A HYBRID EFFECTIVE STRESS – TOTAL STRESS PROCEDURE FOR ANALYZING SOIL EMBANKMENTS SUBJECTED TO POTENTIAL LIQUEFACTION AND FLOW by Ernest Naesgaard B.A.Sc., The University of British Columbia, 1973 M.Eng., The University of British Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2011 © Ernest Naesgaard, 2011 Abstract Seismic design of major civil structures (bridges, dams and embankments) is moving increasingly towards using performance design methodologies which require determination of earthquake induced movements. Development of these numerical design tools and procedures for use in engineering practice for estimating the earthquake induced ground deformations of potentially liquefiable soil is the topic of this dissertation. Fully coupled effective stress numerical analyses procedures developed at the University of British Columbia (UBC) were used to simulate field and centrifuge test case histories. These analyses can offer considerable insight, but due to the complexity of the problem and variability of the parameters involved, there is considerable uncertainty. The author, therefore, recommends that the relatively new state-of-the-art effective stress analyses should be augmented by carrying out an additional analysis compatible with conventional design processes. This latter analysis uses published post-liquefaction “residual” soil strengths derived from back-analysis of field case histories by others. The developed design methodology uses the effective stress (UBCSAND) soil constitutive model for dynamic analyses, and empirical “residual” post-liquefaction soil strengths for a postshaking total stress static analysis. In the proposed approach, the effective stress dynamic analysis is used to determine zones of liquefaction, to quantify earthquake induced deformations, and to provide overall insight. The post-shaking total stress static analysis, with “residual” strength parameters used in elements which liquefied, is carried out to capture the effects of complex stratigraphy and localization that may be missed by the effective stress model. Calibration and validation of the UBCSAND model was undertaken by comparing the model with field case histories and laboratory simple shear, shake table, and centrifuge tests. The measured response of some centrifuge tests being used for validation was indicative of the centrifuge model not being fully saturated. This was problematic as P-wave measurements within the centrifuge model suggested full saturation. A series of triaxial tests with P-wave measurements was carried out. These tests, and the numerical modeling of them, showed that high P-wave velocities were not always indicative of full saturation and they provided a logical explanation for the observed centrifuge response. ii Preface Technical advance evolves over time and involves the contribution of many. This research is no different. Chapter 2 is an overview of my understanding of key aspects of soil liquefaction and related to phenomena. Material included in chapter was previously published in Naesgaard et al. (2005, 2006, 2007, and 2009) with co-authors including Professor P.M. Byrne, Dr. M. Seid-Karbasi, and Dr. M. Beaty. I wrote the papers with input and review from the co-authors. Much insight also comes from the work of Professor T. Kokusho. The material in Chapter 3, “Soil Mixing”, was previously published by Naesgaard and Byrne (2005). Professor Byrne proposed the “mixing” concept in 1988/89 to explain the flow of tailings into the underground Mufulira mine workings. I proposed the concepts of cyclic mixing as a means of inducing soil liquefaction and low strength and flow within sands. Others, Vasquez-Herrera and Dobry (1989), had similar thoughts. I carried out rudimentary cyclic shear tests to study the mixing of layered granular soils. I also made the observation that mixing could be induced by turbulent flow in the vicinity of water films. Chapter 4 is a brief overview on post-liquefaction residual strength that is based on literature review. Chapter 5 is largely a discussion on the constitutive models UBCSAND, UBCTOT, and UBCHYST that were developed at the University of British Columbia by Professor Byrne and his students. UBCTOT was developed by Professor Byrne and Dr. Beaty. UBCSAND was developed by Professor Byrne and several students. I made modifications to UBCSAND described in Section 5.3.4 and compiled a list of limitations and challenges with the program. The algorithm for UBCHYST was developed by Professor Byrne. I coded UBCHYST as a constitutive model in FLAC, made improvements regarding unload/reload loops, developed options for strength reduction as a function of strain or number of cycles, and developed documentation explaining the model. The proposed hybrid effective stress – total stress design approach in Chapter 6 is a new contribution derived from my research and is published in papers by Naesgaard, Byrne, and Beaty (2009), and by Naesgaard and Byrne (2007). iii Chapter 7 contains UBCSAND model calibrations and example analyses that I have either carried out or supervised, unless otherwise indicated. • Section 7.2 provides a summary of recent single element numerical simulations of the simple shear tests by Sriskandakumar (2004). • Section 7.3 describes a calibration of the model to empirical liquefaction triggering charts. This procedure is commonly used for calibrating UBCSAND for commercial design projects. The example shown was calculated by Dr. Beaty. • Section 7.4 describes some early 2004 one dimensional (infinite slope) column tests that I carried out to demonstrate that UBCSAND/FLAC could simulate the post-shaking flow failure phenomena. This may be the first simulation of this in a numerical model. • Section 7.5 describes one of the earlier commercial uses of UBCSAND with a centrifuge test program for validation. The work is described in Naesgaard et al. (2004) and in Yang et al. (2004) with co-authors P.M. Byrne, K. Adalier, T. Abdoun, and B. Gohl. • Section 7.6 is from a paper by Naesgaard, Byrne, Seid-Karbasi, and Park (2005) that described some numerical modeling of centrifuge tests carried out as part of the UBC Liquefaction Initiative. I carried out the numerical modeling described in this section. • Section 7.7 describes numerical modeling that I conducted to simulate the behaviour observed in Professor Kokusho’s shake table tests. The work was previously published in a paper by Naesgaard, Byrne, and Beaty (2009). • Section 7.8 describes my work on numerical modeling of the Lower San Fernando dam in California. This was previously published in Naesgaard and Byrne (2007). This section compares an actual well documented field case history to that obtained using the proposed design methodology. Other, practicing engineers, have also adopted the models described in this dissertation and are using them on similar projects to not only provide insight into liquefaction phenomena, but also to quantify earthquake induced ground movements. The work described in Chapter 8 was previously published by Naesgaard, Byrne and Wijewickreme (2007). I did the laboratory testing and numerical analyses, while Professor Byrne and Professor Wijewickreme provided consultation and review. E. Naesgaard April 2011 iv Table of Contents ABSTRACT .......................................................................................................................ii PREFACE ........................................................................................................................ iii TABLE OF CONTENTS .................................................................................................. v LIST OF TABLES ......................................................................................................... viii LIST OF FIGURES .......................................................................................................... ix LIST OF SYMBOLS .....................................................................................................xvii ACKNOWLEDGEMENTS ............................................................................................ xx DEDICATION ................................................................................................................ xxi 1 INTRODUCTION........................................................................................................ 1 2 SOIL LIQUEFACTION, VOID REDISTRIBUTION AND POSTLIQUEFACTION RESIDUAL SHEAR STRENGTH ............................................ 3 2.1 Overview ............................................................................................................. 3 2.2 Loose Sand Layer underlying Low Permeability Barrier ................................... 7 2.3 Pore Water Void Redistribution within real Embankments and Slopes ........... 10 2.4 Chapter 2 Summary and Conclusions ............................................................... 14 3 SOIL MIXING ........................................................................................................... 25 3.1 Introduction to Mixing Phenomena .................................................................. 25 3.2 Mufulira Mine Case History (mixing due to static slope failure) ..................... 26 3.3 Cyclic Mixing in Rudimentary Laboratory Shear Tests ................................... 27 3.4 Discussion ......................................................................................................... 28 3.5 Chapter 3 Summary and Conclusions ............................................................... 30 4 RESIDUAL STRENGTH FROM BACK-ANALYSES OF CASE HISTORIES................................................................................................................ 43 4.1 Introduction ....................................................................................................... 43 4.2 Seed (1984, 1987) ............................................................................................. 45 4.3 Seed and Harder (1990) .................................................................................... 45 4.4 Olson and Stark (2002) ..................................................................................... 45 4.5 Idriss and Boulanger (2008) .............................................................................. 45 4.6 Discussion ......................................................................................................... 46 4.7 Chapter 4 Summary and Conclusions ............................................................... 46 v 5 ANALYSIS PROCEDURES OVERVIEW ............................................................. 51 5.1 Introduction ....................................................................................................... 51 5.2 State-of-Practice Total Stress Analysis (UBCTOT) ......................................... 53 5.3 UBCSAND Coupled Effective Stress Analysis ................................................ 54 5.3.1 Introduction to UBCSAND ............................................................................... 54 5.3.2 Elastic Response................................................................................................ 56 5.3.3 Plastic Response ................................................................................................ 57 5.3.4 Recent Improvements/Options added to UBCSAND ....................................... 58 5.3.4.1 5.3.4.2 5.3.4.3 5.3.4.4 Pull Down Yield Surface on unloading if no Cross-over .................... 58 Modeling of Dense Soils with Lower Phase Transformation Friction Angle...................................................................................... 58 Delay Dilation following Post-Liquefaction Cross-over ..................... 58 Dilation Reduction or Cut-off upon Expansion of Element ................ 59 5.3.5 UBCSAND/FLAC Limitations and Challenges ............................................... 60 5.3.6 UBCSAND/FLAC Grid-Model Development Considerations ......................... 63 5.4 Hysteretic Model for Non-liquefiable Clay/Silt Soils (UBCHYST) ................ 64 5.4.1 Introduction ....................................................................................................... 64 5.4.2 Non-Linear Hysteretic “UBCHYST” Model .................................................... 65 5.4.3 Implementation of UBCHYST Model .............................................................. 66 5.4.4 UBCHYST Calibration ..................................................................................... 67 5.4.5 Summary ........................................................................................................... 67 5.5 Chapter 5 Summary and Conclusions ............................................................... 67 6 PROPOSED COMBINED EFFECTIVE STRESS – TOTAL STRESS APPROACH ............................................................................................................... 79 6.1 Introduction ....................................................................................................... 79 6.2 Implementation of Procedure in FLAC............................................................. 79 6.3 Discussion ......................................................................................................... 84 7 MODEL CALIBRATION, VALIDATION AND APPLICATION EXAMPLE.................................................................................................................. 89 7.1 Introduction ....................................................................................................... 89 7.2 Calibration to Laboratory Cyclic Simple Shear Tests....................................... 89 7.2.1 Introduction ....................................................................................................... 89 7.2.2 Results and Discussion...................................................................................... 90 7.3 Calibration to “Seed type” Empirical Liquefaction Triggering Charts ............. 91 vi 7.4 One-Dimensional Tests for demonstrating Void Redistribution ...................... 93 7.5 George Massey Tunnel Centrifuge Test Modeling ........................................... 94 7.6 UBC/C-CORE Centrifuge Test Modeling ........................................................ 95 7.7 Kokusho Shake Table Emulation ...................................................................... 96 7.8 Lower San Fernando Dam Example Analysis .................................................. 97 7.8.1 Introduction ....................................................................................................... 97 7.8.2 Coupled Effective Stress Analysis of LSFD – Phases 2 and 3a ....................... 98 7.8.3 Post-Liquefaction Total Stress Analysis of LSFD – Phase 3b .......................... 98 7.8.4 Results of LSFD Example Analysis .................................................................. 99 8 P-WAVE MEASUREMENTS AS AN INDICATOR OF SATURATION IN CENTRIFUGE TESTS ...................................................................................... 131 8.1 Introduction ..................................................................................................... 131 8.2 Laboratory Element Testing Program ............................................................. 133 8.3 Numerical Modeling ....................................................................................... 135 8.4 Discussion ....................................................................................................... 136 8.5 Chapter 8 Summary and Conclusions ............................................................. 137 9 DISCUSSION AND CONCLUSIONS ................................................................... 149 9.1 Summary and Conclusions .............................................................................. 149 9.2 Recommendations ........................................................................................... 156 9.2.1 Recommendations based on the Work carried out for this Dissertation ......... 156 9.2.2 Recommendations for Future Research .......................................................... 157 REFERENCES .............................................................................................................. 160 APPENDIX A UBCSAND1v02 FISH code and Flow Chart ....................................... 169 APPENDIX B Analysis Of A Concrete Gravity Dam Over Potential Liquifiable Soil Illustrating Proposed Hybrid Procedure Methodology .................. 184 vii List of Tables Table 3-1 Summary of Rudimentary Cyclic Shear Tests................................................ 33 Table 7-1 Summary of Undrained Laboratory Simple Shear and UBCSAND Simulations .................................................................................................... 101 Table 7-2 Massey Tunnel Project. “Class A” Numerical Predictions compared to Centrifuge Test Results (in prototype scale) ............................................. 101 Table 7-3 Soil properties used in emulation of Kokusho shake table tests . ...................102 Table 8-1 Summary of Laboratory Test Conditions and Results................................... 140 Table 8-2 Soil and Fluid Properties used in Numerical Analyses ................................. 141 Table B-1 Example Dam analysis model profile and soils types. Soil properties are summarized in Table B-1 ........................................................ 189 viii List of Figures Figure 2-1 Comparison of cyclic drained simple shear response of loose and dense Fraser River sand (from Sriskandakumar 2004). (a) is shear stress versus shear strain, (b) is shear strain versus volumetric strain, and (c) is shear stress versus volumetric strain. (Note labels in (c) refer to the loading phase – the soil is always contractive on unloading). .......................................................................................................17 Figure 2-2 Cyclic (a) stress path and (b) stress-strain response of loose sand without initial static shear stress (from Sriskandakumar 2004). ......................18 Figure 2-3 Postulated response of loose sand to undrained cyclic simple shear loading followed by monotonic loading to large strain. (a) is stress strain space, (b) is stress path space and (c) is void ratio vertical effective confining stress space. “A” is initial state with pore pressure ratio, R u =0, “D” is true liquefaction with R u = 1, and “F” is the strength at the critical or steady state if pore water does not cavitate. ............................................................................................................19 Figure 2-4 Postulated response of loose sand element to cyclic simple shear with “A” to “B” undrained and “B” to “D” with slight pore water inflow. (a) is stress strain space, (b) is stress path space and (c) is void ratio vertical effective confining stress space. “A” is initial state with pore pressure ratio, R u =0, “D” is true liquefaction with R u = 1. At “D” the sample is at the critical state with strength at or near zero........................................................................................................ .......... 20 Figure 2-5 Re-consolidation volumetric strain from dissipation of excess pore pressure following liquefaction. α = 0 is with zero static bias or level ground conditions whereas α = τ static /σ' vo = 0.1 is with a static bias such as that from sloped ground (from triaxial data by Dismuke 2003 as reported by Malvick 2005). .........................................................................21 Figure 2-6 Hammer blow to base of 130mm diameter column of loose sand liquefies it and causes almost immediate formation of water film at boundary with less permeable fine sand (Kokusho and Kabasawa 2004). ...............................................................................................................21 Figure 2-7 Path A is the stress path for a soil element with a static bias and with pore water inflow that is cyclically loaded until flow instability occurs. Path C would be an element with there is pore water outflow and flow instability does not occur (Malvick 2005 – after Kulasingam 2003). ...............................................................................................................22 Figure 2-8 Factors affecting thickness and residual strength of localized shear zone under barrier (a) grain-size (b) leakage through barrier (c) undulations of barrier surface (d) variation of stress on barrier (from Naesgaard et al. 2006). .....................................................................................23 ix Figure 2-9 Flow redistribution mechanisms within an embankment with low permeability barrier and high permeable layer. Vectors show directions of pore water flow (Naesgaard et al. 2006). ....................................24 Figure 3-1 Graphic illustrating the effects of soil mixing. Elements 'A', 'B' and 'C' have the same number and volume of large and small particles. ‘A’ is the element with a fine layer over a coarse layer prior to disturbance. 'B' represents the case where fine particles fall into the voids between large particles and leave a water filled void or interlayer at the top of the element. ‘C’ represents the case where the particles mix evenly and the larger particles move into a looser arrangement. Both ‘B’ and ‘C’ will have significantly lower shear strength then ‘A’ if mixing does not occur (Naesgaard and Byrne 2005). ...............................................................................................................35 Figure 3-2 1970’s Mufulira Mine tailings liquefaction failure. .........................................35 Figure 3-3 Photograph of 1970 Mufulira Mine sink hole (tailings.info 2005). .................36 Figure 3-4 Grain-size of Mufulira Mine tailings (Barrett and Byrne 1988; Horsfield and Been 1989). ...............................................................................36 Figure 3-5 Void ratio vs. undrained residual shear strength (τ res = (σ1-σ3)/2) relationship for sandy silt, silty sand, and mixed residual in-rush tailings. Note: Mixed tailings have essentially zero shear strength at the in-situ void ratio (Naesgaard and Byrne 2005 from triaxial tests by Horsfield and Been 1989). ..........................................................................37 Figure 3-6 Conceptual undrained triaxial compression stress path for silty sand, sandy silt and mixed silt-sand at same void ratio and initial stress state (Naesgaard and Byrne 2005). ...........................................................................37 Figure 3-7 Postulated progressive failure mechanism of 1970 sinkhole at Mufulira Mine. (1) Sides of initially step sided sinkhole fails, (2) as soil slides to bottom of slope, the layers mix and due to this mixing lose nearly all strength (liquefy), (3) the liquefied soil flows away into mine workings, (4) mechanism is repeated giving a progressive failure mechanism, and (5) the final stable slope at shallow 10° angle (Naesgaard and Byrne 2005). .......................................................................... 38 Figure 3-8 Rudimentary cyclic shear test with saturated layered sample (Naesgaard and Byrne 2005). ...........................................................................38 Figure 3-9 Grain-size of fine and coarse sand used in rudimentary simple shear tests (Naesgaard and Byrne 2005)....................................................................39 Figure 3-10 Test “A” before and after cyclic shearing (Naesgaard and Byrne 2005). ...............................................................................................................39 Figure 3-11 Post-shearing for non-layered Tests “B” and “C” (Naesgaard and Byrne 2005). .....................................................................................................40 x Figure 3-12 Before and after cyclic shearing for Test “D” with glass beads and coarse sand (Naesgaard and Byrne 2005). .......................................................40 Figure 3-13 Slide from Kokusho cylinder shake table test video showing zone of mixing between coarse and fine sand caused by pore fluid flow and turbulence (Kokusho 2003) (Video from: http//:www.civil.chuou.ac.jp/lab/doshitu/index.html).........................................................................41 Figure 3-14 Steady state void ratio vs. residual strength (Sus) for reconstituted (remolded) Lower San Fernando Dam (California) soil. Note that the residual strength (S us ) for the combined (mixed) homogeneous specimens is much lower than that of the layered not-mixed specimens (Baziar and Dobry 1995). (Without mixing the strength is about 0.31 tsf whereas with mixing the strength is about 0.06 tsf). .................42 Figure 4-1 Post-liquefaction residual strength form case histories (a) Seed et. al. (1984), (b) Seed and Harder (1990), (c) Olsen and Stark (2002) and (d) Idriss and Boulanger (2008). ......................................................................49 Figure 4-2 Post-liquefaction residual strength form case histories by Idriss and Boulanger (2008) with undrained strength inferred from University of British Columbia laboratory tests by Sriskandakumar (2004) superimposed (Byrne et al., 2008). ..................................................................50 Figure 5-1 In the equivalent linear method initial G(1) and ξ(1) are estimated. An elastic analysis is carried out with these values and from the results a new γ (1) eff is obtained. This γ (1) eff is used to get a new G(2) and ξ(2) which are used for the second elastic analysis iteration. This process is repeated until the process converges and strain compatible GFinal and ξfinal are obtained. ......................................................................... 