections using the method proposed by Green and Ebeling (003) in the FLAC analyses. The resultant force and its point of application on the wall are needed for the structural design of the wall. Average stress distribution was assumed to determine the resultant forces acting on the wall. The detail of the approach is discussed in the following subsection Determination of forces assuming constant stress distribution To determine the forces acting on the wall sections, the average stress distribution across the elements was assumed, as illustrated in figure (6.1). To apply this approach, Gauss integration point s stresses were considered to calculate the average stress within the element. For the assumed constant stress distributions, the forces acting on the top and bottom nodes of each wall element, shown as in figure (6.1), were computed using the following expressions: 63

78 Chapter 6. DIANA results and discussion top 1,j F top 1,j h 1 1 bottom 1,j F bottom 1,j top,j F top,j h bottom,j F bottom,j top F top 3,j 3,j h 3 3 bottom 3,j F bottom 3,j Figure 6.1. Constant stress distribution approximation across the element (from Green & Ebeling 003) Where top 1 top 1 (6.1) F i, j = hiσ i, j = hiσ i, j bottom 1 bottom 1 (6.) F i. j = hiσ i, j = hiσ i, j F, - force acting on the top node of element i and at time increment j top i j h i - length of element i σ - lateral stress acting on the top of element i and at time increment j top i, j σ i, j - average stress acting on the element i and at time increment j F, - force acting on the bottom node of element i an at time increment j bottom i j σ - lateral stress acting on the bottom of element i and at time increment j bottom i, j 64

79 Chapter 6. DIANA results and discussion The total force acting on the stem or heel section P j at time increment j was determined by top bottom (6.3) P j = ( Fi, j + Fi, j ) = hiσ i, j i 6.1. Incremental dynamic forces i In addition to computing the total resultant forces acting on the wall sections, the incremental dynamic forces ΔPj at time increment j were computed. ΔPj is the difference between the total resultant force Pj at time increment j minus the total resultant force prior to shaking (i.e., Pj at j = 0, designated as Pstatic): Δ P = P P (6.4) j j static Reaction height of forces The points of application of the total and incremental dynamic resultant forces were computed for the wall (or the stem and heel section of the cantilever wall) sections in terms of their vertical distances above the base of the wall. For the total resultant forces, the vertical distances Y were computed using the following relation: Where, (6.5) Y j = h i i j σ j i, j h σ y i, j i Yj - vertical distance from the base of the retaining wall to the point of application of the total resultant force acting on the wall section at time increment j y i - vertical distance from the base of the retaining wall to the center of element i The vertical distances ΔY from the base of the retaining wall to the points of application of ΔPj acting on the wall (or the stem and heel section of the cantilever wall) were computed using the following relation: (6.6) Δ Y j = P Y P j j static ΔP j Y static In this equation, Ystatic is the vertical distance from the base of the retaining wall to the point of application of the total resultant force acting on the wall (or the stem and heel section of the cantilever wall) prior to the shaking (i.e., Yj at j = 0) Dynamic earth pressure coefficient Lateral earth pressure coefficient (K j,diana ) can be back-calculated at time increment j from the DIANA results using the following expression: 65

80 Chapter 6. DIANA results and discussion (6.7) K j, DIANA P = γ H j, DIANA ( 1 k ) v, j where k v,j is the vertical inertial coefficient at time increment j (assumed to be zero) The sign convention of horizontal inertial coefficient k h used to analyze the data is shown in following figure (6.). k h can be simply calculated dividing acceleration (a h ) by gravity. (6.8) k h = ah g away from backfill k h towards backfill a h (m/s ) towards backfill away from backfill Figure 6.. Horizontal acceleration a h, corresponding dimensionless horizontal inertial coefficient k h, of a point in the backfill portion of sliding wedge 6. Presentation and discussion of simplified data The following section presents some of the phased analysis results and dynamic analysis results. Using the procedures described in the preceding section, the total and dynamic incremental resultant forces acting on the wall (or the stem and heel section of the cantilever wall) sections were determined from the DIANA computed stresses, as well as the corresponding vertical distance above the base at which the resultant forces act. Additionally, the permanent relative displacements of the wall computed in the DIANA analysis are presented. Finally, a brief discussion is given concerning the deformed shape of the wall-soil system at the end of shaking Phased analysis stress distribution Static horizontal stress that was computed by DIANA distribution along the height of the wall was compared with the theoretical stress distribution that was used to design the wall. This comparison gives the idea of how much shear stress has been mobilized in the static condition. 66

81 Chapter 6. DIANA results and discussion DIANA active pressure DIANA passive pressure Design active pressure Design passive pressure Height (m) Pressure *10^4 (N/m^) Figure 6.3. Pressure distribution along the diaphragm wall in sand at the end of the phased analysis 10 8 DIANA passive pressure DIANA active pressure Theoritical passive pressure Theoritical active pressure Height (m) Pressure *10^4 (N/m^) Figure 6.4. Pressure distribution along the diaphragm wall in clay at the end of the phased analysis 67

82 Chapter 6. DIANA results and discussion From figure (6.3) and (6.4), the DIANA computed active pressures along the height of the diaphragm wall show the good agreement with the theoretical values computed using Rankine (classical method) theory. But the passive pressures do not show a good match with the theoretical values. Although the presence of cohesion indicates that tensile stresses will be develop between the upper portion of the wall and the backfill in active case, tensile stresses do not actually develop in the field Pressure along the stem Pressure along the heel verital section through heel Design pressure Height (m) Pressure N/m^ (10^4) Figure 6.5. Pressure distribution along the cantilever wall in sand at the end of the phased analysis Table 6.1 Pressure values along the height of the cantilever wall Height (m) Sand Clay Stem (kpa) Heel (kpa) Stem (kpa) Heel (kpa)

83 Chapter 6. DIANA results and discussion Pressure along the stem Pressure along the heel verital section through heel Theoretical pressure Height (m) Pressure *10^4 (N/m^) Figure 6.6. Pressure distribution along the cantilever wall in clay at the end of the phased analysis From figure (6.5) and (6.6), the active pressure calculated by DIANA along the section through the heel of the cantilever wall in sand show the good agreement with the theoretical values. But in the case of clay is not true, the stresses along the heel section are higher than the theoretical stresses. It means that the shear strength of the soil is not fully mobilized at this stage DIANA pressure Design pressure Height (m) Pressure *10^4 (N/m^) Figure 6.7. Pressure distribution along the gravity wall in sand at the end of the phased analysis 69

84 Chapter 6. DIANA results and discussion DIANA pressure Theoretical pressure Height (m) Pressure (N/m^) 10^4 Figure 6.8. Pressure distribution along the gravity wall in clay at the end of the phased analysis The figure (6.7) and (6.8) show the DIANA computed horizontal stress and the theoretical stress along the height of the gravity wall for sand and clay. In both cases of sand and clay the stresses computed by DIANA are less than the theoretical stresses. 6.. Dynamic analysis stress distribution The dynamic stresses computed by DIANA and the forces along the wall due to that stresses are of interest for earthquake engineers. The bending moments and shear forces are the quantities needed for structural design of wall. In the case of clayey soil, the tensile stress developed at the top part of the wall is neglected in dynamic force, bending moment and shear force calculation. Only compressive stresses are taken into account for the calculation Pressure distribution along the diaphragm wall This section gives the information about stress distribution along the diaphragm wall at the end of the dynamic analyses for both cases sand and clay. 70

85 Chapter 6. DIANA results and discussion DIANA end of dynamic analysis DIANA end of dynamic analysis DIANA active end of phased analysis DIANA passive of phased analysis Wall Height (m) Pressure *10^4 (N/m^) Figure 6.9. Pressure distribution along the diaphragm wall in sand at the end of the dynamic analysis (EL Centro) DIANA end of dynamic analysis DIANA end of dynamic analysis DIANA active end of phased analysis DIANA passive end of phased analysis Wall Height (m) Pressure *10^4 (N/m^) Figure Pressure distribution along the diaphragm wall in sand at the end of the dynamic analysis (Chi-Chi) 71

86 Chapter 6. DIANA results and discussion DIANA end of dynamic analysis DIANA end of dynamic analysis DIANA active end of phased analysis DIANA passive end of phased analysis Wall Height (m) Pressure *10^4 (N/m^) Figure Pressure distribution along the diaphragm wall in sand at the end of the dynamic analysis (Kobe) DIANA end of dynamic analysis DIANA end of dynamic analysis DIANA active of phased analysis DIANA passive end of phased analysis Wall Height (m) Pressure *10^4 (N/m^) Figure 6.1. Pressure distribution along the diaphragm wall in clay at the end of the dynamic analysis (EL Centro) 7