70 Figure 5-2 Example of the liquefaction triggering methodology in UBCTOT. The CSR at “A” would cause liquefaction in 15 cycles; however, since it is only ½ a cycle this pulse provides 1/15 x ½ = 1/30th of the loading to cause liquefaction. Pulse “B” would cause liquefaction in 2 cycles and, therefore, provides ½ x ½ = ¼ of the loading necessary to cause liquefaction. The full cycle of “A” and “B” would provide ¼ + 1/30 = 17/60 of the loading necessary to liquefy the element. ...................... 70 Figure 5-3 Bilinear response of an element that liquefies when using UBCTOT (Beaty, 2001). ...................................................................................................71 Figure 5-4 Liquefaction triggering chart and equations from Idriss and Boulanger (2006, 2008). ....................................................................................................72 Figure 5-5 Yield surface in UBCSAND. The hardener (dη = d(τ/σ')) that expands the yield surface (i.e., from A to B) is a function of the plastic shear strain and plastic shear modulus as illustrated in Figure 5-8. ...................................................................................................................72 Figure 5-6 Flow Rule in UBCSAND. Below Ф cv or Ф pt shear strain induces volumetric contraction and above it induces volumetric dilation. ...................73 xi Figure 5-7 Stress ratio history showing definition of loading, unloading and reloading UBCSAND (Beaty 2009; Beaty and Byrne 2011)...........................73 Figure 5-8 Plastic strain increment and plastic modulus in UBCSAND. ..........................74 Figure 5-9 Comparison of postulated post-liquefaction response of loose sand and that modeled by UBCSAND. ....................................................................74 Figure 5-10 A dilation delay (proportional to dilation (expansive plastic volumetric strain) in last ½ cycle) is induced at shear stress cross-over in UBCSAND...................................................................................................75 Figure 5-11 Large element(c) emulating the behaviour of a small element (b). If (a) and (b) are at critical state with δl/l = δL/L then (c) can be made to behave similarly to (b) by setting dilation to 0 when the volumetric strain is equal to δl/L ........................................................................................75 Figure 5-12 Loading in simple shear induces hysteretic response in soil with shear modulus (G) and loop-size (damping) varying with strain. .............................76 Figure 5-13 Characteristic behaviour of soil in cyclic shearing. .........................................76 Figure 5-14 UBCHYST model key variables. .....................................................................77 Figure 5-15 Example calibration of UBCHYST constitutive model to G/G max and damping ratio curves. .......................................................................................77 Figure 5-16 Comparison in response between laboratory simple shear test and UBCHYST simulation for case with static bias (note that the scales on plots are not the same).................................................................................78 Figure 6-1 Flow chart showing combined coupled effective stress / total stress analysis procedure (Naesgaard and Byrne 2007). ............................................88 Figure 7-1 Typical grain-size distribution and microscope view of Fraser River Sand used for cyclic shear tests (from Sriskandakumar 2004). .....................104 Figure 7-2 Comparison of undrained simple shear test L1 laboratory test to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response. ................................................105 Figure 7-3 Comparison of undrained simple shear test L2 laboratory test to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response. ................................................106 Figure 7-4 Comparison of undrained simple shear test L2 laboratory test to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response. ................................................107 Figure 7-5 Comparison of Undrained simple shear test L12 laboratory test with initial static bias to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response – using same calibration parameters as those used without static bias. ............108 xii Figure 7-6 Comparison of undrained simple shear test L12 laboratory test with initial static bias to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response – using additional hfac7 calibration parameter to control pull-down prior to triggering. ..........................................................................................109 Figure 7-7 Comparison of cycles to liquefaction between undrained laboratory simple shear tests and UBCSAND simulations. ............................................110 Figure 7-8 Undrained dense sand simple shear stress vs. strain comparison of D3 laboratory test to UBCSAND simulation (a) Laboratory test, and (b) UBCSAND simulation. ..................................................................................111 Figure 7-9 Undrained dense sand simple shear stress path comparison of D3 laboratory test (a) to UBCSAND simulation (b)............................................112 Figure 7-10 Undrained dense sand simple shear pore pressure ratio comparison of D3 laboratory test (a) to UBCSAND simulation (b). .....................................113 Figure 7-11 Empirical liquefaction triggering chart with overlain UBCSAND simulations (from Beaty 2009; Beaty and Byrne 2011).................................114 Figure 7-12 Volumetric strain (εv) within an infinite slope column with a low permeability crust over loose sand. ................................................................115 Figure 7-13 UBCSAND numerical model of infinite slope (1D) column with low permeability barrier cap. At time x-x' zone 'I' has expanded to the critical state, dilation goes to zero, and flow failure is initiated. ...................115 Figure 7-14 Section through Massey Tunnel in Fraser River. ...........................................116 Figure 7-15 Typical transverse FLAC model grid used from Massey Tunnel. Scale: the tunnel is 24m wide - vertical and horizontal scale are the same................................................................................................................116 Figure 7-16 Typical distorted mesh with displacement vectors. Scale: the tunnel is 24m wide - vertical and horizontal scale are the same...................................116 Figure 7-17 Typical distorted mesh with displacement vectors for level ground condition. Scale: the tunnel is 24m wide - vertical and horizontal scale are the same. ..........................................................................................117 Figure 7-18 Layout of Massey project centrifuge model showing location of instrumentation Model 2 (Legend: a3 = accelerometer; P12 = piezometer; Lv and Lh= LVDT). Dimensions in centimeters. Model 1 and Model 3 are similar...............................................................................117 Figure 7-19 Model #1 accelerometer A10 (dark line is FLAC class A prediction and light line is from centrifuge test). ............................................................118 Figure 7-20 Comparison of class A numerical prediction and centrifuge results for Model #2 with 10m wide densification. .........................................................118 xiii Figure 7-21 Comparison of pore pressure in piezometer P5 in Model #3 (dark colour is FLAC class A prediction and lighter colour is centrifuge test result)..............................................................................................................119 Figure 7-22 Typical grid and input time history used for UBC Liquefaction Initiative C-CORE centrifuge tests (shown in prototype scale). ....................119 Figure 7-23 Comparison of Centrifuge tests and numerical results for profile with low permeability silt barrier (COSTA-C): (a) displaced grid, (b) Horizontal displacement of sliding block over barrier (for the centrifuge data the solid line is measured and the dashed line is corrected to better match final displacements), (c) vertical displacement at crest, (d) centrifuge and numerical surface profiles, (e) calculated lateral displacement contours in metres, (f) and (g) pore pressure time histories at P3 and P6, (h) acceleration time history at A6. ..................................................................................................................120 Figure 7-24 (a) Initial and displaced profile of centrifuge test CT5 with three drainage slots (b) numerical analysis of same................................................121 Figure 7-25 Comparison of vertical displacement near crest with (CT5) and without (COSTA-C) drainage slots. Post-shaking flow initiated in the COSTA-C test at approximately 70s..............................................................122 Figure 7-26 Kokusho shake table model at start of test (Kokusho 2003). .........................122 Figure 7-27 Kokusho shake table test profile at end-of-shaking. ......................................123 Figure 7-28 Post-failure profile of Kokusho shake table test. Failure occurred a short time after all shaking had stopped. ........................................................123 Figure 7-29 UBCSAND/FLAC model grid of Kokusho shake table test. .........................124 Figure 7-30 Original boundary of model and displaced shape of FLAC model grid. ................................................................................................................124 Figure 7-31 Contours of horizontal displacement (a) with barrier layers in place and flow on and (b) with barriers but no flow. ..............................................125 Figure 7-32 Calculated displacement time histories of shake table model with and without barrier and with and without flow. ....................................................126 Figure 7-33 Lower San Fernando Dam which failed into the reservoir approximately 30s after end of earthquake shaking. Blue areas are soil that was mixed and deemed to have liquefied. Lower figure is reconstructed profile which shows location of liquefied soil before failure. (Seed et al. 1973). ..............................................................................127 Figure 7-34 Grid, locations of low permeability barriers, (N1) 60 and shear strength input parameters used in Lower San Fernando numerical example. .............128 Figure 7-35 Profile showing zones deemed to have liquefied at end of strong shaking (R u > 0.7). .........................................................................................129 xiv Figure 7-36 Displacement time histories of point near upstream toe of dam for both 3a coupled effective stress analysis (ESA) and 3b total stress analysis (TSA) alternative. Both analyses stopped due to excessive element distortion. ........................................................................................129 Figure 7-37 (a) Horizontal velocity (m/s) and (b) horizontal displacement (m) at end of analyses (40s) for coupled effective stress analysis. .........................130 Figure 8-1 Variation of (a) Skempton ‘B’ with Degree of Saturation (S r ) and (b) P-wave velocity (V p ) with Skempton ‘B’ for Fraser River Sand, with D r =20% K b =92 MPa, G=56 MPa, and porosity = 0.45 in accordance with Equations (1) and (2). ...........................................................................143 Figure 8-2 Normalized cyclic strength ratio (CSR) of sand compared to P-wave velocity (V p ) as measured in laboratory tests conducted with Toyoura sand. Developed using data from Tsukamoto et al. (2002) and Ishihara et al. (1998). CSR is that which causes liquefaction (double amplitude strain of 5%) in 20 cycles (for Relative Density of 40-70%). .......................................................................................................143 Figure 8-3 Laboratory test layout. ..................................................................................144 Figure 8-4 Variation of shear wave velocity (V s ) with effective mean normal confining pressure (σ’ m ) as measured in laboratory tests. ............................144 Figure 8-5 Typical P-wave time histories as measured in the laboratory. Traces (a) is for B = 0.48 with a calculated V p of 1415 m/s and trace (b) is for B = 0.95 with a calculated V p of 1557 m/s. .............................................145 Figure 8-6 Measured shear wave (V s ) and P-wave (V p ) versus Skempton ‘B’. .............146 Figure 8-7 35 by 75 element mesh used for FLAC analyses. ........................................146 Figure 8-8 (a) Represents homogeneous-partial-saturation (HPS) where the air bubbles are small and scattered throughout the void spaces. This is numerically modeled by reducing the fluid modulus (K w ) at all the nodes in the model. (b) Represents non-homogeneous-partialsaturation (NHPS) where there are only a few large bubbles at select locations. This is numerically modeled by reducing the fluid modulus only at select nodes within the model. ...........................................147 Figure 8-9 Comparing V p versus Skempton B from both HPS and NHPS FLAC analyses to that observed in laboratory tests and that calculated from Equation (2). Data from Ishihara et al. (1998) and Tamura et al. (2002) that deviated from the correlation from Equation (2) are also shown. ............................................................................................................ 147 Figure 8-10 Typical variation of fluid modulus within NHPS FLAC grid. Small squares represent locations with low fluid modulus representative of air bubbles, whereas all other elements have a high fluid modulus of 2.2x109 Pa. ....................................................................................................148 xv Figure A-1 Flow diagram for UBCSAND1v02 illustrating how it fits within the FLAC (ITASCA 2008) Mohr Coulomb framework. ......................................18 Figure B-1 Example Dam analysis model profile and soils types. Soil properties are summarized in Table B-1. .......................................................................189 Figure B-2 Example Dam analyses model grid details. ..................................................190 Figure B-3 Example Dam analysis typical post-earthquake horizontal displacement contours. .................................................................................190 Figure B-4 Example Dam analyses showing typical post-earthquake vertical displacement contours. .................................................................................191 Figure B-5 Example Dam analysis showing typical post-earthquake pore pressure ratio contours. .................................................................................191 Figure B-6 Time histories showing displacement of center of intake structure and model base (top of till) for typical analysis (positive is downstream or upward). ...............................................................................192 xvi List of Symbols B: Skempton’s coefficient for pore pressure Be : Elastic bulk modulus CRR: cyclic resistance ratio CRR 15 : cyclic resistance ratio CSR: cyclic stress ratio Dr: relative density D 50 : mean size of soil grains (50% by weight passing) e: void ratio e max : maximum void ratio e min: minimum void ratio G: shear modulus G max : Shear modulus at small strain Ge: elastic shear modulus GP: plastic shear modulus G liq : liquefied soil shear modulus Gp i : initial plastic modulus at η=0 i: hydraulic gradient K o : at-rest soil pressure coefficient K: bulk modulus K: elastic bulk modulus K w : bulk modulus of fluid (water) K mix : bulk modulus of air-fluid mixture Ke G : shear modulus number Ke B : bulk modulus number K α : correction factor for static bias or slope K σ : correction for confining pressure K b: bulk modulus of soil skeleton K m : earthquake magnitude correction factor M: constraint bulk modulus m_hfac1 to m_hfac7: UBCSAND calibration factors Mod1: UBCHYST calibration parameter, - a reduction factor for first-time or virgin loading Mod2:UBCHYST calibration parameter - to account for permanent modulus reduction with large strain (optional) Mod3: UBCHYST calibration parameter - to account for cyclic degradation of modulus with strain or number of cycles, etc. n: porosity n: UBCHYST calibration parameter n 1: UBCHYST calibration parameter (N 1 ) 60 : normalized Standard Penetration Test, N-value for energy and overburden (N 1 ) 60-cs : normalized N-value for clean sand (N 1 ) 60cs-Sr : SPT N value normalized and fines corrected for use in calculating residual shear strength OCR: over-consolidation ratio xvii Pa: atmospheric pressure PI: soil plastic limit PGA: peak horizontal ground acceleration Q c1cs : normalized and fines corrected CPT tip bearing pressure Rf: failure stress ratio R u : ratio of excess pore pressure to initial vertical effective stress R f : UBCHYST calibration parameter constant that truncates hyperbolic curve Sr: saturation S r : post-liquefaction residual strength Sss: steady state strength SPT: Standard Penetration Test t: time TL: thickness of liquefied soil layer beneath barrier layer u: pore pressure ue: excess pore pressure V p : compression wave velocity V s : shear wave velocity t: time $switch: a 0/1 flag indicating whether or not the post liquefaction volumetric strain has reached the desired ε v ∆u: change in pore pressure ∆σ V : decrement of vertical stress α: static bias γp s : plastic shear strains ε: strain ε 1 : axial strain ε a : axial strain εe: elastic strain εp: plastic strain εp v : Plastic volumetric ε v : desired volumetric strain in the liquefied element according to published correlation ε v : volumetric strain η: stress ratio η 1 : change in stress ratio η η 1f : change in stress ratio to reach failure envelope in direction of loading η 1fold : η 1f in previous reversals η 1old : η 1 in previous reversals η f : stress ratio at failure η max : maximum stress ratio (η) at last reversal μ: Poison’s ratio ρ = density σ': effective stress σ' m : mean normal effective stress σ tot : total stress σ V: vertical stress σ' v : vertical effective stress xviii σ v : vertical total stress σ'vo: effective initial vertical stress τ: shear stress τ: shear stress τ res: residual strength τ xy : developed shear stress in horizontal plane τ xy : shear stress on the horizontal (τ xy ) plane ϒ: shear strain ϒ: unit density ϒe: elastic shear strain ϒ max : maximum shear strain during earthquake shaking ϒp: plastic shear strain ϒw: water unit weight Φ: soil friction angle Φcs/ss: quasi/steady state friction angle φ cv : constant volume friction angle φ f : Peak friction angle Ø f : peak friction angle Φ p : peak friction angle Φ pt : phase transformation friction angle Ψ: dilation angle xix Acknowledgements The financial support from the British Columbia Ministry of Transportation - UBC Professional Partnership Program, the Canadian Council of Professional Engineers, Trow Associates Inc., and the National Scientific and Engineering Research Council Strategic Liquefaction Grant No. NSERC 246394 is gratefully acknowledged. The triaxial testing system with bender elements were purchased using a grant from The Canada Foundation for Innovation (CFI) - New Opportunities Infrastructure Funding, Project No. 6869. Assistance from BC Hydro is also much appreciated. Discussion and support from colleagues and friends Ali Amini, Michael Beaty, Jusheng Qian, Ali Khalili, Mavi Sanin, Pascale Rouse, Mahmood Seid Karbasi, David Siu, Alex Sy, Sung Sik Park, (Uthaya) M. Uthayakumar, Somasundaram Sriskandakumar, Sonny Singha, James Wetherill and many others is much appreciated, as is the drafting assistance from Danny Lam, and formatting and editing input from Diane McCulloch and Julie Sedger. Correspondence with Peter Cundall of ITASCA and Ryan Phillips of C-CORE on soil saturation and P-wave correlations and modeling is also greatly appreciated. The author would like to thank the University of British Columbia staff and faculty for their assistance and enlightenment. The author wishes to thank his research committee members Dr. Upul Atukorala and Professor Mahdi Taiebat, and most importantly his supervisors, Professor Peter M. Byrne and Professor Dharma Wijewickreme, for their support and encouragement. xx DEDICATION to Tineke, Daniel, Jasper, Mom, my late Father and Everyone else who made this happen xxi Chapter 1 - Introduction 1 INTRODUCTION Seismic design of major civil structures (bridges, dams and embankments) is moving increasingly toward the use of performance design methodologies. The owner, design code, or other body holding jurisdiction generally specifies performance criteria for the proposed structure. Then it is the designer’s task to demonstrate that the structure will perform in accordance with the design criteria. A key aspect of performance design is to determine the earthquake induced ground and structure movements. To do this, designers are increasingly resorting to the use of dynamic numerical analyses. When the soils or rock involved are dense, these analyses can often be carried out using relatively simple elastic-plastic constitutive models. However, with saturated loose granular soils there is potential for soil liquefaction, related large deformations and possible flow sliding. This behaviour is significantly more complex, and the numerical procedures required reflect this. The research for this dissertation involves development of these numerical design tools and procedures for use in engineering practice for estimating the earthquake induced ground deformations of soil that may potentially liquefy. The developed design methodology uses effective stress and total stress soil constitutive models, UBCSAND and UBCHYST, developed at the University of British Columbia (UBC). There are several assumptions and approximations in these analysis procedures and due to this, empirical soil strengths that have been backcalculated from past earthquake case histories are also used as part of the process. Chapters 2 and 3 of this dissertation give an overview of the behaviour of saturated loose sandy soils, prone to liquefaction during strong earthquake shaking. Chapter 2 discusses liquefaction and pore water void redistribution mechanisms and Chapter 3 focuses on “soil-mixing” as an alternative means for low strengths and flow. Chapter 4 discusses empirical post-liquefaction soil strengths derived from field case histories. Chapter 5 presents an overview of the numerical analyses models while Chapter 6 discusses suggested analysis methodology for use in design work. Related model calibration and analysis examples are given in Chapter 7. Chapter 8 provides a diversion from the main dissertation theme. Some of the work described in Chapter 7 involved centrifuge testing at C-CORE, cyclic simple shear laboratory testing at UBC (Sriskandakumar 2004), and numerical modeling at UBC and Memorial University of Newfoundland (University of British 1 Chapter 1 - Introduction Columbia Liquefaction Initiative research program (NSERC Grant 246394)). Both centrifuge tests and laboratory tests were used for numerical model calibration purposes, and are described in Chapter 7. The response of some of the centrifuge tests was indicative of the centrifuge model not having been fully saturated. This was puzzling as P-wave measurements within the centrifuge model were indicative of full saturation. A series of triaxial tests with P-wave measurement apparatus were carried out at UBC. These tests showed that high P-wave velocities were not always indicative of full saturation (Naesgaard et al. 2007) and provided an explanation of the observed centrifuge response. A summary of this testing program and subsequent conclusions is presented in Chapter 8. An overview, conclusions and recommendations for further research are presented in Chapter 9. 2 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength 2 SOIL LIQUEFACTION, VOID REDISTRIBUTION and POSTLIQUEFACTION RESIDUAL SHEAR STRENGTH 2.1 Overview An overview of the characteristic behaviour of sands when sheared monotonically, cyclically, drained, and undrained has been thoroughly described by Seid Karbasi (2008) and numerous other authors (Idriss and Boulanger 2008; Byrne et al. 2006; Vaid and Chern 1985; Castro 1975 and Casagrande 1936). This will not be reiterated in this dissertation and only an overview of the key aspects related to the proposed modeling is presented. This overview is the author’s interpretation and understanding of key points. Fundamentally, the response of soil is related to the response of particle to particle contacts to internal pore fluid pressures and external boundary stresses. Mathematical and numerical models, such as, PFC3D (ITASCA 2003), that simulate the particulate behaviour are available. These models currently have limited ability but are developing quickly and with advancement in computing power are expected to become important design tools in the near future. Current methodologies, including those discussed in this dissertation, generally treat the soil as a continuum. The intent is that the constitutive model of a single element in the continuum should simulate the soil skeleton – pore fluid interaction with external stresses that are observed in laboratory element tests. When these elements are combined into a group, with suitable compatibility and continuity equations, will simulate the behaviour observed in large physical models and field scenarios. In this chapter, both the behaviour of a single element (drained, undrained, and partially drained response as observed in a simple shear laboratory test) and the behaviour of a soil profile, or in numerical terms a group of elements, are discussed. The purpose of these discussions is to provide a background on behaviour inherent in potentially liquefiable soil deposits. This behaviour should, ideally, be emulated by the proposed numerical model, and understood by the users of the numerical model. Drained Response: When typical loose sand, relative density (D r = 40%), is tested in drained (dry) cyclic simple shear, its stress path is similar to that experienced by a one dimensional soil column during earthquake loading. Initially, the soil is contractive on loading and unloading. However, on loading to large strains, the stress state exceeds a phase 3 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength transformation stress ratio (approximately equal to the tangent of the constant volume friction angle (Φ cv ) in loose sands) and the soil becomes dilative (Figure 2-1) (Ishihara 1993; Vaid et al. 1981). Dense sand (D r = 80%) behaves in a similar manner except the dilative response is more pronounced and phase transformation occurs at lower shear strain (Figure 2-1) and sometimes at lower stress ratio. Both dense and loose sands are contractive on unloading. In most cases, the net result of drained cyclic loading in both dense and loose sands is a reduction in sample volume (Finn et al. 1982). Undrained Response: If the soil pores are filled with water that has been prevented from escaping (undrained condition), then pore pressures will increase when the soil skeleton attempts to contract, and decrease when the soil skeleton attempts to expand (dilate). With repeated cycles the net result is generally an increase in pore pressure and related decrease in effective stress (Vaid and Chern 1985; Sriskandakumar 2004). With a sufficient number of cycles, the stress path may reach the zero effective stress (σ' = 0) and zero shear strength (τ = 0) origin in the τ - σ' stress space, and “true liquefaction” occurs (Figure 2-2). However, with continued shearing following true liquefaction, the soil will usually attempt to dilate and gain strength (Figure 2-2) (Vaid and Chern 1985). At very large strains, the soil will reach a critical or steady state where further shear strain does not induce pore pressure or volume change and further strength gain will not occur (Casagrande 1936; Negussey et al. 1988). The terms critical state and steady state are the same state and are taken to be synonyms in this dissertation. It is postulated that in undrained loading, the post-liquefaction strength of a soil element will not be reached until, (i) the pore water cavitates and, thus, allows the sample to increase in volume (dilate) and reach the critical state, or (ii) the high mean effective stress generated by dilation suppresses the dilation and the soil reaches its critical state strength, or (iii) the sand grains crush and the soil reaches a critical state of the crushed material (Figure 2-3). The strength of the sand reached in (i), (ii) or (iii) is generally much larger than the commonly accepted post-liquefaction “undrained” strengths back-calculated from case histories (Seed 1987; Seed and Harder 1990; Olson and Stark 2002; and Idriss and Boulanger 2008) and is often larger than the drained strength. In this dissertation, these back-calculated post-liquefaction strengths are referred to as the “residual strength” (τ res or S r ) of the liquefied soil. 4 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Response with small inflow (partially drained condition): If, in lieu of undrained loading, a small inflow of water (expansion) occurs, then shear can occur without the strength gain observed in undrained tests. Upon reaching the critical state, further inflow will rapidly cause total loss of strength (Vaid and Eliadorani 1998; Boulanger and Truman 1996; and Sento et al. 2004) (Figure 2-4). This process of void or pore water redistribution was proposed by Whitman (1985) (NCR 1985) and received recent attention within the engineering design community by the shake table tests by Kokusho (1999, 2003). As illustrated in Figure 2-4, with continued water inflow, the shear strength of a soil element will always go to zero (water film condition). “Residual Strength” in the Field: Natural soils and many man-made soils are often layered and have variations in grain-size and permeability. Earthquake shaking and related liquefaction will induce pore water gradients and flow that will result in contraction of the soil skeleton with outflow of pore water in some areas, and expansion with inflow of pore water, in other areas (void redistribution) (NCR 1985; Whitman 1985; Naesgaard et al. 2006). Inflow produces soil strengths considerably lower than those obtained assuming undrained conditions (Vaid and Eliadorani, 1998). When there is a low permeability layer or barrier, the migrating pore water becomes trapped beneath the barrier and forms an area of local expansion with low effective stress (NCR 1985). With sufficient inflow, an actual water interlayer with zero shear strength will develop. The “residual strength” back- calculated from field case histories commonly used for design is not a fundamental soil property nor the strength of a single soil element, but rather the average strength of multiple elements that are deemed to have liquefied. This is discussed further in Section 2.3. Consolidation and other volumetric strain mechanisms within liquefied soil: Volume reduction of the soil skeleton of saturated granular soils subjected to earthquake shaking or other cyclic loading is postulated to occur by several distinct mechanisms. • Mechanism 1 is net volume reduction and/or pore water pressure increase induced by cyclic shear strain and related particle rearrangement (Taylor 1948, Rowe 1962); 5 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength • Mechanism 2 is volume reduction of the soil skeleton due to sedimentation of particles and only occurs at those times in the process when the mean effective stress is essentially zero (Kokusho et al. 2003; Malvick 2005; Dashti 2009); • Mechanism 3 is plastic consolidation of the soil skeleton with mean effective stress greater than zero (this often occurs in combination with Mechanism 4) (Terzaghi and Peck 1967); • Mechanism 4 is elastic consolidation of the soil skeleton due to soil loading (increase in mean effective stress) (Terzaghi and Peck 1967); and, • Mechanism 5 is reduction of the skeleton volume due to mixing of layers of coarse and fine soil (likely more prevalent at low effective stresses) (Byrne 1989, Naesgaard and Byrne 2005). Typical net volume reduction resulting from earthquake shaking and postearthquake consolidation in areas of relatively flat topography (small initial static bias) are in the range of 2-5% for looser sands, and in the range of 0.5% for denser sands (Malvick 2002; Idriss and Boulanger 2008). Observations from physical testing indicates that during cyclic loading much of the particle movement and settlement occurs during the portion of the cycles when the soil is passing through the shear stress/effective stress origin (zero effective stress) and is postulated to take place mainly by Mechanism 2 (Kokusho et al. 2003). When there are significant shear stresses present, the sedimentation (Mechanism 2) does not take place and volume reduction is typically significantly reduced (Malvick 2005). This is illustrated in triaxial test data by Dismuke (2003) (Malvick 2005) and shown in Figure 2-5. Stress Redistribution: When granular soils within an embankment are subjected to earthquake shaking, the shear may induce contraction of the soil skeleton in some areas and corresponding build-up of pore pressure and reduction in effective stress. This reduction in effective stress will not occur uniformly throughout the embankment and will be more prevalent in some areas than others. When the effective stress decreases within a local zone or pocket, the shear modulus of the soil in that zone or pocket will also decrease, and shear stresses that were previously supported by this soil will be shed to adjacent stiffer zones (Naesgaard et al. 2006). Thus, even though the overall static bias within an embankment slope may be significant, there will still be local zones or pockets of liquefied soil with 6 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength limited, or no static bias. With reduced shear stresses, the soil in these pockets will be more contractive, effective stresses lower, and the ability of that zone to resist shear stresses will be further reduced. 2.2 Loose Sand Layer underlying Low Permeability Barrier In this Section, the postulated behaviour of a one-dimensional column of loose (D r = 40%) sand underlying an impermeable barrier layer is discussed. It is assumed that the initial shaking is sufficient to produce true liquefaction (zero effective stress) throughout the soil underlying the barrier for the level ground Case A. For Cases B and C with a static bias, it is assumed that the shaking is sufficient to reduce the effective stress to the minimum value required to support the shear stress associated with the static bias. Also, for Cases B and C, it is assumed that the static bias is lower than the phase transformation friction angle so that shaking produces a net increase in pore pressure. When referring to shear stress (τ) in this Section it is the shear stress on a plane parallel to the ground surface and not necessarily the maximum shear stress (τ max ). Effective stress is the mean normal effective stress (σ' m ) unless indicated otherwise. Case A – Level ground (no static bias): At the on-set of earthquake shaking, the full depth of the loose sand layer attempts to contract, pore pressures increase and effective stresses decrease. During the shaking cycles, there may be alternating shear induced contraction and dilation, and also some expansion of the sand skeleton due to a reduction in effective stress. However, the net result is contraction of the skeleton and an increase in pore pressure throughout the column. In a short time, a pore pressure gradient develops with the highest head at the base of the column and a lower head at the top. This causes pore water to flow upward relative to the soil particles. A soil element at the base of the column has a net outflow of pore water and consolidates relatively quickly, whereas, soil elements between the top and bottom of the column experience little volume change, as water flowing into and out of each element is similar. At the top of the sand column, directly under the impermeable barrier, water flows out. It cannot pass through the impermeable barrier, and a water interlayer develops. If the barrier is totally impermeable, the water incompressible, and the sand layer is totally liquefied (i.e., effective stress is zero), then a water interlayer develops almost immediately. With time, outflow from the bottom soil element diminishes, and the 7 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength next higher element (above the bottom element) consolidates because the outflow now exceeds the inflow. This process advances as a consolidation front, moving from the bottom upward. During the consolidation process, the size of the water layer under the barrier increases. It should be noted that the shear strength of a water interlayer is zero. In addition to settlement due to sedimentation, there will be plastic and elastic settlement due to the restoration of effective stress. The restoration of effective stress does two things, (1) it causes greater consolidation at the bottom layers due to the change in effective stress being greater with depth, and (2) it causes the flow gradient to decrease more in the bottom parts of the layer than in the top parts. The net result of (1) and (2) is greater contraction at the base of the layer and the potential for some expansion in the upper soil layers that are below the barrier water film as proposed by Seid-Karbasi (2008). For very loose sands, it is postulated that sedimentation-type consolidation dominates and that settlement due to the restoration of effective stress is small as the bulk modulus of sands increases rapidly with effective stress. For example, assuming an effective stress change from 0 to 100 kPa and a constrained modulus of 500·Pa·(σ' m / Pa)0.5 (where Pa = 100 kPa) then by integration a volumetric strain due to elastic settlement will be = 0.004 or 0.4%. Typical post-liquefaction settlement from both field case histories and laboratory tests is about 2% to 5% (Idriss and Boulanger 2008) and is about 10 times higher than the elastic portion. In summary, the net result of earthquake shaking on loose sand, which underlies an impermeable barrier, in level ground conditions (no static bias), will be contraction throughout the height of the deposit. A water interlayer will form at the top of the sand column. The thickness of this water interlayer will be equal to the net consolidation in the underlying loose sand and would, typically, range from 2 to 5%, depending on density. This is illustrated by the post-impact formation of localized water films in columnar tube tests by Kokusho (Kokusho and Kabasawa 2003) (Figure 2-6). The water film illustrated in Figure 2-6 occurred almost immediately after the base of the column was struck. Note that if the overlying fine sand in Figure 2-6 was removed, the remaining coarse sand would still settle in a similar manner when struck, but the water would pond on the surface, rather than form an interlayer. 8 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength The inherent challenges of modeling the above process numerically are discussed in Chapter 5. For denser sands, the skeleton volume reduction due to sedimentation would be lower and settlement due to restoration of effective stress would be a larger portion of the total settlement. Case B – Sloping ground with stress path during shaking that pass through the τ - σ' m origin: At the on-set of earthquake shaking, the full depth of the loose sand soil column attempts to contract, pore pressures increase, and effective stresses decrease. (i) Assuming the sand is dense of the critical state (a common situation even for loose sands (Vaid and Thomas 1995)), then the effective stress does not reduce to zero except during cross-over when the stress path passes through the τ - σ' m origin. (ii) Sedimentation settlement (Mechanism 2) only occurs during shaking and only during cross-over (when σ' m is zero) – the amount of settlement that occurs during shaking is largely a function of permeability. The sand will have essentially zero bulk and shear modulus during the cross-over and a large cyclic mobility and movement in direction of static bias may occur during shaking. (iii) Post-shaking vertical or mean effective stress will be limited by the value required to support the static bias and will not go to zero (unless there is sufficient pore water inflow to take the sand to the critical state (see (v) below) (Boulanger and Truman 1996). (iv) Much of the post-shaking settlement will be due to reinstatement of effective stress (Mechanisms 3 and 4) and will be significantly less than that which would occur in the absence of static bias (as indicated in Figure 2-5 by Path C1-C3 with static bias compared to path W0-W3 without bias). (v) With time, the pore pressure gradient will reduce with depth, hence, inflow may exceed outflow in the upper sections. The extreme upper boundary of the sand unit has no outflow due to the impermeable barrier, and, therefore, expands more than the other zones. Expansion reduces shear strength (both because the effective stresses drop, and because the friction angle is decreasing from a peak value toward a critical state value), and shear modulus which leads to further localization. With sufficient inflow, the upper element reaches the critical state and further inflow 9 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength causes flow failure. Failure will occur just before a water film forms due to the static bias (Malvick 2005). Case C – Sloping ground with stress path during shaking not passing through the τ σ' m origin: At the on-set of earthquake shaking, the full depth of the loose sand soil column attempts to contract, pore pressures increase and effective stresses decrease; however, effective stresses do not go below that required to support the static bias (Boulanger and Truman 1996, Malvick 2005, Kulasingam 2003). (i) Assuming the sand is dense of the critical state (a normal condition), then initially the effective stress does not reach zero and is sufficient to support the shear stress in the soil. (ii) Decrease in effective stress leads to some cyclic mobility, especially the case if the stress path intersects the failure plane, however, the deformations will often be less than that for Case B, where the cycles pass through the stress path origin. (iii) A pore pressure gradient will still form within the column. Lower parts of the column will experience a net loss of water (as indicated by Path C in Figure 2-7) and maintain or regain strength whereas soil in the upper part of the column under the barrier will gain the water expelled from the lower parts and expand (as indicated by Path A in Figure 2-7). As indicated by Path A in Figure 2-7, once inflow is sufficient to take the sample to the critical state then shear localization occurs and further inflow leads to flow failure. (iv) Sedimentation settlement (Mechanism 2) will be essentially zero and post-shaking settlements will be due to reinstatement of effective stress only. Post-shaking settlements will be significantly less than those which would occur for Case A with no static bias and less than that which would occur for Case B where there are crossovers. 2.3 Pore Water Void Redistribution within real Embankments and Slopes In real soils, the barrier boundary will not be perfectly plane, of infinite lateral extent, nor will the localization be infinitely thin, nor the barrier totally impermeable. In a real world scenario, there will be undulations, varying normal stress, finite grain-sizes, etc. (Figure 2-8) (Naesgaard et al. 2005, 2006). These items will result in the ‘residual’ shear strength along 10 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength the interface varying both in time and space and will usually result in an average value that is greater than zero. Some key factors postulated to affect the shear strength and behaviour of the barrier interface include: (i) Soil density - Dense soil (at a given confining pressure) has greater affinity for expansion and can accommodate more inflow before reaching the critical state, than does a loose soil (at the same confining pressure) (Seid-Karbasi 2008). (ii) Net volume of inflow - The available pore water inflow will be a function of the volume and relative density of the loose layer feeding the inflow, permeability of the loose layer, pore water flow path, and, earthquake shaking amplitude and duration. Dense soil will contract less and, therefore, have less pore water to contribute to the expansion beneath the barrier than a loose soil. (iii) Grain-size - The larger the grains the greater the thickness of the shear band and the amount of inflow required for steady state shear to be achieved in the shear band (Figure 2-8(a)). Roscoe (1970) suggests that localized shear bands are in the order of 10 to 20 times the particle mean diameter (D 50 ). (iv) Permeability and continuity of the barrier - Leakage through the barrier will reduce the net inflow into the interface (Figure 2-8(b)). (v) Waviness or undulations of barrier interface - These may be undulations that precede earthquake shaking and/or may be due to deformations induced by strong earthquake shaking during the event (Figure 2-8(c)). (vi) Variations in vertical stress - Variations in vertical total stress along the barrier boundary will allow the higher stressed sections (Figure 2-8(d) ‘X’) to maintain some strength while the lower stressed sections (Figure2-8(d) ‘Y’) heave. (vii) Shear stress redistribution will occur and instead of there being an area with uniform static shear stress bias, stronger points will attract stress, and other areas will have low stress with possibly no resulting static bias. The areas with no static bias may have essentially zero shear strength due to water film development and possibly have significant sedimentation settlement, whereas, zones attracting shear will have finite residual shear strengths and significantly less consolidation settlement. 11 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength The above conditions will result in the “residual” shear strength along the interface varying in both time and space, resulting in an average value that is greater than zero. This ‘average value’ is the “residual” shear strength that is believed to be reflected in the low values back-calculated from case histories. When inflow is into a zone with a static shear bias a reduction in effective stress will initially occur with nominal shear strain until the zone hits the failure envelope (in stress path space). Once on the failure envelope a large shear strain that is a function of inflow volume and dilation angle will occur (Boulanger and Truman 1996, Sento et al. 2004) while the shear stress is maintained at the value of the static bias. With continued inflow, the layer will eventually reach the critical state at a shear stress corresponding to the static bias, and any further inflow will lead to strength loss and flow slide. This is illustrated in Figure 2-7. In the one dimensional infinite slope model previously discussed in Section 2.2, the net pore water migration is essentially vertical. However, in embankments and water-edge slopes both vertical and/or lateral flow may be responsible for pore water void redistribution effects (NRC 1985). Figure 2-9 illustrates pore water void redistribution concepts within a hypothetical embankment of loose sand with a single low permeability barrier layer and a high permeability layer. Only flow failure in the upstream direction is being considered. Initially, liquefaction will tend to be more prevalent and trigger earlier near the centre of the embankment where the static bias is lower. The on-set of liquefaction (shaded area around ‘A’ and ‘B’ in Figure 2-9(a)) initiates a complex series of reactions. The author postulates these to include: (1) The liquefied zone will soften and shed load toward the edges of the embankment, thus reducing static bias within the liquefied zone and increasing the shear stress on the outer zones. (2) The deeper portions of the liquefied soil (‘A’ in Figure 2-9(a)) will have a higher hydraulic head than that of the overlying soil (‘B’ in Figure 2-9(a)). Both ‘A’ and ‘B’ will have a higher hydraulic head than that within the shells of the embankment (‘C’ in Figure 2-8(a)). This will initiate upward and outward pore water flow. The outer zones of the embankment are often dilative will also attempt to draw water in from the reservoir. 12 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength (3) The upward rising pore water will become trapped under the low permeability barrier at ‘B’ in Fig 2-9, cause the soil in the vicinity of the barrier to expand, and form a localized weak layer and possible water film at the underside of the barrier. (4) The expansion of the soil under the barrier will cause an increase in permeability at the underside of the barrier which will facilitate a progressive lateral migration of water into the highly stressed outer zones of the embankment. (5) Pore water migrating into the outer embankment zones (both from the central portions of the dam and the reservoir) will reduce effective confining stresses and potentially bring the soil in the shell to failure. Once on the failure envelope a shear strain approximately proportional to the inflow induced volumetric expansion will ensue. This will continue until the inflow subsides or the localized layers expand to the critical state (Sento et al. 2004). Once at the critical state, further inflow causes loss of all strength and possibly flow failure. (6) Lenses of high permeability will also facilitate the migration of pore water from the higher head central portions of the embankment toward the lower head and highly stressed shells. The effects of this are as discussed in (5). (7) The on-set of liquefaction generally occurs during strong earthquake shaking. However, pore water void redistribution takes time and will occur both during and after earthquake shaking. The process may lead to the triggering of a flow slide (during or after end-of-shaking) or cause limited movements that stabilize due to geometry changes or because the source of water driving the expansion is no longer present. (8) Higher volumetric strains and shear localization will occur immediately below barriers where there is pore water inflow. With localization the inflow (volume of water) required for the soils to go to the critical state and to initiate flow failure is reduced. (9) In embankments and water-edge slopes, liquefaction within the centre portion of embankments or back from a water-edge slope will induce high pore pressures approaching the local overburden stress. This initial liquefaction results in (1) the 13 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength liquefied zones shedding shear stress to adjacent non-liquefied zones, and (2) the generation of local high pore pressures and pore pressure gradients. The redistribution of shear stress reduces the static bias within the liquefied soils allowing effective stresses to go essentially to zero and allowing greater potential for volume change. The high pressure water migrates both vertically and laterally towards the edges of the embankment, or towards the water-edge slope (location of lower overburden pressure), whereby expansion and potential for strength reduction will also occur. Since water flow and pressure redistribution takes time, there is often a delay between occurrence of earthquake shaking and time of flow failure. There are numerous case histories where soil liquefaction occurred during earthquake shaking, but related flow failure did not occur until sometime after the shaking terminated. Documented examples include the Lower San Fernando Dam, California, where the upstream face of the dam failed 20 to 30 seconds after the end of earthquake shaking (Seed 1973; Seed 1987), and Niigata, Japan, where eyewitnesses reported that the girders of the Showa Ohashi Bridge fell a few minutes after the earthquake motion had ceased (Hamada 1992). 