87 Chapter 6. DIANA results and discussion DIANA end of dynamic analysis DIANA end of dynamic analysis DIANA active end of phased analysis DIANA passive end of phased analysis Wall Height (m) Pressure *10^4 (N/m^) Figure Pressure distribution along the diaphragm wall in clay at the end of the dynamic analysis (Chi-Chi) DIANA end of dynamic analysis DIANA end of the dynamic analysis DIANA active end of phsed analysis DIANA passive end of phased analysis Wall Height (m) Pressure *10^4 (N/m^) Figure Pressure distribution along the diaphragm wall in clay at the end of the dynamic analysis (Kobe) Figure (6.9) to figure (6.14) show that the passive pressures at the end of the dynamic analysis are greater than those given by the static analysis, but does not fully match the theoretical 73

88 Chapter 6. DIANA results and discussion values. The active pressure shows little or no increase at the end of the dynamic analysis, because active side is already fully mobilized at the end of the phased analysis. It might be expected, in all cases with sand, that the point of rotation of the wall is around 1m above the bottom of the wall, because below 1m, the active region became passive and passive became active. It will be ensured in the section (6..3) describes the deformed shape of the wall Pressure distribution along the stem of the cantilever wall The following figures provide the information about the stress distribution along the stem of the cantilever wall at the end of the dynamic analyses. Additionally, theoretical active pressure line and at-rest pressure line are given for the comparison of stresses at beginning of the dynamic analysis and at the end of the dynamic analysis Ka-pressure DIANA pressure at the end of the phasedanalysis Ko-Pressure DIANA pressure after the dynamic analysis Height (m) Pressure *10^4 (N/m^) Figure Comparison of pressures along the stem of the cantilever wall at the end of the phased and dynamic analysis (sand, EL Centro) 74

89 Chapter 6. DIANA results and discussion Ka-pressure DIANA pressure at the end of the phasedanalysis Ko-Pressure DIANA pressure after the dynamic analysis Height (m) Pressure *10^4 (N/m^) Figure Comparison of pressures along the stem of the cantilever wall at the end of the phased and dynamic analysis (sand, Chi-Chi) Ka-pressure DIANA pressure at the end of the phasedanalysis Ko-Pressure DIANA pressure after the dynamic analysis Height (m) Pressure *10^4 (N/m^) Figure Comparison of pressures along the stem of the cantilever wall at the end of the phased and dynamic analysis (sand, Kobe) 75

90 Chapter 6. DIANA results and discussion Above figures (6.15 to 6.17) show the stress distribution along the stem of the cantilever wall before the dynamic analysis is very close to K a -condition, but at the end of the dynamic analysis, the residual earth pressure is approximately equal to K 0 -condition. Similar increases in the earth pressures were found in other studies, both numerical and laboratory (i.e., centrifuge and shake table), as outlined in Whitman (1990) and also reported by Green and Ebeling from FLAC analyses [19] Pressure distribution along the height of the gravity wall The following figure shows the pressure distribution along the height of the gravity wall. It shows clearly that the pressures at the end of the dynamic analysis are higher than those at the end of the phased analysis and also close to the active pressure values EL centro Chi-Chi Kobe End of phased analysis 5 Height (m) Pressure *10^4 (N/m^) Figure Pressure along the gravity wall at the end of the dynamic analysis (Sand) 76

91 Chapter 6. DIANA results and discussion EL centro Chi-Chi Kobe End of phased analysis 5 Height (m) Pressure *10^4 (N/m^) Figure Pressure along the gravity wall at the end of the dynamic analysis (Clay) In addition to this, some points concerning the lateral earth pressures during the dynamic analysis were observed. a. The maximum earth pressure behind the wall occurs when the wall is at its maximum displacement towards the backfill, which occurs also at the time of a maximum outward (away from backfill) horizontal acceleration at the base b. The minimum earth pressure occurs when the wall is at its maximum displacement away from the backfill, which also occurs at the time of a maximum inward (towards backfill) horizontal acceleration at the base. c. The peak accelerations at the top of the far field and at top of the wall lag those at the base. Similar phasing relations between the different quantities were found in both the results from numerical model and the measurements from the centrifuge test, as outlined in A.S.Al-Homoud and R.V.Whitman (1999) [] Design lateral earth pressure coefficient and DIANA computed lateral earth pressure coefficient Using this expression (6.7), K DIANA values were computed for the wall at the peaks and troughs during the strong motion portion of the k h time-history. The acceleration time-history was computed at the mid-point of the sliding soil wedge and was divided by gravity to 77

92 Chapter 6. DIANA results and discussion compute the k h time-history. The K DIANA values thus computed are plotted as functions of their corresponding absolute values of k h in following figures. Additionally, the active and passive dynamic earth pressure coefficients (KAE and KPE, respectively) computed using the Mononobe-Okabe expression (.4) given in the Literature review are shown in following figures. The shape of the Mononobe-Okabe active and passive dynamic earth pressure curves warrant discussion. As k h increases, Kae increases, while Kpe decreases. For the conditions examined (i.e., horizontal backfill, vertical wall, kv = 0), Kae and Kpe reach the same limiting value. The limiting K value occurs when the angles of the active and passive failure planes (which are assumed to be planar in the Mononobe-Okabe formulation) become horizontal. For comparison purposes, the lateral earth pressure coefficient calculated using Wood (1973) approach for nonyielding backfills is also plotted in the following figures. By treating ΔP E mentioned in equation (.19) as the dynamic incremental force, the equivalent earth pressure coefficient was computed by substituting ΔPE into equation (6.7) for P and adding Ko to the result. The resulting curve, shown in following figures, will likely be a conservative upper bound of the earth pressures Comparison of lateral earth pressure coefficients for diaphragm wall The lateral earth pressure coefficients calculated from the DIANA results were compared with those calculated using the Mononobe-Okabe equation, as explained in the previous section. The results are presented separately for walls in sand and clay. Additionally, the lateral earth pressure coefficients calculated using the DIANA results, with k h towards or away from the backfill, were differentiated in the following figures. 78

93 Chapter 6. DIANA results and discussion 5 Lateral earth pressure coefficient (K) M-O theoretical active M-O theoretical passive Wood (v=0.63) towards backfill active Away from backfil active away from backfill passive towards backfill passive K 0 K a Kh Figure 6.0. Comparison of active and passive lateral earth pressure coefficient (K DIANA ) backcalculated from DIANA results with values computed using the Mononobe-Okabe expressions (Sand) It is observed from figure (6.0), 1. Active pressure coefficient: a. K > K Mononobe-Okabe, for moderate levels of shaking b. K < K Mononobe-Okabe, for larger levels of shaking c. K away from backfill > K towards backfill d. The computed K values show a general scatter around the curve for the Mononobe- Okabe dynamic active earth pressure curve.. Passive pressure coefficient: e. The computed K values for smaller levels of shaking show the very low values f. The computed K increases with level of shaking g. The computed K values do not show a general scatter around the curve for the Mononobe-Okabe dynamic passive earth pressure curve 79

94 Chapter 6. DIANA results and discussion However, at the larger levels of shaking, the Mononobe-Okabe expressions for active pressures failed to predict the induced stresses on the wall. The computed dynamic stresses from numerical analysis are higher than those computed by the Mononobe-Okabe equation for active pressures in the range of small to moderate levels of shaking. At the smaller levels of shaking, the passive pressures are not fully mobilized, therefore the K values computed from the DIANA results are smaller than those calculated from the Mononobe-Okabe expressions. When the level of shaking increases, the mobilization of passive pressure also increases, consequently the K values increases. Lateral earth pressure coefficient (K) M-O theoretical active M-O theoretical passive Wood (v=0.346) Ortigosa towards backfill active Away from backfil active away from backfill passive towards backfill passive Kh Figure 6.1. Comparison of active and passive lateral earth pressure coefficient (K DIANA ) backcalculated from DIANA results with values computed using the Mononobe-Okabe expressions (Clay) It is observed from figure (6.1), 1. Active pressure coefficient a. K K Mononobe-Okabe, for smaller levels of shaking b. K away from backfill K Moninbe-Okabe and K towards backfill < K Mononobe-Okabe, for moderate levels of shaking c. The computed K values show a general scatter around the curve for the Mononobe- Okabe dynamic active earth pressure curve d. The computed K values show a general scatter above the curve for the Ortigosa dynamic active earth pressure curve 80