2.4 Chapter 2 Summary and Conclusions In this Chapter, an overview of items contributing to liquefaction and potential flow sliding are given. These are behaviours that are important for the proposed numerical model to capture and important for the proposed users of the model to understand. It is noted that sands have contractive tendencies when sheared at stress ratio less than the phase transformation stress ratio, have dilative tendencies sheared at higher stress ratio, but always contractive tendencies on unloading. The net response of cyclic shear loading of saturated sands is nearly always a tendency for contraction. This in turn generates pore water pressures, potential liquefaction of the sands (low effective stress and very low shear moduli), and sets up a hydraulic gradient within the deposit. The gradients cause water flow (generally upward or toward a free face) out of some zones and into others. Those zones losing water generally contract and those gaining pore water expand. When there is a low permeability barrier above the sand layer the inflow of pore water into the soil directly below the barrier results in localization of both the expansion and shear strain. As the sand 14 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength expands, its potential shear-induced-dilative-tendencies decrease, and with sufficient expansion, the critical state, at which shear strain will induce no further expansion, is reached. This contraction and expansion can have a dramatic affect on both the effective stress and the shear strength of the sand. This is particularly the case for loose sand under low permeability barriers where expansion tends to be localized and strengths can go to very low values, or at the limit, zero when a water interlayer forms. Part of the behaviour described in this Chapter concerns that of a soil element (i.e., drained and undrained response) and the other part is related to the interaction between zones or groups of elements in numerical terms (i.e., water flow, void and stress redistribution). This distinction is noted when modelling the behaviour numerically using coupled effective stress methods. The soil constitutive model captures the effective stress-strain element behaviour of the soil skeleton, while a suitable computational process captures the coupling of volume changes within the skeleton with pore water pressure, pore water flow, and stress strain interaction of the whole model (Byrne et al. 2006; Taiebat et. al. 2010). Modelling of the above behaviour using the constitutive model UBCSAND (Beaty and Byrne 1998; Byrne et al. 2004) within the commercially available program FLAC (ITASCA 2008) is further discussed in Chapters 5 to 7. Most of the concepts in this Chapter have been discussed by others at some time in the past. However, the framework and emphasis of the discussion is important and is that of the author. 15 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength FIGURES Chapter 2 Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength 16 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-1 Comparison of cyclic drained simple shear response of loose and dense Fraser River sand (from Sriskandakumar 2004). (a) is shear stress versus shear strain, (b) is shear strain versus volumetric strain, and (c) is shear stress versus volumetric strain. (Note labels in (c) refer to the loading phase – the soil is always contractive on unloading). 17 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-2 Cyclic (a) stress path and (b) stress-strain response of loose sand without initial static shear stress (from Sriskandakumar 2004). 18 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-3 Postulated response of loose sand to undrained cyclic simple shear loading followed by monotonic loading to large strain. (a) is stress strain space, (b) is stress path space and (c) is void ratio vertical effective confining stress space. “A” is initial state with pore pressure ratio, Ru=0, “D” is true liquefaction with Ru = 1, and “F” is the strength at the critical or steady state if pore water does not cavitate. 19 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-4 Postulated response of loose sand element to cyclic simple shear with “A” to “B” undrained and “B” to “D” with slight pore water inflow. (a) is stress strain space, (b) is stress path space and (c) is void ratio vertical effective confining stress space. “A” is initial state with pore pressure ratio, Ru=0, “D” is true liquefaction with Ru = 1. At “D” the sample is at the critical state with strength at or near zero. 20 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-5 Re-consolidation volumetric strain from dissipation of excess pore pressure following liquefaction. α = 0 is with zero static bias or level ground conditions whereas α = τstatic/σ'vo = 0.1 is with a static bias such as that from sloped ground (from triaxial data by Dismuke 2003 as reported by Malvick 2005). Figure 2-6 Hammer blow to base of 130mm diameter column of loose sand liquefies it and causes almost immediate formation of water film at boundary with less permeable fine sand (Kokusho and Kabasawa 2004). 21 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-7 Path A is the stress path for a soil element with a static bias and with pore water inflow that is cyclically loaded until flow instability occurs. Path C would be an element with there is pore water outflow and flow instability does not occur (Malvick 2005 – after Kulasingam 2003). 22 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-8 Factors affecting thickness and residual strength of localized shear zone under barrier (a) grain-size (b) leakage through barrier (c) undulations of barrier surface (d) variation of stress on barrier (from Naesgaard et al. 2006). 23 Chapter 2 – Soil Liquefaction, Void Redistribution and Post-Liquefaction Residual Shear Strength Figure 2-9 Flow redistribution mechanisms within an embankment with low permeability barrier and high permeable layer. Vectors show directions of pore water flow (Naesgaard et al. 2006). 24 Chapter 3 – Soil Mixing 3 SOIL MIXING 3.1 Introduction to Mixing Phenomena Another procedure that may result in high void ratios, excess pore water and liquefaction is soil mixing (Naesgaard and Byrne 2005; Vasquez-Herrera and Dobry 1989). Natural waterpluviated soils are often layered and when layers of coarse and fine grained soils are mixed together in an undrained or near-undrained environment then the resulting post-liquefaction shear strength of the mixed soil is much lower than the shear strength of the individual layers (Figure 3-1 A, B, and C). There is also fourth Case D, that is not illustrated in Figure 3-1, which results in the larger particles moving to the top during cyclic shearing (personal communication from Professor D. Chan, University of Alberta). During the soil liquefaction and post-liquefaction shear process there are numerous occasions during which extensive turbulence and mixing of particles occurs. This has the potential for creating zones with post-liquefaction strengths well below those that would be obtained from undrained laboratory tests on the individual non-mixed layers, and is a mechanism for achieving the strength loss required for flow liquefaction. Mixing may act independent of, and/or together with, pore water redistribution phenomena. Mixing may be initiated by several mechanisms including: • Pre-liquefaction mechanical mixing of fine and coarse layers such as induced by shearing during slope failure; • Cyclic mixing where cyclic shearing induces the fine particles to fall into the voids of an adjacent coarse layer; and, • Post-liquefaction mixing, including: - mechanical mixing due to shear failure of liquefied soils; - mixing due to pore water turbulence and failure at interfaces between coarser sand and overlying finer non-plastic silt barrier layers. This may happen upon the initiation of water interlayers or may happen when a water bubble breaks through a confining silt layer (such as that which occurs during the formation of sand boils). 25 Chapter 3 – Soil Mixing The ease with which cyclic mixing occurs is postulated to be a function of the relative particle size of adjacent coarse and fine layers. If the voids in the coarse layer can contain the particles of the fine layer, then upon being disturbed (by landslide or earthquake shaking) may migrate into the voids of the coarse particles as illustrated in Figure 3-1 B and C. Whereas, if the voids in the coarse layer do not freely contain the particles from the fine layer then considerably greater turbulence and energy will be required for the soils to mix. Mixing can produce potential net volumetric strains significantly larger than those which would occur due to shaking of the individual homogeneous layers. This can result in either the formation of a water film (Figure 3-1B) or in a mixed soil with near zero residual shear strength (Figure 3-1C). In the field, mixing has been observed resulting from static slope failure in the Mufulira Mine tailings pond (Naesgaard and Byrne 2005). Mixing has also been observed in videos from shake table tests with non-plastic silt barriers by Kokusho (2003) and in simplistic plastic tube shear tests (Naesgaard and Byrne 2005). 3.2 Mufulira Mine Case History (mixing due to static slope failure) Byrne and Beaty (1997); Barrett and Byrne (1988); and Byrne (1991), postulated soil mixing as the reason for the flow of tailings into the Mufulira Mine in Zambia. At this site, a tailings pond overlaid underground mine workings as shown schematically in Figure 3-2. In 1968, the underground hanging walls of the mine collapsed and sink holes with steep side slopes formed in the tailings pond resulting in two cases of minor inflow of tailings into the underground workings. Following that, in 1970, a large sink hole formed and failed back along a very shallow angle of 10 degrees (Figures 3-2 and 3-3) to develop a crest diameter of over 250m. One million tonnes of tailings flowed into the underground mine workings, killing 89 people (tailings.info 2005). From studies in 1988/89 by Barrett and Byrne (1988), Byrne (1989), and Horsfield and Been (1989), it was noted that the tailings within the overlying pond were comprised of alternating layers of sandy silt and coarser silty sand. Whereas, the tailings within the core of the sink holes and those that had flowed into the underground mine workings (residual in-rush or mixed tailings), contained the same materials, but they were highly mixed and homogeneous (Figure 3-4). The silty sand made up about 65% of the total deposit, and sandy silt the remaining 35%. The layers were clearly 26 Chapter 3 – Soil Mixing visible with thickness averaging 100 to 150mm but varying from less than 50mm to 250mm. Water contents were in the range of 21 to 34% with an average of 25% and average in-situ void ratio was about 0.65 to 0.7. An extensive series of undrained triaxial compression and extension tests on the soil of the individual layers yielded high residual undrained strengths at the in-situ void ratio and could not account for the liquefaction observed at the site. However, when the fine and coarse materials were mixed at the same average in-situ void ratio corresponding to an “average” undrained state, the resulting soil had essentially zero strength (Figures 3-5 and 3-6). It is postulated that the sides of the 1970 sinkhole were initially steep and then failed progressively, as illustrated in Figure 3-7. As the soil on the initially steep-sided sinkhole failed, the silt and sand layers mixed, leading to a near complete loss of strength that allowed the soils to flow as liquefied slurry into the underground mine workings. 3.3 Cyclic Mixing in Rudimentary Laboratory Shear Tests Rudimentary cyclic shear tests have been conducted in order to demonstrate whether an earthquake or other cyclic shear loading may also induce mixing in susceptible layered soils (Naesgaard and Byrne 2005). A clear vinyl tube of 12mm inside diameter with plugged ends, as illustrated in Figure 3-8, was used for the tests. The samples were prepared by removing the plug from one end, filling the tube with water, and pluviating the sand through the water. Once a layer of sufficient thickness was achieved, the surface of the sand was lightly tapped with a rod to render the surface level, then another layer was placed in a similar manner, if required, or the plug was placed and clamped. The sample was then manually sheared by grabbing the top and bottom of the tube and pushing it back and forth while keeping the top and bottom of the sample parallel. Sample heights ranged from 27 to 40mm. Strain on typical cycles was estimated to be in the range of 10% to 20%. Table 3-1 summarizes the configurations and results of six typical tests. The grain-size distributions of the sands and glass beads used are shown in Figure 3-9. Migration of the fine sand particles into the voids of the underlying coarser material was readily evident during Test “A”. The thickness of the water layer that formed at the top of the sample (Figure 3-10) stabilized with approximately 20 shear cycles. When the same test as “A” was carried out with homogeneous fine sand only or coarse sand only (Tests “B” and “C”), the induced 27 Chapter 3 – Soil Mixing volumetric strain was appreciably smaller (Figure 3-11). Tests “D” (Figure 3-12) and “E” show that mixing becomes substantially more difficult as the size difference between the coarse and fine layer decreases. Test “F” showed that cyclic mixing does not readily occur without gravity or some other mechanism to move the fine materials into the coarser layer. 3.4 Discussion In general, the susceptibility to mixing induced strength loss is deemed to be dependent on particle and layer properties (particle shape, grain-size gradation of individual layers, layer thickness), in-situ effective stresses, the disturbing forces (hydraulic gradient, gravity, number and magnitude of shear strain cycles), orientation of layers relative to the disturbing forces, saturation, and drainage constraints (permeability, presence of low permeability barrier and rate of disturbance). Several conditions must be met before mixing, and the resulting dramatic reduction in shear strength, will occur. (1) Alternating layers of coarse and fine grained soil must be present. Ideally, the layer thicknesses should be such that the available volume of fine grained soils will fit within voids of the coarse grained layers. (2) A disturbing force must be present to initiate the mixing. This could be a high hydraulic gradient with associated low effective stress and turbulent flow, gravity induced shear (slope failure), cyclic loading (from an earthquake, wind, waves or machine vibrations), or a combination of these conditions. (3) A substantial volume of water must be present in the voids and this water should not be able to escape quickly so that the mixing results in low inter-particle stresses (low effective stresses) and liquefaction, rather than drainage and consolidation. (4) Fine particles must be able to migrate into the voids of the coarser layer. Ideally, the fine particles should fit between the voids of the coarse layer. The ease with which mixing occurs is proposed to be related to criteria similar to the ((D 15 ) filter /(D 85 ) soil ) < 4 to 5 criteria used to prevent internal erosion when designing granular filters. Preliminary conclusions from the rudimentary cyclic shear tests are that if the ratio ((D 15 ) coarse layer / (D 85 ) fine layer ) > 3 to 4, then cyclic mixing may occur, 28 Chapter 3 – Soil Mixing whereas, if the ratio is less than this, the soils will not mix easily. Cohesive and/or cemented soils also would not mix as readily. In the rudimentary cyclic shear tests, mixing progresses with each loading cycle until the sample is totally mixed. The cyclic shearing action dislodges the fine particles from the soil skeleton in the vicinity of the contact with the coarse layer, and then gravity carries the fine particles into the voids of the coarse layer. When the coarse layer is above the fine layer (Test “F”), the shearing action still dislodges the fine particles from the soil skeleton near the interface with the coarse layer. However, in this case, there is no mechanism to move the loosened fine particles up into the coarse layer and, therefore, mixing either does not occur, or is minimal. It should be noted that the rudimentary shear tests carried out as part of this research were intended to provide qualitative comparisons only. The tubes were of relatively small diameter and wall friction and distortion of the walls would affect the results. The amplitude of the induced shear strain per cycle was also highly variable and relatively large (est. 10-20%). The conclusion from these tests was that cyclic shearing could induce mixing only when specific grain size criteria of the layers were met. For normal earthquake shaking and layering pre-liquefaction cyclic shear induced mixing is probably minimal. Further higher quality laboratory work is warranted to better define the conditions under which cyclic shear induces mixing. In other situations, the fine particles may be moved by a hydraulic gradient and pore water flow, rather than by gravity. With a sufficient flow, mixing could occur between vertical boundaries (such as, the core of a dam), with fine particles moving into a coarse layer. It is postulated that mixing will occur with greater ease when effective stresses are low, as is the case when soil has liquefied. Mixing due to a hydraulic gradient and related turbulent pore water flow generated by soil liquefaction can be observed at some of the boundaries between coarse and fine sand and between sand and non-plastic silt in the movies from the shake table tests by Kokusho (1999, 2003) (Figure 3-13). Baziar and Dobry (1995), Amini and Qi (2000), have conducted cyclic triaxial tests on layered silt-sand samples without having reported any significant mixing effects during the tests. In both cases, the layered samples behaved similar to that of homogeneous samples of the individual layers. The reason for mixing not having occurred in these cases may be 29 Chapter 3 – Soil Mixing due to the grain-size criteria (((D 15 ) coarse layer / (D 85 ) fine layer ) > 3 to 4) not being met, or possibly to the presence of some plasticity, or, possibly, to the triaxial loading, and related stress regime, not being as conducive to mixing as was the case in the elementary shear tests carried out as part of the author’s research. Remoulding (mixing) was postulated by Baziar and Dobry (1995) and Byrne and Beaty (1997), as a possible reason for the low “residual strength” observed at the end of flow failure of the Lower San Fernando Dam. Triaxial tests by Baziar and Dobry (1995) (Figure 3-14) showed that the residual strength of the mixed soil was 4 times less than the residual strength of the layered soil with the same void ratio. Mixing can occur independently, or together with, the pore water migration void redistribution mechanism. Both mechanisms may produce water interlayers and/or very low soil strength zones. If mixing and pore water redistribution occur together, then pronounced strength loss and near zero strengths can be expected. Mixing may result in a water interlayer as observed in the rudimentary cyclic shear tests and illustrated in Figure 3-1 ‘B’, or just in an extremely loose liquefied soil as illustrated in Figure 3-1 ‘C’, in the Mufulira Mine case history, and in the liquefied zones of the Lower San Fernando Dam. 3.5 Chapter 3 Summary and Conclusions Many natural soils and man-made hydraulically placed soils are layered. When the fine and coarse layers mix, the resulting “residual” shear strength is much lower than that of the individual layers at the same void ratio. This mixing provides explanation for the flow liquefaction observed at the Mufulira Mine site and, at least partially, accounts for the low end-of-failure residual shear strength back-calculated for the Lower San Fernando Dam failure. Mixing is not a new concept, but is a mechanism that can both result in liquefaction and lower the undrained strength of liquefied soils. The concept of cyclic mixing was proposed by the author and rudimentary cyclic shear tests were developed by him to review this concept. The rudimentary cyclic shear tests were carried out within a 12mm internal diameter vinyl tube to investigate cyclic shear induced soil mixing. The ratio of the grain-size of the coarse and fine layers is shown to be a key factor in a layered soil’s susceptibility to mixing. A preliminary postulation by the author is that cohesionless soils may be susceptible to cyclic mixing when ((D 15 ) coarse layer / (D 85 ) fine layer ) > 3 to 4. It is also postulated that mixing will 30 Chapter 3 – Soil Mixing more readily occur when effective stresses and related inter-particle contact forces are low. Mixing is probably more common as a post-liquefaction mechanism and helps explain the low post-liquefaction strengths and large run-out distances observed in some instances. Within the low effective stress – low strength environment both mechanical shear and turbulent flow can induce mixing. The occurrence of mixing induced by turbulent flow is a concept postulated by the author following examination of the shake table test videos by Kokusho (http//:www.civil.chuo-u.ac.jp/lab/doshitu/index.html). 31 Chapter 3 – Soil Mixing TABLES Chapter 3 Soil Mixing 32 Chapter 3 – Soil Mixing Table 3-1 Summary of Rudimentary Cyclic Shear Tests Test Sample Description (prior to shearing)(a) A Post-shearing (D 15 ) coarse layer (c) (D 85 ) fine layer Remarks 17 1.7/0.5 = 3.4 Fine and coarse sand mixed in approximately 20 shear cycles (Figure 10). 0.5 1.4 NA Sample given 50 shear cycles (Figure 11). 0.5 1.4 NA Sample given 50 shear cycles (Figure 11). 0.8/0.5 = 1.3 No water layer or mixing with 20 cycles, but sample mixed when given over 100 shear cycles and light tapping of specimen (Figure 12). Water Film thickness (mm) Volumetric Strain (b) (%) 13mm fine sand over 20mm coarse sand 5.5 B 36 mm fine sand C 36mm coarse sand D 15mm glass beads over 25mm coarse sand E 10mm fine sand over 20mm medium coarse (mc) sand 1 to 4 3 to 13 1.0/0.5 = 2 Minimal mixing with 20 cycles, but sample mixed when given over 100 shear cycles and light tapping of specimen. F 24mm coarse sand over 12mm fine sand(d) 1.0 2.8 1.7/0.5 = 3.4 Minimal mixing with over 100 shear cycles. 0 to 3.5 0 to 9 (a) Dimensions given are thickness of layers. Sand gradations are shown in Figure 3-9. (b) Volumetric strain = water film thickness/total thickness x 100 (c) D 15 = particle diameter with 15% of particles smaller; D 85 = particle diameter with 85% of particles smaller (d) Note: In this case, the coarse sand is above rather than below the fine sand. 33 Chapter 3 – Soil Mixing FIGURES Chapter 3 Soil Mixing 34 Chapter 3 – Soil Mixing Figure 3-1 Graphic illustrating the effects of soil mixing. Elements 'A', 'B' and 'C' have the same number and volume of large and small particles. ‘A’ is the element with a fine layer over a coarse layer prior to disturbance. 'B' represents the case where fine particles fall into the voids between large particles and leave a water filled void or interlayer at the top of the element. ‘C’ represents the case where the particles mix evenly and the larger particles move into a looser arrangement. Both ‘B’ and ‘C’ will have significantly lower shear strength then ‘A’ if mixing does not occur (Naesgaard and Byrne 2005). Figure 3-2 1970’s Mufulira Mine tailings liquefaction failure. 35 Chapter 3 – Soil Mixing Figure 3-3 Photograph of 1970 Mufulira Mine sink hole (tailings.info 2005). Figure 3-4 Grain-size of Mufulira Mine tailings (Barrett and Byrne 1988; Horsfield and Been 1989). 36 Chapter 3 – Soil Mixing Figure 3-5 Void ratio vs. undrained residual shear strength (τ res = (σ 1 -σ 3 )/2) relationship for sandy silt, silty sand, and mixed residual in-rush tailings. Note: Mixed tailings have essentially zero shear strength at the in-situ void ratio (Naesgaard and Byrne 2005 from triaxial tests by Horsfield and Been 1989). Figure 3-6 Conceptual undrained triaxial compression stress path for silty sand, sandy silt and mixed silt-sand at same void ratio and initial stress state (Naesgaard and Byrne 2005). 37 Chapter 3 – Soil Mixing Figure 3-7 Postulated progressive failure mechanism of 1970 sinkhole at Mufulira Mine. (1) Sides of initially step sided sinkhole fails, (2) as soil slides to bottom of slope, the layers mix and due to this mixing lose nearly all strength (liquefy), (3) the liquefied soil flows away into mine workings, (4) mechanism is repeated giving a progressive failure mechanism, and (5) the final stable slope at shallow 10° angle (Naesgaard and Byrne 2005). Figure 3-8 Rudimentary cyclic shear test with saturated layered sample (Naesgaard and Byrne 2005). 