95 Chapter 6. DIANA results and discussion. Passive pressure coefficient e. The computed K values for smaller levels of shaking show the very low values f. The computed K increases with level of shaking h. The computed K values do not show a general scatter around the curve for the Mononobe-Okabe dynamic passive earth pressure curve For all the analyses, the computed stresses of the wall are not showing the very good agreement with those predicated by the Ortigosa (005) expressions at the low level of k h as well as the high level of K h. At the smaller levels of shaking, the passive pressures are not fully mobilized, therefore the K values computed from the DIANA results are smaller than those calculated from the Mononobe-Okabe expressions. When the level of shaking increases, the mobilization of passive pressure also increases, consequently the K values increases Comparison of lateral earth pressure coefficients for cantilever wall The lateral earth pressure coefficients calculated from DIANA results were compared with those calculated using the Mononobe-Okabe equation, as explained in the previous section. The lateral earth pressure coefficients were calculated both at the stem section and the heel section. The results were presented separately for walls mounted in sand and clay. Additionally, the lateral earth pressure coefficients calculated using the DIANA results, for k h towards or away from the backfill, were differentiated in the following figures. 81

96 Chapter 6. DIANA results and discussion 5 Lateral earth pressure coefficient (K) M-O theoretical active M-O theoretical passive Wood (v=0.63) towards backfill at heel away from backfill at heel towards backfill at stem away from backfill at stem K 0 K a Kh Figure 6.. Comparison of active lateral earth pressure coefficient (K DIANA ) back-calculated from DIANA results with values computed using the Mononobe-Okabe expressions (Sand) All the calculations were done to make the figure (6.) as explained in section Several distinct trends may be observed from figure (6.): a. Kheel > Kstem when k h < 0 (i.e., when k h is directed toward the backfill). b. Kstem > Kheel when k h > 0 (i.e., when k h is directed away from the backfill). c. The largest Kstem occurs when k h > 0 (i.e., when k h is directed away from the backfill). d. The largest Kheel occurs when k h < 0 (i.e., when k h is directed toward the backfill). e. The computed K values show a general scatter around the curve for the Mononobe- Okabe dynamic active earth pressure curve. Similar behaviour of lateral earth pressure coefficient has been observed in numerical analyses (FLAC) and reported by Green and Ebeling [19]. 8

97 Chapter 6. DIANA results and discussion Lateral earth pressure coefficient (K) M-O theoretical active M-O theoretical passive Wood (v=0.346) Ortigosa towards backfill at heel away from backfill at heel towards backfill at stem away from backfill at stem Kh Figure 6.3. Comparison of active lateral earth pressure coefficient (K DIANA ) back-calculated from DIANA results with values computed using the Mononobe-Okabe expressions (Clay) Similar behaviour were observed as in the case of sand, a. Kheel > Kstem when k h < 0 (i.e., when k h is directed toward the backfill). b. Kstem > Kheel when k h > 0 (i.e., when k h is directed away from the backfill). c. The largest Kstem occurs when k h > 0 (i.e., when k h is directed away from the backfill). d. The largest Kheel occurs when k h < 0 (i.e., when k h is directed toward the backfill). e. The computed K values show a general scatter within Mononobe-Okabe and Ortigosa Unlike in sand, the calculated values of lateral earth pressure coefficients do not show a general scatter around the curve for the Mononobe-Okabe dynamic active earth pressure curve. It shows good agreements with Mononobe-Okabe for very low levels of shaking. But in the case of moderate and larger levels of shaking, the calculated lateral earth pressure coefficients show quite high deviation from Mononobe-Okabe. The stresses from dynamic analyses are higher than the Ortigosa Comparison of lateral earth pressure coefficients for gravity wall The lateral earth pressure coefficients calculated from DIANA results were compared with those calculated using the Mononobe-Okabe equation, as explained in the previous section. The results were presented separately for walls mounted in sand and clay. Additionally, the lateral earth pressure coefficient calculated using DIANA result, when the k h towards backfill or away from backfill, were differentiated in the following figures. 83

98 Chapter 6. DIANA results and discussion M-O active M-O passive Wood (v=0.63) towards the backfill away from backfill Lateral earth pressure coefficient (K) K 0 K a Kh Figure 6.4. Comparison of active lateral earth pressure coefficient (K DIANA ) back-calculated from DIANA results with values computed using the Mononobe-Okabe expressions (Sand) It is observed from figure (6.4), a. K K Mononobe-Okabe, for smaller and moderate levels of shaking b. K away from backfill > K towards backfill c. The computed K values show a general scatter around the curve for the Mononobe- Okabe dynamic active earth pressure curve 84

99 Chapter 6. DIANA results and discussion 3.5 M-O active M-O passive Wood (v=0.346) towards the backfill away from backfill Ortigosa Lateral earth pressure coefficient (K) Kh Figure 6.5. Comparison of active lateral earth pressure coefficient (K DIANA ) back-calculated from DIANA results with values computed using the Mononobe-Okabe expressions (Clay) Similar observation was made as in sand, a. K away from backfill > K towards backfill b. The computed K values show a general scatter around Ortigosa for low level of shaking and below for high level of shaking. Unlike in sand, the calculated values of lateral earth pressure coefficients do not show a general scatter around the curve for the Mononobe-Okabe dynamic active earth pressure curve. It shows acceptable agreements with Ortigosa for very low levels of shaking. But in the case of moderate and larger levels of shaking, the calculated lateral earth pressure coefficients show quite high deviation from Mononobe-Okabe and Ortigosa Point of application of total dynamic forces and incremental dynamic forces The point of application of total dynamic forces and incremental dynamic forces are also interesting. As explained in the previous section the Y and Y were computed. Some of the computed Y and Y were presented in following figures, because the extreme ranges of the Y values and the erratic characteristics of the time-histories of Y made them impossible to present in an intelligible manner. 85

100 Chapter 6. DIANA results and discussion Point of application of total dynamic forces for diaphragm wall The point of application of total dynamic force in diaphragm wall was calculated and presented in figure (6.6). The points of application of total dynamic forces are presented together with point of application of static force calculated at the end of the phased analysis. All the points of application of static or dynamic forces were given as a percentage of wall height. It was clear from the figure (6.6) that that point of application of static force (0.95%) calculated was below the value (0.33%) calculated using triangle stress distribution along the height, because the stresses below the point of rotation of wall had very large passive stresses. The points of application of dynamic forces for smaller levels of shaking show a scatter around 0.5% (H/4), but for larger level of shaking it showed big range of deviation Location of dynamic earth force above wall base %of wall height towards backfill away from backfill (Y/H)static Kh Figure 6.6. Point of application of total active dynamic force for diaphragm wall 86

101 Chapter 6. DIANA results and discussion 0.6 Location of dynamic earth force above wall base %of wall height Away from backfill towards backfill Choudhury (00) upper limit Choudhury (00) lower limit Kh Figure 6.7. Point of application of total passive dynamic force for diaphragm wall in sand 0.55 Location of dynamic earth force above wall base %of wall height away from backfill towards backfill Choudhury (00) upper limite Choudhury (00) lower limit Kh Figure 6.8. Point of application of total passive dynamic force for diaphragm wall in clay 87

102 Chapter 6. DIANA results and discussion From figure (6.7 &6.8), the points of application of passive pressure from the base of the wall are above the upper limit proposed by Choudhury (00). The reason for that is the passive pressure mobilization is higher in the top layer of soil and also below the point of rotation of the wall, the passive pressure become active (low). So the point of application takes higher value than that proposed by Choudhury Point of application of total dynamic forces and incremental dynamic forces for cantilever wall The following figures (6.9 & 6.30) show the point of application of total dynamic and incremental dynamic forces for cantilever wall at the stem and heel section as a percentage of wall height. Figure (6.9) contains also the point of application of static force (0.33%) calculated at the end of the phased analysis. 0.5 Location of dynamic earth force above wall base %of wall height towards backfill at heel away from backfill at heel towards backfill at stem away from backfill at stem (Y/H)static Kh Figure 6.9. Point of application of total dynamic force at stem and heel section of cantilever wall It is observed from figure (6.9) a. Points of application of total dynamic forces at the stem and heel sections show a scatter around the point of application of static force, when k h away from the backfill at stem and heel. b. Points of application of total dynamic forces at stem and heel section show big deviation around the point of application of static force, when k h towards backfill at stem and heel. 88

103 Chapter 6. DIANA results and discussion 0.7 Location of incremental dynamic earth force above wall base %of wall height 0.6 towards backfill at heel away from backfill at heel towards backfill at stem 0.5 away from backfill at stem Seed-Whitman (1970) Kh Figure Point of application of incremental dynamic force at stem and heel section of cantilever wall This clearly shows from figure (6.30) that the point of application of incremental dynamic forces has big deviation among them and with theoretical values (Seed-Whitman, 1970). The point of application of incremental dynamic forces increases with increasing k h, when k h is towards the backfill at the heel Point of application of total dynamic forces and incremental dynamic forces for gravity wall The following figures (6.31 & 6.3) show the point of application of total dynamic and incremental dynamic forces for cantilever wall at the stem and heel sections as a percentage of wall height. Figure (6.31) contains also the point of application of static force (0.33%) calculated at the end of the phased analysis. 89

104 Chapter 6. DIANA results and discussion 0.5 Location of dynamic earth force above wall base %of wall height towards backfill away from backfill (Y/H)static Kh Figure Point of application of total dynamic force of gravity wall It is observed from figure (6.31) a. Points of application of total dynamic forces are above the point of application of static force (0.33%) and below the level of 0.40%, when k h away from backfill. b. Points of application of total dynamic forces show large fluctuation, when k h towards backfill. 90

105 Chapter 6. DIANA results and discussion 0.9 Location of incremental dynamic force above base % of wall height towards backfill away from backfill Seed-Whitman (1970) Kh Figure 6.3. Point of application of incremental dynamic force for gravity wall This clearly shows from figure (6.3) that the point of application of incremental dynamic forces has considerable deviation among them and with theoretical values (Seed-Whitman, 1970) Deformation and displacement of the wall This section gives the deformed shape of the diaphragm wall at different time steps in the dynamic analyses, and also the relative displacement time history at the base of the cantilever wall. It gives the information about relative permanent displacement and permanent tilt of the gravity wall. This information is useful for displacement base design procedures. 91

106 Chapter 6. DIANA results and discussion Deformation of diaphragm wall Height (m) 6 End of phased construction at 1.5sec later the dynamic analysis End of the dynamic analysis Defomation (cm) Figure Deformed shape of the diaphragm wall in sand at different time step (EL Centro) Height (m) 6 at 1.5sec later the dynamic analysis End of the dynamic analysis End of the phased analysis Defomation (cm) Figure Deformed shape of the diaphragm wall in sand at different time step (Chi-Chi) 9

107 Chapter 6. DIANA results and discussion Height (m) 6 End of phased construction at 1.5sec later the dynamic analysis End of the dynamic analysis Defomation (cm) Figure Deformed shape of the diaphragm wall in clay at different time step (EL Centro) Height (m) 6 4 End of phased construction at 1.5sec later the dynamic analysis End of the dynamic analysis Defomation (cm) Figure Deformed shape of the diaphragm wall in clay at different time step (Chi-Chi) The results for sand and clay those were computed using Kobe were very large. Those were not accepted and removed. 93

108 Chapter 6. DIANA results and discussion Deformation of cantilever wall Structural wedge Driving wedge Range of shear band Figure Annotated deform mesh from EL Centro analysis The reason for the deviation of the DIANA computed stresses and those computed by the Mononobe-Okabe expressions can be understood from examining figure (6.37), (For more details see appendix-d figure.14). At large values of k h directed away from the backfill, the induced inertial forces on the structural wedge cause it to simultaneously bend, rotate, and potentially slide away from the backfill, at which time a small wedge of soil moves vertically downward. (The structural wedge consists of cantilever wall and the backfill contained within) As the direction of k h reverses, the small wedge of soil prevents the structural wedge from returning from the bending and rotation of the structural wedge. The initial stresses imposed on the stem of the wall correspond to active conditions. As k h increases in the direction away from the backfill, the stresses on the stem increase according to the Mononobe-Okake expressions for active conditions. However, upon reversal of the direction of k h, the stresses imposed on the stem do not decrease as predicated by Mononobe- Okabe expressions, but rather remain relatively constant. The stepwise increase in the lockedin stresses continues until the residual stresses imposed on the stem correspond to at-rest conditions, while the dynamically induced inertial stresses are superimposed on the locked-in residual stresses. [See figures (6.15 to 6.17)] 94

109 Chapter 6. DIANA results and discussion Figure Relative permanent displacement time-history of base of the cantilever wall in sand (El Centro) Table 6. Relative permanent displacement of base of the cantilever wall Earthquake recode Relative permanent displacement of base of the cantilever wall (cm) Sand Clay EL Centro Chi-Chi Kobe Deformation of gravity wall The relative permanent displacement of the base and the permanent tilt of the wall are interesting for the displacement based design of a gravity walls. The table below has values of relative permanent displacement of the base and permanent tilt of the wall mounted on sand and clay. Table 6.3 Relative permanent displacement of base and permanent tilt of gravity wall Earthquake recode Relative permanent displacement of base (cm) Sand Permanent tilt of wall *10-3 (rad) Relative permanent displacement of base (cm) Clay Permanent tilt of wall *10-3 (rad) EL Centro Chi-Chi Kobe

110 Chapter 6. DIANA results and discussion It was observed by Al-Homoud and Whitman that there is a small amount of permanent outward tilt 1.39*10-3 rad(less than 0.08º) for 0.g El Centro input motion in their study. In this study shows 1.3*10-3 rad Bending moment and shear force This section gives the information about the bending moment and shear force envelop that are useful for the structural design of the walls Bending moment and shear force envelop for diaphragm wall The bending moment and shear force envelope has been drawn using the stresses due to the dynamic forces. The moment on the wall is written as clockwise-positive. The shear forces inducing an anti-clockwise moment are taken as positive Height (m) Maximum envelop Minimum envelop End of phased analysis Moment *10^4 (Nm/m) Figure Bending moment envelop for diaphragm wall in sand (EL Centro) 96

111 Chapter 6. DIANA results and discussion Maximum envelop Minimum envelop End of phased analysis Height (m) Shear force (N/m) Figure Shear force envelop for diaphragm wall in sand (EL Centro) Maximun envelop Minimun envelop End of phased analysis 7 Height (m) Moment *10^4 (Nm/m) Figure Bending moment envelop for diaphragm wall in sand (Chi-Chi) 97

112 Chapter 6. DIANA results and discussion Maximum envelop Minimum envelop End of phased analysis Height (m) Shere force *10^4 (N/m) Figure 6.4. Shear force envelop for diaphragm wall in sand (Chi-Chi) Height (m) Maximum envelop Minimum envelop End of phased analysis Moment *10^4 (Nm/m) Figure Bending moment envelop for diaphragm wall in clay (EL Centro) 98

113 Chapter 6. DIANA results and discussion Maximum envelop Minimum envelop end of phased analysis 7 Height (m) Shear force *10^4 (N/m) Figure Shear force envelop for diaphragm wall in clay (EL Centro) Maximun envelop Minimun envelop End of phased analysis 7 Height (m) Moment *10^4 (Nm/m) Figure Bending moment envelop for diaphragm wall in clay (Chi-Chi) 99

114 Chapter 6. DIANA results and discussion Maximum envelop Minimum envelop End of phased analysis Height (m) Shere force *10^4 (N/m) Figure Shear force envelop for diaphragm wall in clay (Chi-Chi) End of phased analysis EL centro Chi-Chi Kobe Height (m) Moment *10^4 (Nm/m) Figure Bending moment envelop for cantilever wall in sand 100

115 Chapter 6. DIANA results and discussion 7 6 Height (m) End of phased analysis EL Centro Chi-Chi Kobe Moment *10^4 (Nm/m) Figure Bending moment envelop for cantilever wall in clay Height (m) 4 3 End of the phased analysis EL Centre Chi-Chi Kobe Shear force (N/m) Figure Shear force envelop for cantilever wall in sand 101

116 Chapter 6. DIANA results and discussion End of phased analysis EL Centro Chi-Chi Kobe Height (m) Shear force (N/m) Figure Shear force envelop for cantilever wall in clay EL Centro Chi-Chi Kobe End of phased analysis 5 Height (m) Pressure *10^4 (N/m^) Figure Maximum pressure envelop along the height of the gravity wall in sand 10

117 Chapter 6. DIANA results and discussion EL Centro Chi-Chi End of phased analysis 5 Height (m) Pressure *10^4 (N/m^) Figure 6.5. Maximum pressure envelop along the height of the gravity wall in clay 103