38 Chapter 3 – Soil Mixing Figure 3-9 Grain-size of fine and coarse sand used in rudimentary simple shear tests (Naesgaard and Byrne 2005). Figure 3-10 Test “A” before and after cyclic shearing (Naesgaard and Byrne 2005). 39 Chapter 3 – Soil Mixing Figure 3-11 Post-shearing for non-layered Tests “B” and “C” (Naesgaard and Byrne 2005). Figure 3-12 Before and after cyclic shearing for Test “D” with glass beads and coarse sand (Naesgaard and Byrne 2005). 40 Chapter 3 – Soil Mixing Figure 3-13 Slide from Kokusho cylinder shake table test video showing zone of mixing between coarse and fine sand caused by pore fluid flow and turbulence (Kokusho 2003) (Video from: http//:www.civil.chuo-u.ac.jp/lab/doshitu/index.html). 41 Chapter 3 – Soil Mixing Figure 3-14 Steady state void ratio vs. residual strength (S us ) for reconstituted (remolded) Lower San Fernando Dam (California) soil. Note that the residual strength (S us ) for the combined (mixed) homogeneous specimens is much lower than that of the layered notmixed specimens (Baziar and Dobry 1995). (Without mixing the strength is about 0.31 tsf whereas with mixing the strength is about 0.06 tsf). 42 Chapter 4 – Residual Strength from Back Analyses of Case Histories 4 RESIDUAL STRENGTH FROM BACK-ANALYSES OF CASE HISTORIES 4.1 Introduction Field experience from previous earthquakes indicates that residual strengths can be much lower than values obtained from undrained laboratory tests on undisturbed samples. This is proposed to be due to the presence of low permeability barriers, void redistribution, soil mixing, and stress redistribution. Based on back-analysis of field case histories, Seed (1984, 1987), Seed and Harder (1990), Olson and Stark (2002), Idriss and Boulanger (2008), among others, have proposed residual shear strength (τ res or S r ) or strength ratio (S r /p') for liquefied soil as a function of either SPT (N 1 ) 60 or normalized CPT tip resistance Q c1cs (Figure 4.1). The recent work by Idriss and Boulanger (2008) includes different relationships for cases with and without void redistribution (Figure 4.1(d)). The general procedure as described by Idriss and Boulanger (2008) involves performing post-earthquake static limit-equilibrium slope stability analyses. The “residual strength” or “residual strength ratio” of the liquefied portion of the failure surface is adjusted until the calculated factor of safety is one. Non-liquefied portions of the failure surface are assigned best-estimate shear strengths. The use of the initial (pre-failure) slope geometry gives an upper-bound strength while use of the post-failure geometry gives a lower value. Various procedures which account for sliding inertia and evolving geometry have been used to interpolate between these two values. The calculated “residual strength” or “strength ratio” is then correlated to a fines corrected SPT N-value or CPT tip resistance Q c1cs value representative of the liquefied zone within the slope. It should be noted that back-calculating a residual strength is not a simple task. There are many uncertainties, unknowns, and items to consider, including: • Soil stratigraphy, density (or (N 1 ) 60 or Q c1 ) of various zones, water table at time of the earthquake; • Time of occurrence of the flow failure (relative to earthquake shaking), deformed shape at the time of the failure, presence of shaking and inertial effects at time of failure; 43 Chapter 4 – Residual Strength from Back Analyses of Case Histories • Extent of liquefaction and extent of the failure surface where residual strengths are to be back-calculated; • Pore pressure regime and soil strengths to be used in the non-liquefied zones of the back-calculation failure surface - should peak undrained strengths or strengths with steady state pore pressures, etc. be used in the non-liquefied zones?; • Variation of residual strength during the failure process – mixing and pore water void redistribution is likely during the failure process and strengths may be less than the initial value when failure initiated; • Methods for using residual strengths in design. For example, if circular limitequilibrium analyses without any inertial effects are to be used for design, should the same be used for back-analyses? Whereas, if the residual strengths are to be used to calculate run-out distances of the failure, then residual strengths back-analyzed with consideration of momentum and post-failure geometry should be used; • Variable(s) to be used to correlate the residual strength (i.e., initial effective stress (σ' vo ), fines corrected and normalized SPT N-value (N 1 ) 60cs , fines corrected and normalized CPT tip resistance Q c1cs , static bias, shaking duration, factor of safety against liquefaction, etc.). As previously discussed, the residual strength back-calculated from case histories is not a soil property, but an averaged strength achieved within the liquefied portion of the failed slope. There is a significant variation and uncertainty in this parameter, and more than one solution. There is also a problem in terms of similarity between the design problem considered and the case histories upon which the correlations are based. For example, using residual strengths back-calculated from failed embankments may not be applicable for analyzing the lateral movement of piles through liquefied soil or using the residual strength back-calculated from embankments with steep slopes may not be applicable for use on very shallow slopes, etc. However, even though there are inherent uncertainties in these values, designers have to manage with the tools and correlations at their disposal. Presently, the back-calculated residual strength correlations may be the best information available. They 44 Chapter 4 – Residual Strength from Back Analyses of Case Histories are being used in the design of both minor and major structures. When designers use backcalculated strengths they should endeavour to understand how they were derived and accept that they will be working with some degree of variability and uncertainty. Some of the key correlations that have been used in the past, or currently in use, are briefly discussed below. 4.2 Seed (1984, 1987) H.B. Seed back-calculated seventeen field failures and correlated the residual strength (τ res ) SPT (N 1 ) 60cs (Figure 4-1(a)). For his correlations, Seed used the pre-failure geometry of the slope or embankment, and two dimensional limit equilibrium slope analyses. 4.3 Seed and Harder (1990) R.B. Seed and L.F. Harder repeated the back-calculated case histories carried out by H.B. Seed but considered post-failure geometry and momentum effects (Figure 4-1(b)). They also correlated the residual strength (τ res ) to overburden corrected and fines corrected SPT Nvalues (N 1 ) 60cs . The values by Seed and Harder (1990) were lower than those of Seed (1987) due to the inclusion of momentum and evolution of slope geometry effects. 4.4 Olson and Stark (2002) Olson and Stark updated the previous work of Stark and Mesri (1992) who argued that the use of a residual shear strength ratio (residual shear strength divided by the initial vertical effective stress) (τ res /σ' vo ) should be employed rather than just τ res (Figure 4-1(c)). They carried out similar back-calculations as those conducted by Seed and Harder (1990), except that they added more data points and equated a residual shear strength stress ratio (τ res /σ' vo ) to normalized and fines corrected SPT penetration resistance (N 1 ) 60cs . Like Seed and Harder (1990), Olson and Stark considered momentum and the evolution of slope geometry effects. Others who have developed (τ res /σ' vo ) correlations include Ishihara (1993), Wride et al. (1999), and Yoshimine et al. (1999). 4.5 Idriss and Boulanger (2008) Idriss and Boulanger (2008) plotted selected residual strength correlations derived from back-calculations by Seed (1987), Seed and Harder (1990), and Olson and Stark (2002) and 45 Chapter 4 – Residual Strength from Back Analyses of Case Histories then fitted suggested curves for use in design to the data (Figure 4-1(d)). Idriss and Boulanger provide both correlations for residual strength (τ res ) and residual strength ratio (τ res /σ' vo ) versus fines corrected normalized SPT N-value (N 1 ) 60cs , and also a correlation for residual strength ratio (τ res /σ' vo ) versus fines corrected normalized CPT tip resistance (Q c ) 1cs ). Suggested design curves are provided for use in situations where pore water void redistribution would and would not occur. 4.6 Discussion It is noted that in the above data sets the correlation between (N 1 ) 60cs (or (Q c ) 1cs )) and (τ res ) or (τ res /σ' vo ) is poor. Below a threshold, of an ((N 1 ) 60cs ) of about 12 to 15, there is little correlation with the initial pre-liquefaction void ratio or relative density (possibly due to void redistribution effects) and above this value residual strengths are high and flow failure unlikely. This is a somewhat similar result to the lateral spreading estimates by Youd et al 2002 which indicate minimal displacements if (N 1 ) 60 is greater than 15. It is also postulated that if a soil is homogeneous and expected to behave in a truly undrained manner, then the strengths obtained from quality cyclic undrained laboratory tests (with similar stress path and void ratio to that which would occur in the field) should be applicable. This contention supports the use of residual strength ratios as proposed by Idriss and Boulanger (2008), (Figure 4-1(d)) for cases without void redistribution. In Figure 4-2, a conservative estimate of the post-liquefaction strength from undrained cyclic simple shear tests on Fraser River Sand by Sriskandakumar (2004) (Test L14 and undrained monotonic test) was added to the Idriss and Boulanger plot. In the simple shear test, a relative density (D R ) of 40% (indicative of (N 1 ) 60cs ≈ 7.5 assuming D R = of ((N 1 ) 60cs /46)0.5) and initial vertical effective stress of 100 kPa, a post-liquefaction strength of about 20 kPa was derived. 4.7 Chapter 4 Summary and Conclusions For design, the strength of liquefied soil has been based on values back-calculated from field case histories because alternative methodologies have not been available. Experience has shown that strengths back-calculated from laboratory tests tend to give strengths that are too high. This is believed to be due to the laboratory testing process not accounting for void redistribution and mixing effects. If the in-situ soil is truly believed to behave in an 46 Chapter 4 – Residual Strength from Back Analyses of Case Histories undrained manner then the strengths from laboratory tests should be representative. In the field, the soil is generally not truly “undrained” and the “residual” strength is dependent on many variables (soil density, soil fabric, stratigraphy, initial stresses, duration of shaking, permeability/groundwater flow regime, method of analyses used, etc.) and is as much a site property as it is a soil property. Correlating the “residual” strength or strength ratio with soil density or penetration resistance is a gross simplification and consequently there is much scatter and uncertainty in the data. The discussions in this Chapter are important in that the residual strengths backcalculated from case histories are used in current engineering practice, and also because they provide the input parameters for the proposed hybrid numerical procedure that is the topic of this dissertation. In current numerical applications, the author is suggesting using a “residual” strength ratio based on the charts and equations by Idriss and Boulanger (2008). The methodology of applying these equations in the numerical modelling is discussed in Section 6.2. 47 Chapter 4 – Residual Strength from Back Analyses of Case Histories FIGURES Chapter 4 Residual Strength from Back Analyses of Case Histories 48 Chapter 4 – Residual Strength from Back Analyses of Case Histories Figure 4-1 Post-liquefaction residual strength form case histories (a) Seed et. al. (1984), (b) Seed and Harder (1990), (c) Olsen and Stark (2002) and (d) Idriss and Boulanger (2008). 49 Chapter 4 – Residual Strength from Back Analyses of Case Histories Figure 4-2 Post-liquefaction residual strength form case histories by Idriss and Boulanger (2008) with undrained strength inferred from University of British Columbia laboratory tests by Sriskandakumar (2004) superimposed (Byrne et al., 2008). 50 Chapter 5 – Analysis Procedures Overview 5 ANALYSIS PROCEDURES OVERVIEW 5.1 Introduction Key aspects to be addressed in the design of earth structures subjected to earthquake shaking are (i) will liquefaction triggering occur?, (ii) is post-liquefaction stability adequate to prevent a flow slide?, and (iii) are the induced deformations acceptable? For many years, the state of design practice for this assessment has been based on procedures developed by Seed and colleagues in the 1970’s and 1980’s (Seed and Idriss 1971; and Seed 1987). In Seed’s approach, the design problem was divided into separate analysis steps approximately as follows: (1) Carry out an equivalent-linear elastic dynamic analysis to determine maximum cyclic stress ratio or earthquake loading applied to the soil; (2) Assess whether liquefaction will be triggered using empirical correlations between cyclic stress ratio and soil penetration resistance (N 1 ) 60 values; (3) Assign an empirical residual strength using correlations as presented in Section 4 to the soil zones which liquefied. Next carry out a limit equilibrium stability analysis. If the factor of safety is less than 1.0 to 1.2, then a flow slide is assumed to occur and remedial measures to prevent this may be required. (4) If a factor of safety greater than 1.0 to 1.2 was calculated in (3), flow sliding was deemed not to be a problem; however, significant deformations could still occur during shaking. Deformations from shaking (cyclic mobility deformations) were traditionally estimated using empirical correlations (Youd et al. 2002) or using limit equilibrium stability analyses with Newmark-type displacement estimates (Newmark 1965; Bray and Travasarou 2007)). If the calculated deformations were excessive remedial measures were deemed necessary. For many years, the standard for ground response analyses to simulate the site effects of earthquake shaking has been to use the equivalent linear method with the computer program SHAKE (one dimension) (Schnabel et al. 1972; Idriss and Sun 1992), FLUSH (two dimensions) (Lysmer et al. 1975) or variants of them. The equivalent linear method is an 51 Chapter 5 – Analysis Procedures Overview iterative linear-elastic dynamic analysis process which indirectly incorporates the hysteretic and modulus reduction with strain soil behaviour by adjusting the modulus and viscous damping to match pre-determined modulus versus strain and damping versus strain relationships (Figure 5-1). The equivalent-linear method has several shortcomings including: (i) The soil will be over-damped and soft during low-amplitude cycles and underdamped and overly stiff during large amplitude cycles; (ii) The procedure is elastic and does not correctly simulate soil yielding or allow irreversible permanent displacements to occur. This is especially important when there is a static bias. Then plastic deformation will cause deformation “marching” to occur in the direction of the bias; (iii) Unnatural resonance may occur; (iv) The pre-determined modulus versus strain and damping versus strain relationships are based on uniform loading cycles and the correlation used to convert peak strain from earthquake loading to equivalent uniform cycles is only approximate; and, (v) Pore pressure generation (liquefaction triggering and consequences) is not accounted for and must be carried out as a separate analysis (i.e., liquefaction triggering assessment, etc.). In recent years, the use of fundamentally more correct, generally more complex, hysteretic non-linear analyses has become more common (DESRA (Finn et al. 1977), TARA (Finn et al. 1986), FLAC with hysteretic damping (Itasca 2008), D-MOD2000 (Matasovic and Ordonez 2007), UBCHYST (this dissertation), UBCSAND (Byrne et al. 2004), SANISAND (Taiebat et al. 2010), etc.) and are replacing the equivalent-linear method. The complex hysteretic non-linear models can be subdivided into “effective stress” and “total stress” categories. The effective stress models account for shear-induced pore pressure changes during the dynamic analyses, whereas, the total stress models assume constant pore pressure during the dynamic analyses. The current state-of-the-art/practice and the effective stress procedures proposed in this dissertation, involve carrying out a single dynamic analysis where pore-pressure development, potential liquefaction triggering, and resulting deformations are all determined 52 Chapter 5 – Analysis Procedures Overview in chronological sequence in a single analysis, analogous to real behaviour. Alternative approaches are available for this single analysis. One approach is to carry out total stress analyses where there are no changes in pore water pressure during earthquake shaking, and liquefaction is triggered when a threshold of cycles of cyclic stress is reached. The UBCTOT model (Beaty and Byrne 1999, Beaty 2001) is one example of this approach. Another approach is to carry out a coupled effective stress analysis using models or programs such as UBCSAND (Byrne et al. 2004), TARA-3 (Finn et al. 1986), CYCLIC (Para 1996; Elgamal et al. 1999), DYNAFLOW (Prevost 2002) and SANISAND (Taiebat et al. 2010). In the effective stress approach, the constitutive model couples shear strain with soil skeleton volume change. If the soil is saturated then a decrease in skeleton volume (contraction) induces a pore pressure increase and expansion of skeleton volume (dilation) induces pore pressure decrease. Change in pore pressure results in changes in effective stress, soil strength, and soil modulus. The UBCSAND model has been used for the work in this dissertation, and is discussed in more detail later in this Section. Simpler total stress constitutive models may be used for portions of the numerical model where soil liquefaction is not deemed to occur (such as, clay, clayey silt or granular soils above the water table). The UBCHYST model has been developed for this purpose. The advantage of UBCHYST over even simpler elastic plastic models, such as, Mohr Coulomb, is that it provides a more realistic replication of soil behaviour. The modulus varies with stress ratio and damping is a direct consequence of the stress-strain loops. It does not require the use of high Rayleigh damping values that greatly decreases run speed in the program FLAC (ITASCA 2008), will not cause un-natural resonance and does not require modulus versus strain relationships as part of the input. 5.2 State-of-Practice Total Stress Analysis (UBCTOT) UBCTOT (Beaty and Byrne 1999; and Beaty 2001) addresses the liquefaction response taking pre-triggering, triggering, and post-triggering aspects into account in a single analysis. The earth structure and interacting structural elements are modeled in two dimensions as a collection of discrete elements. Initially, stiff moduli, representing pre-liquefaction- triggering conditions are specified. The FLAC Mohr Coulomb model with equivalent linear shear modulus (G) and undrained bulk modulus (K) are used. Then, during dynamic 53 Chapter 5 – Analysis Procedures Overview shaking, the model tracks the dynamic shear stress history τ cyc within each element, where τ cyc = │τ st - τ xy │; τ st equals the static shear prior to dynamic excitation and τ xy is the shear stress on the horizontal plane. The irregular shear stress history caused by the earthquake is interpreted as a succession of half cycles. Each half cycle of cyclic shear stress is transformed into an equivalent number of cycles N eq at τ 15 , where τ 15 is the cyclic shear stress required to cause liquefaction in 15 cycles. So a small pulse may account for a portion of a reference cycle, whereas, a large pulse may account for several reference cycles. If the reference cycle count threshold is reached (ΣN eq ≥ 15), then liquefaction is triggered in the element by changing the soil strength and stiffness to the “post-liquefaction” values. Figure 5-2 illustrates the weighting curve used to establish the correlation between τ cyc and N eq . Figure 5-3 illustrates the stress strain response used with the UBCTOT model before and following liquefaction. Typically, Rayleigh damping of 4-8% is used with UBCTOT. With the model, the most severely loaded or looser zones will liquefy first, the extent of liquefaction will expand with further shaking, and stress redistribution and baseisolation effects will occur in a somewhat similar manner to that which would occur in the field. If sufficient elements liquefy and their residual strength is not adequate for stability, then large deformations (flow slide) will occur. The reference cyclic stress ratio to cause liquefaction in 15 cycles is obtained from state-of-practice liquefaction triggering charts such as that shown in Figure 5-4 and the post-triggering strength is obtained from back-analyzed case history residual strengths (Figure 4-1). The procedure makes use of both state-of-practice triggering and back-analyzed residual strength charts and is a logical extension of the current state-of-practice Seed approach, discussed in Section 5.1. Pore pressure changes are not computed in this approach, but are indirectly accounted for by prescribing much softer post-liquefaction stress-strain relations and residual strength values after liquefaction has been triggered. 5.3 UBCSAND Coupled Effective Stress Analysis 5.3.1 Introduction to UBCSAND A coupled effective stress constitutive model (UBCSAND) and analysis procedure for modeling earthquake shaking and liquefaction has been developed at the University of 54 Chapter 5 – Analysis Procedures Overview British Columbia. UBCSAND is an elasto-plastic effective stress model with the mechanical behaviour of the sand skeleton and pore water flow fully coupled (Beaty and Byrne 1998; Byrne et al. 2004; and Seid-Karbasi 2008). Plastic shear induces a change in the soil skeleton volume and, if the pores are filled with water, a change in pore-pressure and effective stress is induced. The model includes a yield surface related to the developed friction angle or stress ratio (Figure 5-5) and a non-associative flow rule with the soil contractive when the stress ratio (shear stress on plane of maximum shear divided by mean effective stress (τ/σ' m ) where (σ' m =σ' x + σ' y )/2) is less than the phase transformation friction angle and dilative when the stress ratio is greater than the phase transformation friction angle (Figure 5-6). The soil is loading when the stress ratio increases, unloading when it decreases, and cross-over (change from unloading on one side to loading on the other) occurs when the shear stress on the horizontal plane (τ xy ) changes sign (Figure 5-7). On unloading (and reloading on some versions of UBCSAND) the model is elastic with bulk and shear moduli that are a function of mean effective stress. Increases in the mobilized stress ratio (dη) is a function of plastic shear modulus (Gp/σ') and the hardener (dϒp), as illustrated in Figure 5.8. Key elastic and plastic parameters used are adjusted so as to give a good match with simple shear laboratory tests as the loading path of this test, including rotation of principal stress axes, closely approximates that which occurs during earthquake loading. The UBCSAND constitutive model is run within the finite difference program FLAC (ITASCA 2008). In the FLAC program, dynamic analyses are carried out in the time domain with full coupling between groundwater flow and mechanical loading. The FLAC fish code for UBCSAND1ver2 and a flow diagram illustrating UBCSAND are given in the attached Appendix A. The UBCSAND model has been calibrated against simple shear laboratory tests, centrifuge tests with and without impermeable silt barriers (Yang et al., 2004; Phillips et al. 2004; Phillips and Coulter 2005; Seid-Karbasi et al. 2005; and Park 2005) and the empirical liquefaction triggering charts such as those by Idriss and Boulanger (2008). The model is able to simulate both the observed drained behaviour of loose sand soils (contraction when sheared below the Φ cv , (or Φ pt ) and dilative above the Φ cv (or Φ pt )) and the build-up of pore pressure and soil liquefaction that occurs in undrained simple shear tests. The model is also 55 Chapter 5 – Analysis Procedures Overview able to simulate the behaviour of a flow failure when a low permeability barrier was present and no failure when the barrier was absent or when drains were installed through the barrier (Byrne et al. 2006; Naesgaard et al. 2005) and able to simulate the post-shaking failure of the Lower San Fernando Dam (Naesgaard et al. 2006). Atigh and Byrne (2004), showed that UBCSAND could simulate the behaviour of the triaxial tests with fluid inflow carried out by Vaid and Eliadorani (1998). A DLL version of UBCSAND (Version 904aR) and documentation regarding its use has been uploaded to the ITASCA UDM website (Beaty and Byrne 2011). 5.3.2 Elastic Response As indicated above, unloading and reloading (Figure 5-7) are elastic with shear (Ge) and bulk (Be) moduli that are isotropic and a function of mean normal effective stress (σ') as follows: [5-1] Ge = Ke G · P a ·(σ'/P a )n [5-2] Be = Ke B · P a ·(σ'/P a )m where: Ge = elastic shear modulus Ke G = shear modulus number that is a function of relative density Often taken as 21.7∙ A· ((N 1 ) 60 )0.33 where A = 15 to 20 or can also be back-calculated from shear wave velocity (V s ) data as follows: n Ke G = ρ V s 2 / [Pa ∙ (σ' / Pa) ] where ρ = soil density Ke B = bulk modulus number = α· Ke G where α = function of Poisson’s ratio and ranges from 0.67 to 1.33. Pa = atmospheric pressure σ' = mean normal effective stress = (σ' x + σ' y )/2 n ≈ m ≈ 0.5 56 Chapter 5 – Analysis Procedures Overview 5.3.3 Plastic Response Plastic volumetric (εp v ) and shear (γp s ) strains may occur when on the yield surface. Plastic volumetric strains are shear induced and are contractive below the phase transformation stress ratio (φ pt ) (or φ cv ) and dilative above, as illustrated on Figure 5-6 and by the flow rule, Equation 5-3 below: [5-3] d(εp v )/d(γp s ) = -tan(ψ) and -sin(ψ) = (sin(φ cv ) – η) where: ψ = dilation angle φ cv = phase transformation friction angle or stress ratio η = developed stress ratio ≤ sin(φ f ) where φ f = peak friction angle Raising of the yield surface (dη) is carried out through a plastic shear modulus (Gp) and the plastic shear strain increment (d(γp s )) (the hardener) as follows: [5-4] dη = d(γp s )·Gp/σ' The plastic shear modulus is related to η and γp s through a hyperbolic relationship which for first time or virgin loading is as decribed below and illustrated in Figure 5-8. [5-5] Gp where: Gp i = Gp i · (1 - η/η f · R f )2 = plastic modulus at η=0 and equals function of (Ke G and stress level) η = developed stress ratio ηf = stress ratio at failure = sin(φ f ) Rf = constant that truncates hyperbolic curve (between 0.7 and 1.0) For subsequent loading, plastic strains still occur but the plastic shear modulus (Gp i ) is stiffer and increases with the number of cycles. On the other hand, following significant dilation during the last one half cycle, Gp i is greatly reduced to compensate for the plastic volume change (contraction) that occurs on real soil on stress ratio unloading but does not occur in UBCSAND due to it having elastic unloading (Figure 5-9). 57 Chapter 5 – Analysis Procedures Overview 5.3.4 Recent Improvements/Options added to UBCSAND 5.3.4.1 Pull Down Yield Surface on unloading if no Cross-over In past versions of UBCSAND, unloading (Figure 5-9) and reloading (on same side) would be elastic unless there had been a cross-over (going through zero shear stress on the horizontal (τ xy ) plane). With the proposed modification, the yield surface (Figure 5-5) is partially pulled down (drops with the decreasing stress ratio) on unloading, even if there is no cross-over. This results in increased plastic deformation on reloading (as the yield surface is pushed up) and more permanent deformation in the direction of the static bias. In order to eliminate numerical instability, a switch was added which did not allow the yield surface pull-down to occur when effective stresses were very small. 5.3.4.2 Modeling of Dense Soils with Lower Phase Transformation Friction Angle In previous versions of UBCSAND, the constant volume or phase transformation friction angle was usually fixed (typically at 33 degrees) and the peak friction angle was set equal to the phase transformation friction angle plus an increment (generally set equal ∆ϕ to (N 1 ) 60 / 10). With the previous model, dense sands could be calibrated to trigger to liquefy at the correct number of cycles for a given stress ratio. However, the pre- and post-triggering response was too stiff and soft, respectively. Improved response for dense sands was obtained by the author by reducing the phase transformation friction angle and increasing the peak friction angle for denser sands. Suggested current correlations for use in UBCSAND1v02 are: [5-6] Phase transformation friction angle (φ pt ) = 40.0 – ((N 1 ) 60 ) · 0.65) ≤ 33º [5-7] Peak friction angle (φ f ) = 33 + (((N 1 ) 60 ) / 10)1.65 5.3.4.3 Delay Dilation following Post-Liquefaction Cross-over When loose to medium dense soils liquefy, there is a period following cross-over during which the sample strains with very little or no gain in shear strength (Figure 5-10). However, at some limit, strain dilation kicks in, reduces pore pressure and increases strength. This behaviour is simulated by putting in a dilation delay, with zero dilation until the cumulative 58 Chapter 5 – Analysis Procedures Overview plastic shear strain since last cross-over, is greater than the cumulative plastic shear strain of the previous half cycle times a calibration factor. 5.3.4.4 Dilation Reduction or Cut-off upon Expansion of Element When elements expand due to shear induced dilation or due to the flow of pore water into an element, the dilation angle should reduce and eventually go to the critical or steady state value of zero. This concept has been incorporated by setting the dilation angle to zero upon the element volume change exceeding some trigger volume change. This trigger volume change is less than that which would normally cause an element to go to the critical state (Figure 5-11) due to localization and element size effects (Section 5.3.5), due to UBCSAND using an elastic bulk modulus, due to numerical limitation within FLAC causing it to not recognize a bulk modulus less than approximately 1/20th of the bulk modulus of the fluid (ITASCA 2008) (Section 5.3.5) and due to practical limitations on the number of elements that can be placed below an impermeable barrier (Section 5.3.5.5). In a real sand element, the dilation would be a function of void ratio and effective confining pressure and would decrease (less dilative) progressively as void ratio and confining pressure increase, until reaching the critical state, at which time it would be zero. Due to the limitations discussed above and to keep the model simple, it has been decided to cut the dilation in one step upon reaching a prescribed volume change threshold. In UBCSAND/FLAC, this is done by setting the negative of the sine of the dilation angle (“m_dt” in UBCSAND) to be greater than or equal to zero (does not permit shear induced dilation but will allow contraction) when the volumetric strain (“vol_strain” parameter in FLAC) increment since the start of earthquake shaking reaches a prescribed limit. Due to localization and limitations in UBCSAND and FLAC, the threshold has, to date, largely been selected by trial in one dimensional columns and initial runs of two dimensional models. In UBCSAND analyses, sand subjected to earthquake shaking initially has a net contraction, and then expands if there is fluid inflow and/or continuous shear in one direction. The model seems to approximately match model tests and field case histories when the dilation cut-off trigger volume is set close to the original volume prior to the on-set of earthquake shaking (i.e., ∆vol_strain≈0). This gives realistic response for sand with an (N 1 ) 60 of about 10 and elements of about one metre height. 59 Chapter 5 – Analysis Procedures Overview 5.3.5 UBCSAND/FLAC Limitations and Challenges Like all constitutive models, UBCSAND is a simplification and compromise. It is designed for modeling saturated sand soils subjected to earthquake shaking and related potential liquefaction triggering. The model is reasonably good at predicting skeleton volume changes, build up in pore pressures, and number of cycles to liquefaction triggering. There are, however, limitations within the model of which the user should be aware. Some of these limitations are related to the constitutive model, some to the workings of FLAC, and some due to modeling three dimensional phenomena in two dimensions. Some of the key items are: Low Water Compressibility UBCSAND is, typically, run using a fluid modulus that is about one quarter that of the actual fully saturated fluid modulus. It has been found that this value is a good compromise between realistic modeling of liquefaction triggering and numerical time step (proportional to run time). A portion of the modulus reduction can be justified if the actual pore water is not fully saturated. The use of the lower modulus is partially mitigated by using the same low fluid modulus when calibrating the model for liquefaction triggering. Localization and Element Size Dependency As previously mentioned, pore water void redistribution in soil profiles with low permeability barriers can cause localization below the barrier and the analysis results become element size dependent (Yang and Elgamal 2002; Seid-Karbasi and Byrne 2004; Naesgaard et al. 2005; and Seid-Karbasi 2008)). Smaller (thinner) elements will give more realistic behaviour; however, when using FLAC this is often impractical due to analysis time constraints. Analysis time step is a function of element stiffness and minimum dimension. Analysis time is a function of the time step and number of elements in the model. To counter this effect, large elements can be made to behave like small elements by curtailing dilation in the large element at a void ratio that is considerably less than that calculated from critical state theory (Naesgaard et al. 2005; Seid-Kabasi 2008) (Figure 5-11). 60 Chapter 5 – Analysis Procedures Overview Post-liquefaction Settlement When a sand soil is shaken, the skeleton volume contracts slightly and load is transferred to the pore water. This process occurs with very small volume change because water is almost incompressible. However, with time, often after end-of-shaking, the excess pore water pressure dissipates, the effective stress increases, and the soil consolidates. As indicated in Sections 2.1 and 2.2, if the soil has liquefied and effective stresses are zero or near zero (little or no static bias) then much of the post-liquefaction settlement will be plastic (i.e., non-recoverable). In current versions of UBCSAND and FLAC, there are two aspects that lead to the calculated post-liquefaction settlement being too small (in some cases by a factor of about 5 to 10). One problem is lack of volumetric plasticity. UBCSAND has no volumetric yield cap and the bulk modulus is elastic. The modulus is a function of mean normal effective stress and when the effective stress goes to zero the modulus goes to a low default value. However, with dissipation of pore pressure, the modulus starts increasing immediately, whereas, in real liquefied soil, a significant portion of the volumetric deformation would be plastic and occur with no or little change in effective stress. It should be noted that this error is mainly a problem when there is little or no static bias. As indicated in Figure 2-5, when there is a significant static bias, sedimentation settlement does not occur and the elastic response as assumed in UBCSAND, will be more appropriate. In FLAC, there is also a numerical problem when the skeleton bulk modulus is much lower (if more than approximately 20 times less) than the fluid bulk modulus. The program will run; however, the low skeleton bulk modulus will not be recognized by FLAC and consolidation settlements will be underestimated. One option for mitigating this is to reduce the pore fluid modulus as the skeleton modulus decreases. This, however, is fraught with other problems, one being that the fluid modulus is a nodal property, while the volumetric strain is an element property. Therefore, changing modulus in one element affects adjacent elements. The other problem, if the fluid modulus is unrealistically low, is that unrealistic expansion of the fluid will occur when there is flow from zones with high pressure to zones of low pressure. Addition of a plastic volumetric yield envelope to UBCSAND may help to alleviate some of these problems. A new version of UBCSAND with this is currently under 61 Chapter 5 – Analysis Procedures Overview development by the author. However, even if UBCSAND–FLAC can be “fixed” to simulate the post-liquefaction consolidation correctly, there is still a run-time issue. Post-liquefaction consolidation can take several hours or days to occur in the real world. In dynamic analyses, current versions of FLAC/UBCSAND, typically take approximately 15 minutes to run one second of dynamic time. At this rate, a day of real-world consolidation would take approximately 900 days of computer time, which is obviously impractical. In some cases, FLAC’S fast-flow logic will be applicable and can be used to reduce run times to reasonable values. Increasing the permeability throughout the model can also decrease the problem run time, in some cases. Multi-directional Shaking not captured in 2D Model As in all two dimensional modeling, there is no applied out-of-plane earthquake motion. This additional motion should accelerate the on-set of liquefaction and lead to higher overall pore pressures for locations where liquefaction is not triggered. Stress redistribution and pore pressure migration effects in the out-of-plane direction will also not be detected by the model. The out-of-plane behaviour can be partially compensated for by calibrating the triggering model to field case history triggering charts, such as that shown in Figure 5-4; however, out-of-plane stress and pore pressure redistribution effects will not be detected. In the future, these problems can be mitigated by adapting the model to three dimensions and using a program, such as, FLAC 3D (ITASCA 2008). FLAC Element – Node Property Errors FLAC stores pore pressures, fluid modulus, fluid and soil mass at the element nodes, and then calculates element properties for use in the constitutive model by averaging the values at the nodes. This leads to a smearing of properties at internal boundaries. For example, when a sand layer underlying a lower permeability silt layer has an influx of pore water and the element truly liquefies (R u =1.0) then the pore pressure at the upper boundary of the sand layer should also go to R u =1.0. However, the pore pressure at the boundary node will be an average value between the pore pressure in the silt and that in the sand causing R u in the element under the barrier to be less than one. The problem can be mitigated by using multiple smaller (thinner) elements. However, analysis speed is inversely proportional to the 62 Chapter 5 – Analysis Procedures Overview minimum elements dimension. Therefore, the modeling is often a compromise between reasonable run times and representative soil layer dimensions. 5.3.6 UBCSAND/FLAC Grid-Model Development Considerations Developing a UBCSAND/FLAC grid and model is a compromise between computer run times and model accuracy. Some considerations include: Maximum grid size for wave transmission: ITASCA 2008 recommends that the dimension (width or height) of the element should be less than about 1/10th of the wavelength of the highest frequency component that contains appreciable energy. For many earthquake scenarios, this frequency can be taken as 8 to 15 hz and element sizes in the range of one/half to two metres work well. The input record should be filtered of motions with frequencies higher than the desired value. Number of sand layers underlying an impermeable barrier: Seid-Karbasi 2008 suggested that there should be multiple (≈ 10) elements underlying a barrier in order to properly model pore water void redistribution effects. However, unless the spacing of low permeability layers is in the range of 5m or more this grid density is not possible. Two options are available; use fewer elements between the barriers and accept that pore water / void redistribution affects may not be fully captured and/or artificially space the barrier layers further apart than they are in the field and accept that capturing one barrier correctly is better than having multiple layers that are not captured correctly. By using an appropriate dilation cut-off for location with expanding elements, pore water / void redistribution effects can be captured with fewer than 10 elements underlying the barrier. The Lower San Fernando Dam had alternative coarse and fine sand with spacing of less than the 0.5m element size used in the back-analyses (described in Section 7.8). In order to capture the void redistribution effects not all the silt barrier layers were included. In the lower portion of the dam, five sand elements (approximately a combined 2.5m thickness) were placed below the lower silt barrier. These five layers, plus an appropriate dilation cut-off allowed localization and post-shaking failure to be captured by the UBCSAND/FLAC model. 63 Chapter 5 – Analysis Procedures Overview 5.4 Hysteretic Model for Non-liquefiable Clay/Silt Soils (UBCHYST) 5.4.1 Introduction When a soil mass is subjected to earthquake shaking, the primary loading is cyclic shearing in the horizontal plane. The cyclic shearing induces shear stress – strain behaviour that is hysteretic in nature with characteristics as illustrated in Figures 5-12 and 5-13. These features include: • Increasing hysteresis and reduction in secant modulus with greater strain (Figure 5-13a); • Increasing hysteresis and reduction in secant modulus with number of cycles (Figure 5-13b); • Permanent strains bias “ratchetting” when loaded with a static bias (Figure 5-13c); • Potential pore pressure generation that is a function of soil properties, cycle amplitude and/or number of loading cycles. Increased pore pressure results in increasing hysteresis, modulus reduction, and in the limiting condition soil liquefaction and flow (Figure 5-13d); • Permanent secant modulus reduction with “damage” and pore pressure buildup; • Strength reduction with plastic strain. In this Section, a robust, relatively simple, total stress model, UBCHYST developed at the University of British Columbia for dynamic analyses of soil subjected to earthquake loading, is presented. The model is intended to be used with “undrained” strength parameters in low permeability clayey and silty soils, or in highly permeable granular soils, where excess pore water would dissipate as generated. The model has been developed for use in the two dimensional finite difference program FLAC (ITASCA 2008) but can be adapted to other finite difference or finite element platforms. 64 Chapter 5 – Analysis Procedures Overview 5.4.2 Non-Linear Hysteretic “UBCHYST” Model The essence of the proposed hysteretic model is that the tangent shear modulus (G t ) is a function of the peak shear modulus (G max ) times a reduction factor that is a function of the developed stress ratio and the change in stress ratio to reach failure. This function is shown in Equation (5-8) and illustrated in Figure 5-14. [5-8] G t = G max * (1 - (η 1 /η 1f )*R f )n * Mod1 * Mod2 where: η η1 = developed stress ratio = (τ xy /σ v ’) = change in stress ratio η (τ xy /σ v ’) since last reversal = η - η max η max = maximum stress ratio (η) at last reversal η 1f = change in stress ratio to reach failure envelope in direction of loading = η f - η max ηf = (sin(Ø f ) + Cohesion * cos(Ø f )/ σ v ’) τ xy = developed shear stress in horizontal plane σv’ = vertical effective stress Øf = peak friction angle R f , and n = calibration parameters with default values of 1 and 2, respectively Mod1 = a reduction factor for first-time or virgin loading (typically 0.6 to 0.8) Mod2 = optional function to account for permanent modulus reduction with large strain = (1-η 1 /η 1f )rm)*dfac ≥ 0.1 where “rm” and “dfac” are calibration parameters Stress reversals occur when the absolute value of the developed stress ratio (η) is less than the previous value and a cross-over occurs if τ xy changes sign. A stress reversal causes η 1 to be reset to 0 and η 1f to be re-calculated. However, the program remembers the previous reversals (η 1old and η 1fold ) so that small hysteretic loops that are subsets of larger loops do not change the behaviour of the large loop (Figure 5-13a). In the equation above, the tangent shear modulus varies throughout the loading cycle to give hysteretic stress-strain loops with the characteristics illustrated in Figure 5-13. 65 Chapter 5 – Analysis Procedures Overview 5.4.3 Implementation of UBCHYST Model The equation above has been combined with the Mohr Coulomb failure criterion and incorporated into FLAC as a user defined constitutive model (other platforms could be used in lieu of FLAC). Typically, the model is used for fine grained soils (silts and clays) with undrained strength parameters or with free draining granular soil with drained strength parameters. Often it is used in combination with other models, such as, the coupled effective stress model UBCSAND, where UBCHYST is used for the clayey non-liquefiable soils, while UBCSAND is used for the sandy soils that are potentially liquefiable. Steps for implementing the UBCHYST model are typically: (1) Calibration of the model by cyclically loading a single element (this can be done within FLAC or using a spreadsheet); (2) Constructing a one or two dimensional model grid with appropriate soil properties. Typically, the model is initially taken to static equilibrium in stages; initially, using an elastic model, followed by a Mohr Coulomb model, and finally, using a UBCHYST model; (3) FLAC is changed to dynamic configuration and the model is brought to static equilibrium by stepping the model with zero input motion; (4) Earthquake motion is applied to the base of the model and dynamic analysis is performed to the end of the record; (5) Deformations, time histories, response spectra, etc. are obtained. Input parameters for the model include: the maximum shear modulus (G max ) (derived from in-situ shear wave velocity measurements), bulk modulus (set equal or slightly greater than the G max ), soil density, and Mohr Coulomb failure criterian parameters (c and Ф f ). A small viscous (Rayleigh) damping component (fraction of critical damping of 0.5% to 1%) is used to give numerical stability at small strains. Calibration parameters (R f , n, dfac, and rm) vary the shape of the stress strain loops and are adjusted so the model response approximately matches published modulus reduction and damping data or laboratory test results. 66 Chapter 5 – Analysis Procedures Overview 5.4.4 UBCHYST Calibration The model can be calibrated by comparing uniform cyclic response to that inferred from published modulus reduction and damping curves similar to that shown in Figure 5-15 and/or by comparison to the results of cyclic simple shear laboratory tests (Figure 5-16). The simple shear test is preferred over triaxial loading because the loading path with rotation of principal axes, etc. more closely resembles the stress path from earthquake loading. 5.4.5 Summary A simple hysteretic total stress constitutive model (UBCHYST) has been developed at the University of British Columbia for one and two-dimensional earthquake ground response analyses. In the proposed hysteretic model, the shear modulus is a function of stress ratio and varies throughout the loading cycle to give hysteretic stress-strain loops of varying amplitude and area (damping) throughout the earthquake excitation. The magnitude of the stress ratio is limited by a Mohr Coulomb failure envelope. The model parameters allow calibration against laboratory data and/or published modulus-strain and damping-strain correlations. To date, the model has been used as a user defined constitutive model within the commercially available program FLAC. However, it could easily be adapted to other program platforms. The model is relatively simple, robust, runs quickly and has been used effectively on several large engineering projects. 5.5 Chapter 5 Summary and Conclusions This Chapter introduces the constitutive models UBCSAND and UBCHYST that are key parts (especially UBCSAND) of the proposed hybrid analysis procedure. UBCSAND when run within the program FLAC is a coupled effective stress model that considers shear induced soil skeleton volume changes, their effect on pore water pressure, and the resultant pore water flow from one element to another. As indicated in Chapter 2, these are key behaviours needed to capture soil liquefaction and related soil strength changes. The UBCSAND program was developed by Dr. Byrne and his students. New features added to UBCSAND as part of this research include: • Elimination of numerical instability for certain cases with static bias, • Improvement in the simulation of cases with large static bias, 67 Chapter 5 – Analysis Procedures Overview • Improvement in the liquefaction behaviour by adding a dilation delay (period during which shear does not induce volume change) thus allowing large shear strains to develop when the normal effective stress and shear stress is near zero, in agreement with behaviour observed in laboratory test data (Figure 5-10), • Eliminated shear induced dilation when an element reaches a critical volume or void ratio thus making the model compatible with critical state theory and allowing post-liquefaction flow, and • Improvements in the modelling of dense soils by making the phase transformation stress ratio a function of soil density and changing the correlation of peak friction angle with soil density. In Section 5.3.5 is a list of some limitations, and challenges with the UBCSAND/FLAC modelling is also given. UBCHYST is a simple robust constitutive model often used together with UBCSAND for the numerical modelling of soil structures subjected to earthquake shaking. UBCSAND is used for portions of the soil profile deemed susceptible to liquefaction and UBCHYST is used for those deemed not susceptible. The advantage of the UBCHYST model over a simpler Mohr Coulomb model is the non-linear hysteretic loops developed by varying the tangent shear modulus during loading and unloading. This more closely replicates the behaviour of real soil and reduces the Rayleigh damping requirements; thus increasing analysis speed. The initial algorithm for UBCHYST was developed by Professor Byrne while the author coded UBCHYST as a constitutive model in FLAC, made improvements regarding unload/reload loops, developed options for strength reduction as a function of strain or number of cycles, and developed documentation explaining the model. 