118 Chapter 7. Summary and conclusions 7. SUMMARY AND CONCLUSIONS The preceding sections showed the procedures used to simplify the DIANA data and presented results calculated from simplified data for diaphragm wall, cantilever wall, and gravity wall. This section contains some conclusions drawn from the above studies. 7.1 Diaphragm wall From the result computed by DIANA for diaphragm wall, 1. The static stresses induced in the active side match, to an acceptable level, the stresses computed using Rankine theory. Since the passive side is not fully mobilized, it does not match the Rankine predictions The Rankin theory overestimates the passive pressure. The point of rotation of the diaphragm wall is above the base.. Using the Mononobe-Okabe equation to design the diaphragm wall will underestimate the dynamic stresses on the active side. Conversely, it will overestimate dynamic stresses in the passive side. 3. The point of application of total dynamic forces in active side is below the point of application of static forces, around 0.5H, because of the large passive pressure at the bottom part of the diaphragm wall. 4. The point of application of total dynamic forces in the passive side is above the upper limit theoretical value (0.4H) proposed by Choudhury (00). Because the passive pressure is not fully mobilized, small active pressure develop at the base of the diaphragm wall. 7. Cantilever wall For the cantilever retaining wall numerically modeled and analyzed in sand, the stresses induced on the stem of the wall did not correspond with those predicted by the Mononobe- Okabe method. The reason for this deviation is attributed to the relative flexibility of the structural wedge and the non-monolithic motion of the driving soil wedge, both of which violate assumptions inherent in the Mononobe-Okabe method. The points of application of the total dynamic forces, when the k h was away from backfill, were very close to the point of application of static force, conversely, when the k h was towards backfill the point of application of total forces showed large fluctuation from static case, but it was below 0.5H even for k h =0.6. The points of application of incremental dynamic forces showed greater fluctuation from 0.6H proposed by Seed-Whitman (1970). The dynamic response of the wall-backfill system was such that there was an incremental increase from the active to at-rest stress conditions in the residual stresses imposed on the stem of the retaining wall. 104

119 Chapter 7. Summary and conclusions 7.3 Gravity wall On the basis of the results and discussions of the current study of an 8.0 m high and 3.0 m wide gravity wall, the following conclusions can be made: 1. The stresses induced on the wall corresponded with those predicated by the Mononobe-Okabe method, when the levels of shaking were small. The stresses induced deviated with those predicted by Mononobe-Okabe method, when the levels of shaking were large.. The points of application of incremental dynamic forces showed considerable fluctuation about the value (0.6H) proposed by Seed-Whitman (1970). 3. The points of application of total dynamic forces were above the point of application of static force and also below 0.5H. 4. The phasing relations between the different quantities obtained from the analysis are the same as those in Andersen et al. tests. [See section 6...3] The observation from all the walls in the clay soil, it is clear that the cohesion play an important role in earthquake induced stresses on the wall. That means the earthquake induced stresses reduced due to cohesive force. The Mononobe-Okabe equation does not account the effect of cohesion, because of that the lateral earth pressure coefficients calculated from dynamic analysis are less than those calculated using Mononobe-Okabe method. The solution has been given by Ortigosa (005) for clay predicts low value of lateral earth pressure. The conclusion drawn from this study may not apply to retaining wall systems of differing geometry and/or material properties. Further research is required in order to draw more general conclusions regarding the appropriateness of the Mononobe-Okabe method to evaluate the dynamic pressures induced on retaining walls. 7.4 Problems encountered in DIANA modelling This section gives ideas and precautions that can help to overcome some of the numerical problems in convergence. It may also help to get quick convergence of numerical models. Selection of element type and material constitutive model are discussed to keep the model stable and capture stress variation within the element. Additionally, it explains the problems in interface element modeling and the transmitting boundary modeling Type of element and material constitutive model Initially the numerical model was created using Q8EPS - 4 nodes plane strain elements for the wall and the soil. The shear strain is constant over the element area, because of that it cannot capture the bending behavior of the wall. In addition to that, it disturbed the convergence of the model even in the phased analyses. Because the strain variation within the element is linear or constant, it cannot capture the second order or the higher order stress variation within the element. But in the case of retaining wall analyses the stress variation within the soil element depends on the wall moment and conversely, the wall movement depends on the soil 105

120 Chapter 7. Summary and conclusions pressure. So it creates the higher order of stress variation within the soil element. To overcome the above problem CQ16E - 8 nodes element was selected for numerical model. Seemingly there is no special hysteretic constitutive model for dynamic analyses in DIANA; it is very difficult to capture the dynamic behavior of soil in the transient analyses. As usual the Mohr-Coulomb model was selected for the soil element. As explained in section 5.7, very low Rayleigh damping was assumed to soil to dissipate energy when the material in the linear range. Using Rayleigh damping proportional initial mass and stiffness matrix gives quick convergence in dynamic analyses; however, using Rayleigh damping proportional to tangent mass and stiffness matrix prevented convergence Interface element and transmitting boundary Usually working with interface elements in numerical models gives lot of problem, such as convergence. DIANA is also not exempt for that. In reality the interface elements are important in between the wall and the soil element to capture the actual behavior of the system. DIANA allows the user to model the interface element but it is poor in material constitutive models for interface element. It has a few stranded models. Unlike other geotechnical specialized software such FLAC and FLEX, DIANA dose not have option to select and fix the transmitting boundary to simulate the infinite soil medium in the numerical model. Because of this, the transmitting boundary should be made by using discrete dashpot elements to dissipate the energy from the system. DIANA performs displacement-based analysis, and therefore does not report the damping forces correctly. This is not an indication that the calculations performed by DIANA are incorrect, but are simply an artifact of how the results are reported. Therefore, it is reasonable to use the dashpot elements. In general DIANA does not show stable convergence for nonlinear systems. Normally the number of steps to convergence in each loading steps vary randomly, up to very high values. Because of this, large numbers of convergence steps are required. Regular Newton-Raphson works quite well for comparably high nonlinear systems or use regular Newton-Raphson with line search option. 106

121 References REFERENCES [1]Alampalli, S., and Elgamel, A. W. (1990). Dynamic response of retaining walls including supported soil backfill- A computational model Proc., 4th U.S National Conf. on earthquake Engre, Earthquake Eng ineering research Institute, Palm Springs, California, vol.3, pp []Al-Homoud, A. S., and Whitman, R.V (1999). Seismic analysis and design of rigid bridge abutments considering rotation and sliding incorporating non-linear soil behavior Soil dynamics and earthquake engineering [3]Basu, U., and Chopra, A.K., (003), Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation Computer methods in applied mechanics and engineering vol.19, [4]Buckle IG. The Hashin-Awaji earthquake on January 17, Performance of lifelines: performsnce of highways and bridges. Technical Report NCEER , National Center for Earthquake Engineering Research, State University of New York at Buffalo, November 1995, pp [5]Cameron, W.I., and Green, R.A Development of engineering procedure for evaluating lateral earth pressures for seismic design of cantilever retaining walls [6]Choudhury, D., Subba Rao, K. S., and Ghosh, S. (00). Passive earth pressure distribution under seismic condition 15th Engineering Mechanics Conference of ASCE, Columbia University, New York,00. [7]Choudhury, D., and Nimbalkar, S. (005). Seismic passive resistance by pseudo-dynamic method J.Ge otechnique 55, No. 9, [8]Collino, F., and Tsogka, C., (1998), Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media Institut National de Recherche en Informatique et en Automatique [9]Degrande, G., Praet, E., Van Zegbroeck, B., and Van Marcke, P. (00). Dynamic interaction between the soil and an anchored sheet pile during seismic excitation J.Numerical and analytical methods in geomechanics vol.6, [10]Euro code 8, EN Design provisions for earthquake resistance of structures. [11]Finn, W. D. L., Yogendrakumar, M., Otsu, H., and Seedman, R. S., (1989). Seismic response of a cantilever retaining wall: Centrifuge model test and dynamic analysis Proc., 4 th Int. Conf. on Soil Dyn. and earthquake Engrg., Computational Mechanics Inc., Southampton, [1]Finn, W. D. L., Wu, G., and Yoshida, N., (199). Seismic response of sheet pile walls Proc., 10 th World Conf., on Earthquake Engrg, Madrid, vol. 3, pp

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123 References [30]Mononobe, N., and Matuo, H, (199). On the determination of earth pressures during earthquakes Proc. World Engrg, Congr., Tokyo, Japan, vol.9, paper no.388. [31]Nadim, F., and Whitman, R, V, (1983). Seismically induced movement of retaining walls J.Geotech Engrg, ASCE, 109(7), [3]Nason J. McCullough, and Stephen E. Dickenson, (1998). Estimation of seismically induced lateral deformations for anchored sheetpile bulkheads conference proceedings of Geotechnical Earthquake Engineering and Soil Dynamics III, August 3-6, 1998, Seattle, WA, USA. pp [33]Nazarian, H. N., and Hadjian, A. H, (1979). Earthquake-induced lateral soil pressures on structures J.Geotech Engrg. Div, ASCE, 105(9), [34]Okabe, S, (194). General theory of earth pressure and seismic stability of retaining wall and dam J.Japan Soc. Civ Engrs, Tokyo, Japan, 1(1). [35]Ortigosa, P., (005) Seismic earth pressure including soil cohesion The 16 th international conference on soil mechanics and geotechnical engineering, Osaka. [36]Ostadan, F (005). Seismic soil pressure for building walls-an updated approach J.Soil dynamic and earthquake engineering vol.5, [37]Piratheepan, P., and Andrus, R.D. (001), Estimating shear-wave velocity from cone penetration resistance and geologic age Program and Abstracts, 73rd Annual Meeting of the Eastern Section Seismological Society of America, Columbia, SC, October [38]Prakash, S., (1981). Analysis of rigid retaining walls during earthquakes Proc., Int. Conf. on Recent Adv. in Geotech. Earthquake Engrg., and Soil Dyn., Univ. of Missouri, Rolla, Mo., vol III, 1-8. [39]Prakash, S., and Basavanna, B. M., (1969). Earth pressure distribution behind retaining wall during earthquake Proc., 4 th world Conf. on Earthquake Engrg, Satiago, Chile. [40]Psarropoulos, P.N., Klonaris.G., and Gazetas.G. (005), Seismic earth pressures on rigid and flexible retaining walls J.Soil dynamics and earthquake engineering vol.5, [41]Richard, R., and Elms, D.G (1979). Seismic behavior of gravity retaining walls J.Geotec Engrg., ASCE, 105 (GT4). [4]Richards, R., and X. Shi, (1994), Seismic Lateral Pressures in soils with Cohesion J.Geotechnical engineering, ASCE vol.10, N 7. [43]Seed, H.B., and Whitman, R.V., (1970) Design of earth retaining structures for dynamic loads, ASCE Spec.Conf. Lateral Stresses in the ground and design of retaining structures,cornell,pp [44]Siller. T. J., Christiano, P. P., and Bielak, J. (1991). Seismic response of tied-block retaining walls Earthquake Engrg. And Struct. Dyn. 0(7), [45]Steedman, R.S., and Zeng, X., (1990) The influence of phase on the calculation of pseudo-static earth pressure on a retaining wall Geotechnique, 40 (1), [46]Steedman, R.S. (1998). Seismic design of retaining walls Proc. Instn. Civ. Engrs. Geotech. Engng, [47]Steedman, R.S., (1999) Seismic soil-structure interaction of rigid and flexible retaining walls Earthquake geotechnical engineering, 1999 Balkema, Rotterdam, ISBN

124 References [48]Subba Rao, K.S., and Choudhury, D. (005). Seismic passive earth pressures in soils J.Geotechnical and geoenvironmental engineering, Vol. 131, No 1, [49]Sun, K., and Lin, G. (1995). Dynamic response of soil pressure on retaining wall Proc., 3 rd Int. Conf. on Recent Adv. in Geotech. Earthquake Engrg., and Soil Dyn., Univ. of Missouri, Rolla, Mo., [50]Terzaghi, K., (1943). Theoretical soil mechanics John Wiley & Sons, Inc, New York. [51]Towhata, I., and Islam, S (1986). Prediction of lateral displacement of anchored bulkheads induced by seismic liquefaction Japanese society of soil mechanics and foundation engineering. [5]Veletsos, A. S., and Younan, A.H (1994b). Dynamic modeling and response of soil-wall systems J.Geotech. Engrg.,ASCE, 10 (1) [53]Veletsos, A. S., and Younan, A.H (1995). Dynamic soil pressures on vertical walls Proc., 3 rd Int. Conf. on Recent Adv. in Geotech. Earthquake Engrg., and Soil Dyn., Univ. of Missouri, Rolla, Mo., [54]Veletsos, A. S., and Younan, A.H (1996). Dynamic response of cantilever retaining walls J.Geotech Engrg., ASCE, 13 () [55]Wang, Y. Z. (000), "Distribution of earth pressure on a retaining wall" Geotechnique, 50(1), [56]Whitman, R, V., (1990). Seismic design and behavior of gravity retaining walls Proc., Spec. Conf. on Des. And Constr. Of Earth Retaining Struct., ASCE, New York, N.Y., [57]Whitman, R, V., (1991). Seismic design of earth retaining structures Proc., nd Int. Conf. on Recent Adv. in Geotech. Earthquake Engr and Soil Dyn. Univ. of Missouri, Rolla, Mo., vol II, [58]Wood, J. H., (1973). Earthquake-induced soil pressures on structures Rep. EERL 73-05, Earthquake Engineering Research Laboratory, California Inst. of Technol., Pasadena, Calif. [59]Wu, Y., and Prakash, S., (1999) Effect of submergence on seismic displacement of rigid walls Earthquake geotechnical engineering, 1999 Balkema, Rotterdam, ISBN [60]Younan, A.H., and Veletsos, A.S., (000). Dynamic response of flexible retaining walls J.Earthquake engineering and structural dynamics vol.9, [61]Zeng, X., and. Steedman, R.S., (1993) On the behaviour of quay walls in earthquakes Geotechnique, 43 (1), [6]Zerfa, F.Z., and Loret, B (003). Coupled dynamic elastic-plastic analysis of earth structures J.Soil dynamic and earthquake engineering vol [63]Zhang, J.M., Shamoto, Y, and Tokimatsu, K (1998). Seismic earth pressure theory for retaining walls under any lateral displacement Japanese geotechnical society Soils and foundations vol.38 No.,

125 Appendix - A 8. APPENDIX A STATIC DESIGN OF DIAPHRAGM WALL Introduction This appendix illustrates the sizing and structural design for usual loading (i.e., static) of the 6m free-height diaphragm earth retaining wall that is analyzed dynamically in the main body of this report. The wall design is performed in two stages. The first stage consists of sizing the wall to satisfy global stability requirements. The global stability requirements are expressed in terms of the factor of safety against rotation or overturning stability of the wall. The second stage of the wall design entails the dimensioning of the components of the concrete wall. Structural elements are designed by the strength-design method for flexure loading. The design loads are those determined in the first design stage, with appropriate factors applied, thus ensuring that the serviceability requirements are satisfied. Stage 1 Sizing of diaphragm wall As stated previously, the first design stage consists of sizing the diaphragm wall such that global stability requirements are satisfied. Structural part of the proposed wall and backfill is shown in figure A-1, as well as the backfill and foundation material properties. To assess the global stability of the wall, the external forces and corresponding points of action acting on the wall need to be determined. The external forces include the resultant of the lateral earth pressure and reactionary forces (due to passive earth pressure) acting along embedded depth of the wall. R etaining W all P AS P PS Fig.A-1 External forces acting on the diaphragm wall γ s =19.5kN/m 3 φ =40º A1

126 Appendix - A Active earth pressure coefficient Passive earth pressure coefficient ' o φ k a = tan 45 =0.17 ' o φ k p = tan 45 + =4.598 For permanent structures reduction factor for passive pressure is 1.5 k p Mobilized passive earth pressure coefficient k p = = = kaγ t 6 Resultant force due to active pressure ( ) F a + D Resultant force due to passive pressure Moment equilibrium about O 6 D D F + F = a p F p = 1 k γ D p t 1 D 1 D D = 4. 4m D is normally increased by 0% ( 6 + D ) D = 0 Embedded depth D = 5. 0m Stage structural design of diaphragm wall As stated in the introduction to this appendix, the second stage of the wall design entails the structural design of the concrete wall, including the dimensioning of the concrete wall. All structures must satisfy the strength requirements. In the strength design method, this is accomplished by multiplying the service loads by appropriate load factors. Moment capacity check To find the zero shear place X, F F = X 1 ( 6 + X ) = 0 a p A