68 Chapter 5 – Analysis Procedures Overview FIGURES Chapter 5 Analysis Procedures Overview 69 Chapter 5 – Analysis Procedures Overview Figure 5-1 In the equivalent linear method initial G(1) and ξ(1) are estimated. An elastic analysis is carried out with these values and from the results a new γ(1)eff is obtained. This γ(1)eff is used to get a new G(2) and ξ(2) which are used for the second elastic analysis iteration. This process is repeated until the process converges and strain compatible GFinal and ξfinal are obtained. Figure 5-2 Example of the liquefaction triggering methodology in UBCTOT. The CSR at “A” would cause liquefaction in 15 cycles; however, since it is only ½ a cycle this pulse provides 1/15 x ½ = 1/30th of the loading to cause liquefaction. Pulse “B” would cause liquefaction in 2 cycles and, therefore, provides ½ x ½ = ¼ of the loading necessary to cause liquefaction. The full cycle of “A” and “B” would provide ¼ + 1/30 = 17/60 of the loading necessary to liquefy the element. 70 Chapter 5 – Analysis Procedures Overview Figure 5-3 Bilinear response of an element that liquefies when using UBCTOT (Beaty, 2001). 71 Chapter 5 – Analysis Procedures Overview Figure 5-4 Figure 5-5 Liquefaction triggering chart and equations from Idriss and Boulanger (2006, 2008). Yield surface in UBCSAND. The hardener (dη = d(τ/σ')) that expands the yield surface (i.e., from A to B) is a function of the plastic shear strain and plastic shear modulus as illustrated in Figure 5-8. 72 Chapter 5 – Analysis Procedures Overview Figure 5-6 Flow Rule in UBCSAND. Below Ф cv or Ф pt shear strain induces volumetric contraction and above it induces volumetric dilation. Figure 5-7 Stress ratio history showing definition of loading, unloading and reloading UBCSAND (Beaty 2009; Beaty and Byrne 2011). 73 Chapter 5 – Analysis Procedures Overview Figure 5-8 Plastic strain increment and plastic modulus in UBCSAND. Figure 5-9 Comparison of postulated post-liquefaction response of loose sand and that modeled by UBCSAND. 74 Chapter 5 – Analysis Procedures Overview Figure 5-10 A dilation delay (proportional to dilation (expansive plastic volumetric strain) in last ½ cycle) is induced at shear stress cross-over in UBCSAND. Figure 5-11 Large element(c) emulating the behaviour of a small element (b). If (a) and (b) are at critical state with δl/l = δL/L then (c) can be made to behave similarly to (b) by setting dilation to 0 when the volumetric strain is equal to δl/L . 75 Chapter 5 – Analysis Procedures Overview Figure 5-12 Loading in simple shear induces hysteretic response in soil with shear modulus (G) and loop-size (damping) varying with strain. Figure 5-13 Characteristic behaviour of soil in cyclic shearing. 76 Chapter 5 – Analysis Procedures Overview Figure 5-14 UBCHYST model key variables. 25 1 15 0.8 10 0.6 G/Gmax τ (kPa) 20 5 0 -0.5 -5 0 -10 0.5 γ (%) 0.4 0.2 0 0.0001 -15 0.01 0.1 UBCHYST Constitutive Model -20 γ (%) 1 Seed & Idriss (1970) Sand Upper Bound -25 Seed & Idriss (1970) Sand Lower Bound 34.0 0.0 40.0 141.7 2041.0 41000.0 2.5 0.8 Damping ratio (%) peak φ= cohesion (kPa)= σ'v (kPa)= Vs (m/s)= density (kg.m3)= Gmax (kPa)= n= Rf= 0.001 45 40 35 30 25 20 15 10 5 0 0.0001 0.001 0.01 0.1 1 γ (%) Figure 5-15 Example calibration of UBCHYST constitutive model to G/G max and damping ratio curves. 77 Chapter 5 – Analysis Procedures Overview Figure 5-16 Comparison in response between laboratory simple shear test and UBCHYST simulation for case with static bias (note that the scales on plots are not the same) 78 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach 6 PROPOSED COMBINED EFFECTIVE STRESS – TOTAL STRESS APPROACH 6.1 Introduction As discussed in Chapter 5, fully coupled effective stress numerical analyses have been developed (Naesgaard and Byrne 2007; Naesgaard et al. 2009). These can model the triggering of liquefaction, the cyclic mobility deformations that occur during shaking, and simulate the pore pressure migration and trapping of water under low permeability barriers as observed in the field. They have also captured the behaviour observed in centrifuge test case histories. However, there are some numerical difficulties and potential errors, including localization-element size effects, and post-liquefaction consolidation problems, as discussed in Section 5.3.5. There is also often limited information, and, therefore, uncertainty regarding soil stratigraphy, soil properties, and whether key aspects are being picked up by the numerical model. As a tool for use in current design practice, it is proposed to augment the UBCSAND coupled effective stress analyses with a post-shaking total stress analysis which uses liquefied (residual) strengths back-calculated from case histories. With this approach, both the state-of-art effective stress analyses and evidence from past case histories are considered for design. The analyses initiate as a coupled effective stress analysis that is run up to the end of earthquake shaking. At that point, the analysis is bifurcated with one branch being a continuation of the effective stress analysis and the other being switched to a total stress analysis where zones that have been triggered to liquefy in the effective stress analysis are given a residual strength derived from back-analysis of case histories. A flow diagram illustrating the procedure is presented in Figure 6-1. If desired, post-liquefaction consolidation settlement based on published correlations with SPT N-value ((N 1 ) 60cs ) or CPT tip resistance (Q c1cs ) can be added to the procedure. 6.2 Implementation of Procedure in FLAC The proposed numerical process is as follows: (1) The model is built and brought to static equilibrium, initially using a Mohr Coulomb constitutive model, within the program FLAC. 79 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach (2) Potentially liquefiable soil zones in the model are changed to the UBCSAND effective stress model and non-liquefiable zones are changed to the non-linear total stress UBCHYST constitutive model. The overall model is taken to equilibrium, first with the FLAC static configuration and then with the FLAC dynamic configuration. Drained strengths within granular soils and undrained strengths within fine grained soils are calculated and saved for later use. Then a dynamic coupled effective stress analysis is carried out until the end of strong earthquake shaking. During the analyses, zones where liquefaction has been triggered (pore pressure ratio R u (u/σ' vo ) > 0.70 to 1.0) are tracked. Following the end of strong earthquake shaking, a data file is saved (FILE A). Following this, the analyses bifurcate and are continued as two separate cases (3a and 3b on Figure 6-1). (3a) On Path 3a the coupled effective stress analysis is continued (until motion has stopped and pore pressures have dissipated) to a level where flow failure from further pore water / void redistribution is unlikely. This analysis should bring insight into earthquake induced deformations during and following shaking and any related localization and potential post-liquefaction flow based on the assumed stratigraphy. (3b) On Path 3b, the model is changed back to a total stress Mohr Coulomb model but maintained in dynamic mode. Dilation and pore-fluid modulus are set to zero so further shear induced pore pressure changes will not occur. The shear and bulk modulus are left at the same value as they were in the last iteration of the previous effective stress analysis. Elements that are deemed to have liquefied (R u > 0.70 to 1.0) during step (2) are given a ‘residual’ strength that is derived from the backanalysis of case histories and is a function of SPT (N 1 ) 60cs and possibly also vertical effective stress. For example, using Idriss and Boulanger 2008 equation (81); if an element has an SPT (N 1 ) 60cs-Sr = 12 and an initial vertical effective confining pressure of 80 kPa then: S r /σ' vo = exp(12/16 + ((12-16)/21.2)3 – 3) ≤ tan 33 = 0.105 ≤ 0.65 and, therefore, S r = 0.105 x 80 kPa = 8.4 kPa 80 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach where: Sr = “residual” post-liquefaction strength σ' vo = Initial (pre-shaking) vertical effective stress (N 1 ) 60cs-Sr = SPT N value normalized and fines corrected in accordance with Table 4 in Idriss and Boulanger 2008 If the element is liquefied during step (2) then the cohesion (in this dissertation, cohesion refers to the cohesion intercept in the Mohr Coulomb model) in the element is set to 8.4 kPa and friction angle is set to zero. If desired a factor of safety can be applied to the “residual” strength by dividing it by the factor of safety. Other, non-liquefied, elements, if fine-grained, are given their undrained shear strengths or undrained shear strengths times a reduction factor to account for the effects of earthquake shaking (i.e. cohesion=0.8·Su and friction angle=0). Nonliquefied granular soils are given zero cohesion and a friction angle equal to the peak friction angle. However, if the pore-pressure in an element is less than the initial pre-shaking static pore pressure then the elements are given an undrained strength equal to the pre-shaking drained strength (saved at the end of step (2)). The Path 3b analysis is continued (in dynamic mode, but with zero earthquake input) until deformations have stopped and static equilibrium is reached. If deformations are minimal during this phase then flow failure with traditional residual strengths is deemed not to occur, whereas, if deformations are large, the analyses give an indication of the potential consequences of liquefaction and flow failure. Analysis 3b runs relatively quickly and can easily be repeated with alternative ‘residual’ strength parameters. (4) The results from both Path 3a, the coupled effective stress analysis, and Path 3b, the combined effective stress/total stress analysis, should be considered for design. (5) If desired, post-liquefaction consolidation settlement can be included in the model in an approximate manner by using correlations between post-liquefaction volumetric strain (ε v ) and SPT (N 1 ) 60 such as those developed by Wu (2002), Tokimatsu and Seed (1987), or Ishihara and Yoshimine (1992). Generally, the additional 81 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach volumetric strain is only induced in elements that have liquefied. Two alternative procedures similar to that given by Beaty (2001) are proposed. The two procedures are similar, excepting that in one procedure, the desired volumetric strain is forced into the liquefied element, and in the other, only the strain potential is enforced. In an infinite slope with horizontal layers, the two procedures should give the same volumetric strain; however, when only a local pocket liquefies, the strain potential procedure will give less volumetric strain. The proposed implementation is as follows: (i) At the end of the Path 3b (or 3a) analysis (Figure 6-1) the model is changed to a Mohr Coulomb model, as in step 3b (a file saved at the end of the step 3b can be used). FLAC can also be changed from dynamic to static configuration. (ii) The volumetric strain (ε v ) is then calculated for elements that are deemed to have liquefied (R u > 0.70-1.0) using published correlations as indicated above. An example is equation 6-1 below from Yoshimine et al. (2006) as given by Idriss and Boulanger (2008). [6-1] (ε v ) = 1.5 · exp(-0.369· √((N 1 ) 60cs )) · min(0.08,ϒ max ) where: (ε v ) = volumetric strain (N 1 ) 60cs = normalized and corrected SPT N-value ϒ max = maximum shear strain during earthquake shaking (iii) The pore pressures in the model are set to their pre-earthquake static equilibrium values, pore fluid modulus is set to zero to simulate drained conditions, and shear and bulk moduli are set to the values from the last previous iteration of the previous effective stress analysis. (iv) A vertical stress decrement that will be used to generate the desired volumetric strain is calculated. When a FLAC model is in static equilibrium, changing the element modulus does not induce any strain. To get strain there 82 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach has to be a change in stress. When there is a change in stress then an element will strain in accordance with the current modulus, as FLAC is stepped back to equilibrium. Thus, the calculated volumetric strain can be induced in the liquefied elements by dropping the vertical total stress and letting FLAC return (step) to equilibrium. If this is carried out in one single decrement there is a large dynamic inertia generated and the strain will overshoot the desired value. To minimize this dynamic effect several smaller decrements of stress with associated stepping to equilibrium are carried out rather than one large decrement. The vertical stress decrement to be applied to each liquefied element is calculated as follows: [6-2] ∆σ V = (ε v ) · M / I · $switch where: M = constrained modulus = K + 4/3 G K = elastic bulk modulus G = elastic shear modulus εv = desired volumetric strain in the liquefied element according to published correlation σV = vertical stress in FLAC model (compression is negative) ∆σ V = decrement of vertical stress to apply in FLAC (a positive number because compression is negative in FLAC) I = number of iterations to be applied in FLAC as necessary to minimize dynamic inertia (i.e. between 10 and 100). $switch = a conditional value that is either 1 or 0. It is one when either (i) the volumetric strain has not reached the desired ε v or (ii) when number of steps completed is less than ‘I’. If the preceding conditions are met $switch is set to zero. Condition (i) is used if it is desired to enforce a specific strain amount into all the liquefied elements, and condition (ii) is used if it is only desired to enforce the strain potential into the elements. 83 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach (v) A numerical loop consisting of applying the vertical stress decrement, stepping to equilibrium, and checking the $switch condition, is repeated until all values of $switch are zero. At that point, the desired volumetric strain or volumetric strain potential, depending on the condition used for $switch, will be induced in all elements that were deemed to have liquefied. 6.3 Discussion As indicated in Section 6.1, the combined effective stress total stress analysis is undertaken due to the possibility that the effective-stress analysis alone may not capture the full response. The total-stress analysis on its own also is fraught with uncertainty; however, it is tied to actual field case histories and is accepted by the engineering community as current design practice. When the two methods are combined, the envelope of results should be more conservative and chances of capturing the correct response improved. In the author’s opinion, the coupled effective stress dynamic analyses is the state-ofthe-art and for slopes subjected to earthquake shaking and, generally, provides the most reliable estimates of deformation available. The procedure also provides much insight into potential mechanisms. When the earthquake record is provided and actual stratigraphy is as indicated in the model, then the predicted deformation (for cases where there is no flow failure) is probably within a range of one-half to double the actual deformation. Class ‘A’ predictions of centrifuge test deformations have been in this order. For cases with flow failure, the deformation may not be as reliable mainly due to “bad geometry” modelling problems. Adding the “post-shaking” with “residual” strength in liquefied zones adds some confidence to the results and it is suggested that this check should always be carried out. For cases where the consequences of flow failure are severe, it may be prudent for the designer to also consider the probability of getting a lower bound of the case history “residual” strength data. One option is to progressively reduce the “residual” strength or strength ratio used in the analyses until failure occurs (or the “residual” strength goes to zero). If failure occurs then one needs to assess whether the probability of getting that value is acceptable. It should be noted that the proposed total stress residual soil strengths were originally developed using limit equilibrium slope stability methods and would not necessarily be the same had they been calculated using the numerical methods as proposed. 84 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach The differences would be mainly due to (i) deformations during shaking that may reduce the driving stresses that may cause failure in the post-shaking total stress analysis, and (ii) pore water pressures in non-liquefied zones may differ from the assumptions made when the residual strengths were derived. However, there is such large scatter or variation in the backcalculated strength parameters that the differences are not deemed to be significant. If deformations during shaking are very large, then consideration could be given to using the pre-earthquake geometry for the post-liquefaction residual strength analysis. The simplified empirical post-liquefaction consolidation strains are believed to be representative of level ground conditions and the calculated settlements likely over-estimate settlements for conditions where there is a static bias. Incorporating these in the analysis process as discussed in Section 6.2, Item (5) for cases where there is a significant static bias (a relatively common situation) is expected to give conservatively large post-liquefaction consolidation settlements. Forcing the total post-liquefaction strain (ε v ) into each liquefied element is expected to give a conservative estimate of post-liquefaction differential settlement. Enforcing only the strain potential may give more realistic differential settlements. However, when using only strain potential, the resulting differential settlements are sensitive to the chosen soil modulus and strength of the liquefied soil, and care must be taken in choosing these parameters. Selecting elements deemed to have liquefied by monitoring the pore pressure ratio (R u = ∆u dyn /σ' vo ) is not always elementary. Sometimes, there are pore pressure spikes that will cause an excessively high R u for a very short duration that are not representative of the general state of pore pressure within the element, and, sometimes, there are large deformations that cause the initial vertical effective stress (σ' vo ) not to be representative of the current depth and stress state of the element. To mitigate these problems, a dual check, using both the maximum pore pressure ratio during shaking (R u(max) ) and the pore pressure ratio at the end of the earthquake shaking (R u(current) ) is used. Elements having both R u(max) and R u(current) above specific thresholds are deemed to have liquefied. Typically, the R u(max) threshold is set between 0.7 and 0.9 and the and R u(current) is set between 0.5 and 0.7. 85 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach 6.4 Chapter 6 Summary and Conclusions Dynamic numerical analyses using the UBCSAND constitutive model as described in Chapter 5 may be the state-of-the-art in predicting liquefaction induced deformations. However, as indicated in Section 5.3.5 there are still shortcomings. In addition to the modelling shortcomings, there is also the difficulty of characterizing the soil stratigraphy with accuracy at any given real site. As indicated in Chapter 4, the use of these case history derived post-liquefaction strength is current engineering practice, even though there are large uncertainties in the back-calculated strengths. The reason the relatively inaccurate field derived strengths are used is because there are no easy to use readily available alternatives. When considering all this, it was proposed that a good design should consider both approaches, and the hybrid effective stress – total stress design approach described in this Chapter is the result. A description of the procedure and step-by-step details on implementing it are given in this Chapter. A limitation on the UBCSAND model is that bulk modulus is elastic without any volumetric yielding cap. This results in an underestimate of post-liquefaction consolidation settlement. To compensate for this, a procedure for implementing empirically derived post-liquefaction consolidation settlements is also given in this Chapter. An example analysis illustrating the application of the procedure is given in Appendix B. 86 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach FIGURES Chapter 6 Proposed Combined Effective Stress – Total Stress Approach 87 Chapter 6 – Proposed Combined Effective Stress – Total Stress Approach Figure 6-1 Flow chart showing combined coupled effective stress / total stress analysis procedure (Naesgaard and Byrne 2007). 88 Chapter 7– Model Calibration, Validation and Application Example 7 MODEL CALIBRATION, VALIDATION and APPLICATION EXAMPLE 7.1 Introduction An important aspect of numerical modeling is calibrating and verifying the response of the model. The UBCSAND effective stress constitutive model has been calibrated both to replicate the behaviour of undrained simple shear laboratory tests and replicate the liquefaction triggering response implied by the case-history derived liquefaction triggering design charts. The model and proposed analysis methodology has also been verified by comparing the numerical results against those of triaxial tests with inflow (Atigh and Byrne 2004), shake table tests, centrifuge tests, and field case histories. 7.2 Calibration to Laboratory Cyclic Simple Shear Tests 7.2.1 Introduction A key part of developing the UBCSAND model has been making it simulate the behaviour observed in monotonic and cyclic simple shear tests. This is both for the purpose of verifying that UBCSAND can simulate the element behaviour observed in the laboratory and for calibrating post-liquefaction triggering response for use in design. As part of the research of this dissertation, calibrations have been carried out to show that UBCSAND can simulate the behaviour observed in the laboratory testing program by Sriskandakumar (2004). These tests were part of the UBC Liquefaction Initiative (NSERC Grant 246394), a research program of laboratory and centrifuge testing and numerical modeling carried out jointly by the University of British Columbia and C-CORE, Newfoundland. The Sriskandakumar (2004) test program included undrained monotonic, undrained cyclic, and drained cyclic tests on loose sands (Relative Density (D r = 38-44%) and dense sands (Relative Density 80-81%). The tests were carried out on an NGI-type simple shear device. The same Fraser River Sand and air pluviation preparation method being used in the centrifuge tests at C-CORE, was used for the laboratory testing program. Fraser River sand is fluvial sand deposited near the mouth of the Fraser River. The grains are angular to sub-rounded and are composed of approximately 40% quartz, 11% feldspar, 89 Chapter 7– Model Calibration, Validation and Application Example and remainder other minerals and rock fragments (Sriskandakumar 2004). The natural sand typically has approximately 4-10% silt content; however, the fines were removed for the laboratory and centrifuge testing. Figure 7-1 shows a typical grain size curve of the tested material. Numerical modeling of select tests from the work by Sriskandakumar (2004) was undertaken using the UBCSAND1ver2 constitutive model. A FLAC ‘fish’ code for the model is in the attached Appendix. A single element was numerically exercised with the same loading path as the laboratory test. The results were then compared, calibration parameters within the numerical model were adjusted, and the process repeated until a reasonable match between the laboratory and numerical response was achieved. 7.2.2 Results and Discussion Figures 7-2 to 7-10 illustrate comparisons between the numerical and laboratory tests. Table 7-1 summarizes the properties, calibration factors, and cycles to liquefaction from the laboratory tests and numerical analyses. The calibration factors were generally fixed for tests with similar initial conditions, and factors were only changed when known initial conditions changed. It is important to note that calibration factors can be a function of or vary with known initial conditions, such as, initial confining pressure or initial relative density but should not be functions of parameters (such as, cyclic stress ratio) that are not known at the on-set of earthquake shaking. As shown on Figure 7-7 and Table 7-1, the UBCSAND model (as calibrated) tended to trigger liquefaction later than the laboratory tests at high cyclic stress ratios and trigger earlier at lower stress ratio. Increasing the initial elastic shear modulus parameter m_kge improved the ability to calibrate the model to match the laboratory test results at a range of cyclic stress ratio. It is noted in Figure 7-7 that, for a given stress ratio, liquefaction resistance increases with confining pressure. This is believed to be due to stress densification causing the higher relative density with increase in confining pressure. This seems to override the k σ effect which typically would cause liquefaction resistance (CRR) to decrease with increasing confining pressure. When calibrating to the laboratory data with varying confining pressures the use of a k σ factor is not required as the UBCSAND/FLAC model and laboratory test are using the same boundary stresses and relative density. 