127 Appendix - A X =. 18m Moment at X, M = 85.05kNm / m Required section modulus M Sre = = = m / m σ all Thickness of the wall is 0.5m Section modulus of the wall, Required reinforcement S = f S re Ok 3 M A = = =.78mm m re 0.9 f l 0.9 (460/1.15) / Where l lever arm y Use minimum reinforcement amount 100A s bd = mm As = 750 / m 16mm diameter bars in 00mm spacing. Method II: Geometry Pressure Diagram H Active d x Passive Passive Point of Rotation Active Fig A- Geometry and pressure diagram A3

128 Appendix - A For design it is necessary to determine the required depth of penetration for stability and then to size the wall to resist the maximum moment. To determine the depth of penetration required for a given height H we need to consider both moment and force equilibrium: Σ F = 0 Σ M = 0 If the soil is dry the pressures and forces are as shown below σ = γ h K p d x P A1 σ = γ + h K a d ( x H) P P1 σ = γ + h K p d ( x H) P A P P σ = K γ d σ = K γ ( d+ H) h a d h p d Fig A-3 Pressure diagram Where P P P P 1 1 = K A 1 = K aγ d ( x+ H ) P1 P γ x d 1 A = Kaγ d x( d x) + K aγ d ( d x) 1 P = K pγ d ( x+ H )( d x) + K pγ d ( d x) From equilibrium ΣF = 0 : P A1 + P P - P P1 - P A = 0 This gives a quadratic equation involving terms in x and d ΣM = 0: Taking moments about the point of rotation x+ H + d x x d x PA 1 PA P1 + P 3 3 A4

129 Appendix - A This gives a cubic equation involving terms in x 3 and d 3. Two equations with unknowns, x and d, and hence we can determine the required depth of penetration for the wall. The equations can be solved graphically Force equilibrium Moment equilibrium 3.8 x (m) D (m) 4.1 Fig A-4 Graphical solutions for sand Force equilibrium Moment equilibrium x (m) D (m) Fig A-5 Graphical solutions for clay A5

130 Appendix - B 9. APPENDIX B STATIC DESIGN OF CANTILEVER RETAINING WALL Introduction This appendix illustrates the sizing and structural design for usual loading (i.e., static) of the 6.1m-high cantilever retaining wall that is analyzed dynamically in the first part of this report. The wall design is performed in two stages. The first stage consists of sizing the wall to satisfy global stability requirements, The global stability requirements are expressed in terms of the factor of safety against sliding; the factor of safety against bearing capacity failure; and the percentage of the base area in compression, with the latter quantifying the overturning stability of the wall. The second stage of the wall design entails the dimensioning of the components of the concrete wall (i.e., stem and base slab, toe and heel elements) and detailing of the reinforcing steel. Each of the three structural elements is designed by the strengthdesign method as a cantilever, one-way slab for flexure and shear loadings. However, both structural and geotechnical calculations are presented in this appendix. Sizing of cantilever wall As stated previously, the first design stage consists of sizing the cantilever wall such that global stability requirements are satisfied (i.e., sliding, overturning, and bearing capacity). The structural wedge of the proposed wall and backfill is shown in figure B-1, as well as the backfill and foundation material properties. To assess the global stability of the wall, the external forces and corresponding points of action acting on the structural wedge need to be determined. The external forces include the resultant of the lateral earth pressure and reactionary forces acting along the base of the wall. However, before the reactionary forces can be determined, the weights and centers of gravity of the concrete and soil composing the structural wedge are required. 0.9 m 0.5 m.4 m Stem Back Fill 6.1m Toe Base 4 m 0.6 m Heel Fig B-1 Cantilever wall system B1

131 Appendix - B To simplify the determination of the weight and center of gravity of the structural wedge, it is divided into subsections having uniform unit weights and simple geometries, as shown in figure B-. Once the weights and centers of gravity of each subsection are determined, the weight and center of gravity of the entire structural wedge are easily determined, as illustrated in the accompanying sketches. Stem 1 Back Fill Stem 1 Toe Base Fig B- Division of wall and backfill into subsections Weight and centre of gravity from toe of the soil on top of the base ( ) 58.7kN m Ws = γ tlh = = /.4 X s = =. 8m Back Fill 5.5m.4 m c.g. Toe.8 m Weight and centre of gravity from toe of the stem (1) 1 Wstem ( 1) = Astem(1) γ c = 0. = / 0.1 X stem( 1) = = 1. 03m 3 ( ) kN m Toe Stem 1 0. m 1.03 m 5.5m c.g. B

132 Appendix - B Weight and centre of gravity from toe of the stem () 1 Wstem ( ) = Astem() γ c = = / 0.5 X stem( ) = = 1. 35m ( ) kN m Stem 0.5 m c.g. 5.5m Toe 1.35 m Weight and centre of gravity from toe of base ( 0.6 4) 56.64kN m Wbase = γ clbasehbase = 3.6 = / 4 X base = = m Base Toe 1.35 m 4.0m c.g. 0.6 m Total weight of the wall W W + W + W c = stem( 1) stem() base = = 134.5kN / m Centre of gravity of the wall from toe X c = Wi i i X W i i = = 1.593m Weight of entire system, W = W s + Wc = = 393.kN / m Centre of gravity from toe of entire system, X Ws X s + Wc X = W + W s c = 393. =.387m c In a cantilever wall analysis the retaining wall system is divided into two or three wedges: the structural wedge; the driving wedge, and the resisting wedge, when present. Figure B-1 shows the structural wedge, which is defined by the outline of the B3

133 Appendix - B cantilever retaining wall. The lateral extent of the structural wedge is defined by imaginary vertical sections made through the heel of the wall and the toe of the wall. The soil mass contained within this region is also considered part of the structural wedge. The driving soil wedge on the retained soil side (to the right of the Figure B-1 structural wedge and not shown in this figure) generates earth pressure forces tending to destabilize the structural wedge. No resisting wedge is present to the left of the Figure B-1 structural wedge in this case. The general wedge method of analysis is used to calculate the lateral earth pressure force acting on the structural wedge. A sliding stability analysis is conducted of the driving wedge, structural wedge, and resisting wedge (when present) to determine a common factor of safety against sliding for the entire retaining structural system (of the three wedges). The procedure uses limit equilibrium and is iterative in nature. Critical failure angles for the driving side and resisting side (when present) potential planar slip planes are sought for an assumed sliding factor of safety value. The resulting earth forces are summed. If the sum is zero, the system is in equilibrium and the critical sliding factor of safety value has been found. The shear mobilization factor (SMF) is set equal to /3. The SMF and factor of safety are inverses of each other. An SMF of /3 is equivalent to a factor of safety equal to 1.5. The determination of the lateral earth forces and pressures are illustrated in the following equations in simplified hand computations, Shear mobilization factor (SMF) = tan ' ' o ( φ ) = SMF tanφ = tan( 35 ) mob ' φ mob = o The angle of friction alongside the structural wedge defined by the vertical section through the heel of the wall is assumed equal to zero (i.e., δ = 0 ). Thus, for the case being considered (i.e. homogenous backfill), the wedge analysis procedure reverts to the classical Rankine procedure, which is illustrated in the following sketches. K a σ ' v ' o φ = tan 45 mob 5 = tan o 45 = 0.41 = γ h m = 119.6kN / m o 6.1m toe = y ' h Fh,static B4

134 Appendix - B σ = K σ ' h a ' v = = 49.0kN / m 1 ' Fh, static = σ hh 1 = = 149.5kN / m F h, static Δytoe = = kN h Δ y toe = = 3 =.03m The remaining, yet-to-be-determined, forces acting on the structural wedge are the reactionary shear and normal forces (i.e., T and N, respectively) acting on the base of the wall. Figure B-3 shows a free body diagram of the structural wedge with the known and unknown forces identified. As illustrated in the following equations, the magnitude of T and N are determined by summing the forces in the horizontal and vertical directions, respectively, while the point at which N acts is determined by summing the moments around the toe. F v = 0 = N W c W s N = W c + W s = = 393.kN / m F h = 0 = T = F h, static T F h, static = 149.5kN / m B5

135 Appendix - B M toe = 0 = X ' N + F N h, static Δy toe W X c c W X s s X ' N = F N h, satic Δy toe + W X c c + W X s s X ' N F = h, static Δy toe + W X N c c + W X s s = 393. = 1.615m 4.0 m 0 W s 6.1m Fh,static Toe X N ' T W c N Fig B-3 Free body diagram of the structural wedge With all external forces acting on the structural wedge determined, the global stability of the wall is assessed by computing the factor of safety against sliding; the percentage of the base area in compression (overturning stability); and the factor of safety against bearing capacity failure. B6

136 Appendix - B Factor of safety against sliding Minimum factor of safety against sliding (FS sliding ) for cantilever wall in usual loading is 1.5. The FS sliding for usual loadings is computed as follows: T ult = N tan( δ ) δ base = φ base base o = 393. tan(40 ) = 39.95kN / m FS O.K sliding = Tult T = =.0 f 1.5 Percentage of the base area in compression (Overturning stability) The global stability of cantilever retaining walls to overturning is quantified by the percentage of the base area in compression. The 100% percentage of base area is required to be in compression in usual loadings for cantilever walls on soil foundations: B e = X N ' 4 = = 0.385m σ max N 6 e = 1+ B B = = kN / m B/ m Toe Lc Heel σ mim σ min 0 N 6 e = 1 B B e = 1 4 B = 41.53kN / m max X N ' N ' e min 100% base area in compression B7

137 Appendix - B Factor of safety against bearing capacity failure The minimum factor of safety against bearing capacity failure FS bc for cantilever retaining walls in usual loading is 3.0. The following expression for the normal component to the base of the structure of the ultimate bearing capacity for strip footings: Q = B ( ξ ξ ξ ξ c N ) + ( ξ ξ ξ ξ q N ) cd ci ct cg c qd qi qt qg 0 q + ( ξ ξ ξ ξ B γ N ) However, for the wall being analyzed, only the last term is nonzero, and thus, this expression reduce to: ( ξ ξ ξ ξ B γ N ) γd γi γt γg γ Q = B The FS bc for usual loading is computed as follows: γd γi γt γg γ B = B e = = 3.3 ξ γi δ = 1 φ o 0.8 = 1 40 o = 0.3 ( ξ ξ ξ ξ B γ N ) γd γi γt γg γ Q = B = 3.3 = 03.0kN / m ( ) 1 T δ = tan N tan 1 = 393. = o 0.8 ξ = ξ t = ξ g = 1 γd γ Q FS bc = N O.K γ 03.0 = 393. = Stage : structural design of concrete cantilever retaining wall As stated in the introduction to this appendix, the second stage of the wall design entails the structural design of the concrete wall, to include the dimensioning of the concrete base slab (the toe and heel elements) and stem, and the detailing of the reinforcing steel. All reinforced-concrete structures must satisfy both strength and serviceability requirements. In the strength design method, this is accomplished by multiplying the service loads by appropriate load. Thereby reducing steel stresses at service loads. The service loads are those determined in the first design stage presented previously. B8

138 Appendix - B Each of the three structural elements is designed as a cantilever, one-way slab for flexure and shear loadings. The following example uses Grade 40 steel. 100 mm cover is used in the example. Figure B-4 shows the structural wedge and the externally imposed stresses determined in the first design stage. Also shown in Figure B-4 are the critical locations for evaluating shear and bending moment for the stem, heel, and toe elements. 4.0 m Ws 6.1m 1. m kN/m^ 49.0kN/m^ 41.53kN/m^ Critical section for shear Critical section for moment Fig B-4 Critical locations for shear and bending moment Moment capacity of the stem The stem is analyzed as being singly reinforced with the critical section for moment capacity being at the base of the stem, as illustrated below. M stem = =.46kN M u = = 378.1kN M u M n = = = 40.14kN m Fh,moment =11.54kN/m^1 h =1.83m 44.kN/m^ M stem B9

139 Appendix - B Minimum required reinforcing steel: k u = 1 h M n + Pn d f b d ' c = = ' 0.85 fc ku As = f y b d = = mm / m Use 5mm diameter steel 50mm c-c (conservative) A s = mm / m Shear capacity of the stem The critical section for shear in the stem is taken as 0.7m above the interface of the base and stem, where 0.7m is d at the base of the stem. However, the d at the critical section is only 0.675m, due to the taper of the wall. V u = 1.7 ( 9.57) = kN / m φ V c = φ shear 0.17 f ' c b d 4.8m Fh,shear =9.57kN/m^1 φv s = = kN / m 1. 3 ( V φv ) u c m Vstem 38.6kN/m^ 1.3 p 0 O. K ( V φv ) u c = 1.3 ( ) B10

140 Appendix - B Moment and shear capacity of the heel The heel is analyzed as being singly reinforced, with the steel along the top face and 100 mm. coverage. The critical section for both moment and shear capacity in the heel is at the interface of the heel and the stem..4 m Ws =58.7kN/m^1 M heel Vheel Wc =34.0kN/m^ kN/m^ kN/m^ M heel 1. = ( ) ( ) M = 0 M heel = 166.3kNm/ m heel M u = = 8.59kN M u M n = φ 8.59 = 0.9 = 314.0kN Minimum required reinforcing steel: k u = 1 h M n + Pn d f b d ' c = = B11

141 Appendix - B A s 0.85 = ' fc k f y u b d = = 190.5mm / m Use 5mm diameter steel 50mm c-c (conservative) A s = mm / m Shear capacity of heel: V u ( ) = 1.7 = kN / m φ V c = φ shear 0.17 f ' c b d φv s 1.3 p 0 O. K = = kN / m 1. 3 ( V φv ) u ( V φv ) u c c = 1.3 ( ) B1

142 Appendix - B Figure B-5a,b shows the steel reinforcing detailing, determined previously, wherein the stem, heel, and toe were treated as singly reinforced members. a) # 50mm c-c # 50mm c-c # 50mm c-c b) use standard hook # 50mm c-c # 50mm c-c # 50mm c-c Fig B-5 Proposed steel reinforcement detailing. A minimum 100mm cover is required B13

143 Appendix - C 10. APPENDIX C DESIGN OF GRAVITY WALL The proposed model is used to carry out a parametric study on the dynamic response of an 8.0 m high and 3.0 m wide gravity retaining wall proportioned using the traditional approach to seismic design for an earthquake with a peak acceleration of 0.g. It is the wall analyzed by Whitman. The width at the top of the wall is chosen to be 0.80 m. In order to choose the proper wall width (3.0 m in this case) the traditional approach to seismic design is used with an acceleration coefficient equal to 0.5 of the peak acceleration of the design earthquake. Moreover, a safety factor between 1.1 and 1. is chosen based on recommended factors of safety by NAVFAC (198 design manual). The chosen wall has static safety factor against sliding and overturning of.88 and.50, respectively. The corresponding total (static plus dynamic) safety factors are 1.7 and 1.9 respectively. This wall is designed using the traditional approach to design for a horizontal acceleration coefficient of 0.1g, which is chosen correspond to 0.5 of a peak acceleration of 0.g for the design earthquake. C1

144 Appendix - D 11. APPENNDIX D ADDITIONAL RESULTS FROM DIANA ANALYSES Introduction This section provides additional results from DIANA analyses to give a clear picture of whole analyses carried out in this phase of research work. Following figures are the free field acceleration time-histories recorded at the top node of the numerical models Fig D-1 Free field motion recorded from El Centro analysis for diaphragm wall in sand Fig D- Free field motion recorded from Chi-Chi analysis for diaphragm wall in sand D1

145 Appendix - D Fig D-3 Free field motion recorded from El Centro analysis for diaphragm wall in clay Fig D-4 Free field motion recorded from Chi-Chi analysis for diaphragm wall in clay Fig D-5 Free field motion recorded from El Centro analysis for cantilever wall in sand D

146 Appendix - D Fig D-6 Free field motion recorded from Chi-Chi analysis for cantilever wall in sand Fig D-7 Free field motion recorded from Kobe analysis for cantilever wall in sand Fig D-8 Free field motion recorded from Chi-Chi analysis for cantilever wall in clay D3

147 Appendix - D Fig D-9 Free field motion recorded from El Centro analysis for gravity wall in sand Following figures are the displacement time-histories of the cantilever wall base Fig D-10 Relative permanent displacement time-history of base from Chi-Chi analysis for cantilever wall in sand Fig D-11 Relative permanent displacement time-history of base from Kobe analysis for cantilever wall in sand D4

148 Appendix - D Fig D-1 Relative permanent displacement time-history of base from Chi-Chi analysis for cantilever wall in clay Following figures are the deform shape of the cantilever wall at the end of the dynamic analyses Fig D-13 Deformed mesh from Chi-Chi analysis for cantilever wall in sand Fig D-14 Deformed mesh from Kobe analysis for cantilever wall in sand D5

149 Appendix - D Fig D-15 Deformed meshes from Chi-Chi analysis for cantilever wall in clay, deformations magnified by a factor of 0. (Note: Toe of the wall not initially embedded) The following figures give the bending moment time history of diaphragm m D6