90 Chapter 7– Model Calibration, Validation and Application Example Figures 7-5 and 7-6 show the L12 loose sand sample with a significant initial static bias. If the same calibration that was used for the tests without initial static bias is used for this test the results are as indicated on Figure 7-5 and liquefaction is triggered too early; however, the post-triggering response is reasonable. Adding an additional calibration factor (m_hfac7) which adjusts the pull-down of the yield envelope prior to liquefaction triggering) in addition to m_hfac6 which adjusts the pull-down of the yield envelope following “liquefaction” triggering, allowed a better match to the data as shown in Figure 7-6. The model pore pressure response and amount of permanent strain per cycle (marching) is very sensitive to the amount of yield envelope pull-down allowed. Thus, while earlier versions of UBCSAND, which do not allow for pull-down of the yield envelope, would under-predict pore pressure build-up when there was no cross-over, the current UBCSAND1v02, which pulls down the yield envelope on unloading and then generates significant pore pressures on reloading may trigger liquefaction prior to that indicated by the laboratory tests. Figures 7-8 to 7-10 shows the simulation by the current model of a cyclic simple shear test on dense sand. Earlier versions of UBCSAND would tend to be too stiff prior to liquefaction triggering and too soft following triggering. Modifications to the model, including increasing peak friction angle, lowering of the phase transformation friction angle, dilation delay following post-liquefaction cross-over, as described in Sections 5.3.4.1, 5.3.4.2, and 5.3.4.3, allows the model to cyclically soften gradually as observed in the laboratory tests. 7.3 Calibration to “Seed type” Empirical Liquefaction Triggering Charts Determining liquefaction triggering response from site specific laboratory tests is often not possible or impractical due to sample disturbance problems. It is common practice, therefore, to rely on empirical “Seed type” (Seed et al 1984; NCEER 1997; Cetin et al. 2004; Idriss and Boulanger 2008) empirical liquefaction triggering charts and in-situ testing such as Standard Penetration Test N-value or cone penetration test (CPT) tip bearing (Q c1cs ) values when assessing liquefaction triggering. This is also the recommended procedure for use with UBCSAND; however, for post-triggering behaviour and behaviour associated with a static bias, reliance is still placed on laboratory tests. 91 Chapter 7– Model Calibration, Validation and Application Example In the calibration process, a single undrained soil element is exercised so as to trigger liquefaction in the correct number of cycles and to give post-liquefaction stress-strain behaviour consistent with that observed in laboratory simple shear tests. The recommended calibration procedure is as follows: (1) Set up the 2D FLAC profile and bring it to static equilibrium. Generally, there will be a range of soil types, confining pressures, k o values, static bias, and soil densities (density is often represented by penetration resistance of in-situ tests). The model is then broken into zones with similar properties (effective stresses, G max and (N 1 ) 60 , Q t , etc.) and representative soil elements from the cohesionless zones are then selected from each zone for calibration. The vertical and horizontal effective confining pressure, small strain shear modulus, and density (generally, normalized and fines corrected SPT N-value (N 1 ) 60-CS or normalized and fines corrected CPT tip resistance (Q c1cs )) are recorded for each element to be calibrated. (2) An undrained single element model is set up in FLAC and is initialized with the representative vertical and horizontal effective confining pressure, small strain shear modulus, and (N 1 ) 60-CS . (3) A cyclic shear stress (τ xy ) compatible with a cyclic resistance ratio (CRR) that will liquefy (pore pressure ratio (R u ) near 1.0 or shear strain ( ϒ = 3.75% )) in 15 cycles (CRR 15 ), from an empirical liquefaction triggering chart, is then applied to the element to be calibrated. For example, the chart and equations by Idriss and Boulanger, 2008 as shown in Figure 5-4 may be used and τ xy calculated as follows: τ xy = σ vo ' * CRR 15 * k σ where: τ xy = applied cyclic shear stress σ vo ' = vertical effective stress CRR 15 = cyclic resistance ratio from chart or equation kσ correction factor for confining pressure (Idriss and Boulanger = 2008) (4) The single element is then repeatedly cyclically loaded with the τ xy from step (3) and calibration parameters are adjusted until the element liquefies in 15 cycles and 92 Chapter 7– Model Calibration, Validation and Application Example the post-liquefaction stress-strain cycles are compatible with typical laboratory simple shear tests. This process is then repeated for all the elements to be calibrated and a matrix of calibration parameters (m_hfac1 to m_hfac4 or m_hfac6) for the model is obtained. (5) The calibration parameters or equations representing the parameters (as functions of effective confining pressure, penetration resistance, etc.) are then introduced as material parameters within the applicable zones in the larger 2D model and the dynamic analysis continued. Figure 7-11 compares UBCSAND element data to empirical design triggering charts from calibrations by Beaty (2009), Beaty and Byrne (2011). 7.4 One-Dimensional Tests for demonstrating Void Redistribution Infinite slope one-dimensional numerical analyses are useful for developing insights into the behaviour of the low permeability barrier and flow slide mechanisms. Figure 7-12 (Naesgaard et al. 2005) illustrates a typical one dimensional column analysis with typical volumetric strain time histories at various locations within the column. Figure 7-13 shows a displaced grid with velocity time histories above and below the barrier. Note how a flow slide or flow failure condition is initiated (increasing velocity) at time x-x’. This is the critical state when shear induced dilation goes to zero. Shear strain is concentrated (localized) immediately below the low permeability barrier, and the flow failure is independent of the inertial forces from strong shaking. Seid-Karbasi (2008) conducted a detail numerical study of the infinite slope (one dimensional) column of sand soil underlying a barrier. Seid-Karbasi (2008) noted that the upper approximately one-third thickness of the sand layer (Zone E Figure 7-12) tended to be expansive due to inflow, whereas, the soil below this was contractive (Zone C Figure 7-12). However, as discussed in Section 2.2, for level ground conditions this may not be the case and the whole soil column may be contractive excepting for the water layer directly under the barrier. Seid-Karbasi (2008) showed by analyses that drainage columns installed through the low permeability barrier were an effective mitigation measure against post-liquefaction flow failure. However, even though the drains prevented flow failure, some movements still occurred during strong 93 Chapter 7– Model Calibration, Validation and Application Example shaking. Taiebat (2008) also was able to generate post-liquefaction post-shaking deformations in a 1-D columns using the effective stress program SANISAND. 7.5 George Massey Tunnel Centrifuge Test Modeling Dynamic numerical analyses were used for the seismic retrofit design of the forty four yearold immersed-tube George Massey Tunnel. The 1.3 km long tunnel carries four lanes of traffic under the Fraser River just south of Vancouver, British Columbia. Design criteria were that the retrofitted tunnel should withstand both a 0.25g magnitude 7.0 non-subduction earthquake, and a 0.15g magnitude 8.2 distant subduction earthquake without collapse or loss of life, but with possible damage to a repairable level including controllable water leakage. Soil liquefaction, its consequences, and mitigation were the key design challenges in this project. Two-dimensional dynamic analyses using the program FLAC were a prime geotechnical analyses and design tool. Displacements from the numerical analyses were used as input into three-dimensional static structural analyses using non-linear soil springs and nonlinear moment-curvature section properties. The structural analyses were used to assess and mitigate potential cracking in the tunnel. In the 2D FLAC analyses, transverse and longitudinal sections were studied using total and effective stress constitutive models (UBCTOT and UBCSAND) developed at the University of British Columbia. Dynamic shaking, liquefaction triggering, consequences of liquefaction and soil-structure interaction were addressed in each of the models. Analyses were carried out with and without retrofit measures. A centrifuge-testing program was carried out to check the numerical model. Pre-test (Class ‘A’) predictions of the centrifuge test showed good agreement between the FLAC/UBCSAND numerical model and centrifuge test results. The geotechnical design considerations, numerical modeling and centrifuge test program are discussed in detail in Yang et al. (2005), Naesgaard et al (2004), Yang et al, (2003), and Adalier et al (2003). Figure 7-14 shows a transverse cross-section of the approximately one kilometre long tunnel, Figure 7-15 shows an example numerical mesh, Figures 7-16 and 7-17 show calculated displacement vectors from the FLAC/UBCSAND model for sloped river bottom and level river bottom, respectively. Figure 7-18 shows the layout of the centrifuge model. Centrifuge analyses were carried out for three scenarios. One scenario was with no ground improvement around the tunnel, the other was with 10m 94 Chapter 7– Model Calibration, Validation and Application Example wide bands of densification on either side of the tunnel, as indicated in Figure 7-18, and another was with 10m wide drainage zones (zones with same density as remainder of the sand but higher permeability. Measured displacements and accelerations are compared to numerical predictions in Table 7-2 and in Figures 7-19 to 7-20. The numerical analyses of the centrifuge model were carried out in prototype scale. 7.6 UBC/C-CORE Centrifuge Test Modeling A series of eight centrifuge tests were carried out at the C-CORE facility in St. John’s, Newfoundland (Phillips and Coulter 2005; Phillips et al 2004). The centrifuge tests modeled submerged slope configurations with and without: low permeability silt barrier, soil densification dyke, and drainage trenches. Air pluviated Fraser River sand with a relative density of approximately 40% and minimum and maximum void ratio of 0.62 and 0.94 was used (Figure 7-1). Non-plastic commercial ground silica silt was used for the low permeability barrier and clear uniform coarse sand was used for the drainage layers. D 10 and D 50 were 0.16mm and 0.26mm for the loose sand, 0.005mm and 0.016mm for the silt, and 2.2mm and 2.9 mm for the drainage sand, respectively. The centrifuge tests were at 70g with a water plus hydroxypropyl-methylcellulose fluid with a viscosity of 35 times that of water. Simulated earthquake motion was applied during flight using a hydraulically actuated shaker. All dimensions, time histories, etc. given in this section are in the scaled prototype dimensions rather than the actual centrifuge dimensions. During centrifuge spin-up there are large changes in effective stress which result in ‘stress densification’ of loose sandy soils (Park and Byrne 2004). This was accounted for in the analyses. A typical grid and input earthquake record is shown in prototype scale in Figure 7-22. The side forces that would occur within the centrifuge box were accounted for during the dynamic analysis by applying internal nodal forces that were a function of the out-of-plane effective stress (σ΄ z ) times the sidewall friction coefficient. Normalized velocities were used to indicate the direction of the internal nodal forces. Liquefaction flow failure was observed in the tests which included the low permeability silt barrier and higher levels of shaking. Flow failure, generally, did not occur when the barrier was absent or if drainage trenches were placed through the barrier. Figure 7-23 compares centrifuge and numerical results for the COSTA-C test (Phillips and Coulter 95 Chapter 7– Model Calibration, Validation and Application Example 2004) that included a low permeability barrier. In the COSTA-C test, a flow slide occurred at the barrier interface at approximately 50s after end of strong shaking. Figure 7-24 shows displaced profiles for a similar model, CT5 (Phillips et al 2004) that had permeable drainage slots through the silt barrier. With drainage slots (Figures 7-24 and 7-25) all deformation occurred during strong shaking (t < 20s) and flow deformation is prevented. The response of some of the centrifuge tests was indicative of the centrifuge model not being fully saturated. This was a quandary as P-wave measurements within the centrifuge model were indicative of full saturation. To investigate this, a series of triaxial tests with P-wave measurement apparatus were carried out at UBC. This testing program is discussed in Chapter 8. 7.7 Kokusho Shake Table Emulation Kokusho (1999), Kokusho (2000), Kokusho and Kabasawa (2003), carried out a series of shake table tests on both cylinders of sand (Figure 2-6) and on shake table model soil embankments for the purpose of studying the effects of low permeability barriers and pore water migration on slope stability. Figure 7-26 shows a photograph of one of the tests (from videos at http//:www.civil.chuo-u.ac.jp/lab/doshitu/index.html). The rectangular Lucite shake table soil box was 1100mm long, 600mm wide and 800mm high. The sand was placed at a relative density of approximately 27-34% and the model was shaken in the transverse direction only to minimize inertial forces on the slope. Two thin horizontal non-plastic silt barriers were placed in the slope as shown in the figure (Kokusho and Kabasawa, 2003). Shaking induced pore water pressure build-up within the slope and mild movements as shown in the end-of-shaking profile shown in Figure 7-27; however, at this stage, the model appeared stable. Then, approximately four seconds after end-of-shaking, a flow failure initiated with the effects of pore water interacting with the silt barriers clearly visible. Figure 7-28 shows the post-failure profile. A UBCSAND/FLAC model of a similar test to the shake table test was set up to demonstrate that the numerical models could simulate the failure mechanisms observed in the shake table tests. The FLAC model slope was 16m high instead of 600mm to minimize problems with effective stress defaults in the UBCSAND program. The model mesh with two barriers is shown in Figure 7-29. Dynamic analyses were carried out for three conditions 96 Chapter 7– Model Calibration, Validation and Application Example (1) with low permeability layers or barriers and pore water flow allowed, (2) with low permeability barriers but no flow allowed, and (3) with no barrier but flow allowed. Soil properties used in the numerical model are summarized in Table 7-3. Figure 7-30 shows the deformed grid at the end of the analyses (30 s) for the case with barrier and flow. The localization of shearing under the barrier is clearly visible. Figures 7-31 shows horizontal movement contours for the case with barrier and flow and case with barrier and flow not allowed. The effects of pore water redistribution (flow) and the barrier are clear and also illustrated in Figure 7-32 which shows displacement time histories for the three cases modeled. For all cases, similar deformation occurs during shaking; however, only the case with the barrier and pore water flow allowed deforms significantly after end of shaking. A key parameter in getting the model to flow after end-of-shaking is the dilation cut-off that is added to the UBCSAND to simulate the behaviour when a local zone reaches the critical. In these simulations, dilation was cut-off when the volume of any given element expanded beyond its original volume prior to start of shaking. When this was triggered the dilation was set to a value that was contractive to simulate some mixing occurring between the sand and silt barrier after the soil reaches the critical state. 7.8 Lower San Fernando Dam Example Analysis 7.8.1 Introduction In 1971, the Lower San Fernando Dam (LSFD) was shaken by a large earthquake with a peak velocity pulse of around 0.6m/s and peak ground acceleration of approximately 0.5 g (Seed 1973; Seed et al. 1989; Castro et al. 1989; and Castro 1995). Approximately 20 to 30 seconds after the end of the earthquake shaking, the upstream face of the dam failed catastrophically, leaving only 1.5m of freeboard and placing a large population at risk. Extensive investigation and analyses of the dam were conducted following the event. It was concluded from the studies that the hydraulic fill soil within the lower and central portions of the dam had liquefied and overlying portions of the dam had flowed out riding on the liquefied soil (Figure 7-33). In the current example, the liquefiable portions of hydraulic fill soils were given (N 1 ) 60 values of 12 while the clayey core of the dam was given undrained shear strength of 18% of the initial vertical overburden pressure (Figure 7-34). The UBCSAND constitutive 97 Chapter 7– Model Calibration, Validation and Application Example model was used for the potentially liquefiable sandy soil portions of the dam while a Mohr Coulomb model was used for the clayey core and portions of the dam above the water table. The effective stress analyses were fully-coupled (mechanical - pore water flow) and run within the program FLAC. In the actual dam, the spacing of the low permeability layers is much finer than in the numerical model. Limitations on smallest element size are necessary for analysis time to be in a practical range and the behaviour of finely spaced layers, therefore, must be simulated with larger elements by increasing the distance between silt barrier layers and by introducing a dilation cut-off in any elements expanding beyond their original volume. 7.8.2 Coupled Effective Stress Analysis of LSFD – Phases 2 and 3a In the example LSFD analysis, the coupled effective stress analysis was carried out using the UBCSAND constitutive model within the program FLAC as described in Section 5.3 for all saturated sand elements while a total stress Mohr Coulomb model was used for the clay/silt core portion of the dam. Initially, the whole model was brought to static equilibrium using a Mohr Coulomb model and drained stiffness parameters (Phase 1). Then prior to earthquake shaking, the sand elements were changed to the UBCSAND model while portions of the clay/silt core were left with a Mohr Coulomb model but given undrained stiffness parameters. The model was then brought to equilibrium in dynamic mode and all displacements reset to zero (end of Phase 1). Following this, the dynamic analysis with earthquake motion input at the base was run until excessive grid distortion stopped the analysis (≈ 40 s). This constitutes Phase 2 and 3a in Figure 6-1. A file was saved at after the end of strong shaking (15 s) for use in the Phase 3b Total Stress Alternate Analysis. 7.8.3 Post-Liquefaction Total Stress Analysis of LSFD – Phase 3b Elements which had liquefied (deemed to occur when pore pressure ratio (R u ) max > 0.7)) during the strong shaking portion of the analysis (Phase 2) where changed to a Mohr Coulomb model with zero friction angle, zero dilation, and a cohesion set equal to a residual strength (S r ) calculated using the initial pre-shaking effective stress (σ') and the S r /σ' ratio 98 Chapter 7– Model Calibration, Validation and Application Example recommended by Idriss and Boulanger (2007) (Figure 4-1(d)). The dynamic analysis was then continued until stopped by excessive grid distortion (≈ 18 s). 7.8.4 Results of LSFD Example Analysis During strong shaking (Phase 2), the upstream shell moved approximately 3m upstream and the downstream shell moved approximately 1.5m downstream. Liquefaction occurred both within the upstream and downstream portions of the dam as shown in Figure 7-35. Strong shaking ended at approximately 10 seconds (Figure 7-36). Continuing the coupled effective stress analysis (Phase 3a in Figure 6-1) resulted in very little movement within the dam until approximately 32 seconds at which time the upstream shell proceeded to fail in a similar manner to that reported for the actual dam (Figure 7-36). Figure 7-37 shows displacement and velocity of the dam at 40 seconds at which time the program stopped due to excessive distortion of some of the elements. The downstream face did not move significantly following end of strong shaking as was the case for the actual dam. In the Phase 3b total stress alternative, the dam proceeded to fail once the liquefied elements where changed to a Mohr Coulomb model with residual strength S r as given by the Idriss and Boulanger (2008) relationship. This was not unexpected as the Lower San Fernando Dam was one of the key case histories used in deriving the relationship. Again, the dam only failed in the upstream direction as was the case with the actual dam. 99 Chapter 7– Model Calibration, Validation and Application Example TABLES Chapter 7 Model Calibration, Validation and Application Example 100 Chapter 7– Model Calibration, Validation and Application Example Table 7-1 TEST Summary of Undrained Laboratory Simple Shear and UBCSAND Simulations σ'vo DR CSR τST/σ'vo kPa Note 2 100 100 102 100 200 200 200 200 50 50 50 100 100 100 (%) Note 3 40 40 40 40 44 44 44 44 38 38 38 40 40 80 (-) Note 4 0.08 0.1 0.12 0.15 0.08 0.1 0.12 0.15 0.08 0.1 0.12 0.065 0.065 0.35 (-) Note 5 0 0 0 0 0 0 0 0 0 0 0 0.1 0.1 0 No. Note 1 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L12c D3 Notes: 12345- UBCSAND1v02 CALIBRATION PARAMETERS NLIQ LAB (-) UBCSAND (-) Note 6 17.0 16.0 6.5 7.5 2.5 4.0 1.0 2.0 33.0 15.0 7.5 7.0 3.5 4.0 1.0 2.0 12.0 12.0 3.0 6.0 1.5 3.5 15.5 4.0 15.5 12.0 8 8 hfac1 (-) hfac2 (-) hfac3 (-) hfac4 (-) hfac5 (-) hfac6 (-) 0.65 0.65 0.65 0.65 0.40 0.40 0.40 0.40 0.30 0.30 0.30 0.65 0.65 0.05 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.20 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.80 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 3.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.60 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 1.00 Test numbers refer to tests by Sriskandakumar (2004), Chapter 4. "L" signifies loose and "D" signifies dense samples. Initial effective vertical stress DR= Relative density CSR=Ratio of single amplitude shear stress to the initial effective vertical stress TST=Static Shear Bias 6- NLIQ= Number of cycles required to reach pore pressure ratio near 1 for loose samples or number of cycles required to reach 3.75% shear strain for dense samples. 7- For test L12c and additional hfac7 was added to control pretriggering pull-down of yield envelope Table 7-2 Massey Tunnel Project. “Class A” Numerical Predictions compared to Centrifuge Test Results (in prototype scale) Output Parameter Peak tunnel heave (m) Peak tunnel lateral movement (m) Maximum soil displacement (m) Peak tunnel horizontal acceleration (g) Note: Scaling Factor (model:prototype) Model 1(1) Model 2(2) Model 3(3) 1:100 1:100 1:100 100:1 0.25 (0.27) 0.59 (0.68) 1.52 (1.50) 0.10 (0.11) 0.13 (0.14) 0.50 (0.35) 1.20 (1.30) 0.13 (0.08) 0.12 (0.04) 0.40 (0.30) 1.03 (1.10) 0.095 (0.095) Numbers given in brackets are the results from the centrifuge tests. (1) No ground improvement – loose sand around tunnel (2) 10m wide densification on each side of tunnel (3) 10m wide drainage zone on each side of tunnel 101 Chapter 7– Model Calibration, Validation and Application Example Table 7.3 Soil Properties used in simulation of Kokusho Shake Table Tests Constitutive Dry model Density φcv φf (tonnes/m3) SAND UBCSAND SILT BARRIER MOHR COULOMB cohesion (N1)60cs (kPa) blows/ft Bulk modulus shear modulus Porosity Permeability Remarks Kge Gmax (kPa) G (kPa) B (kPa) n k cm/s 1.5 33 34 0 10 15 (1) - 2 Gmax 0.45 5x10-3 1.45 - 25 0 12 12.5 - (2) 2G 0.55 1x10-7 (3) (4) (1) Gmax = 21.7 x Kge x ((N1) 60cs)0.33 x Pa x (σ'm/Pa) 0.5 where Pa = 100 kPa and σ'm = mean normal effective stress in kPa (2) G = (21.7 x Kge x ((N1) 60cs)0.33 x Pa x (σ'm/Pa) 0.5) / 5 (3) UBCSAND calibration factors for SAND layers: m_hfac1 = 0.5, m_hfac2 = 1.0, m_hfac3 = 1.0 (used UBCSAND VERSION 15oo.FIS) (4) Dilation cut-off in UBCSAND model: When incremental volumetric strain since start of shaking = 0.0 dilation m_dt set ≥ 0.001 gave 4m displ & if set to 0.25 gave 7m disp (contractive dilation angles used to account for mixing) 102 Chapter 7– Model Calibration, Validation and Application Example FIGURES Chapter 7 Model Calibration, Validation and Application Example 103 Chapter 7– Model Calibration, Validation and Application Example Figure 7-1 Typical grain-size distribution and microscope view of Fraser River Sand used for cyclic shear tests (from Sriskandakumar 2004). 104 Chapter 7– Model Calibration, Validation and Application Example 1 Lab data ∆u/σVC' 0.8 UBCSAND1v02 0.6 0.4 (a) 0.2 0 0 2 4 6 8 10 12 14 16 18 20 NO OF CYCLES (b) 30 Shear Stress, τ (kPa) Lab data FLAC σ'vc=100kPa; Drc=40% τcyc/σ'vc=0.08; τst/σ'vc =0.0 20 10 0 -15 -10 -5 0 5 10 15 -10 -20 (c) -30 Shear Strain, γ (%) Figure 7-2 Comparison of undrained simple shear test L1 laboratory test to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response. 105 Chapter 7– Model Calibration, Validation and Application Example ∆u/σVC' 1 0.8 Lab data 0.6 UBCSANDv02 0.4 0.2 0 0 2 4 6 8 10 NO OF CYCLES 30 Shear Stress, τ (kPa) Lab data UBCSAND1v02 σ'vc=100kPa; Drc=40% τcyc/σ'vc=0.10; τst/σ'vc =0.0 20 10 0 -15 -10 -5 0 5 10 15 -10 -20 -30 Shear Strain, γ (%) Figure 7-3 Comparison of undrained simple shear test L2 laboratory test to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response. 106 Chapter 7– Model Calibration, Validation and Application Example 1 (a) ∆u/σVC' 0.8 0.6 0.4 Lab data UBCSAND1v02 0.2 0 0 1 2 3 4 5 6 NO OF CYCLES (b) 30 Shear Stress, τ (kPa) Lab data UBCSAND1v02 σ'vc=102kPa; Drc=40% τcyc/σ'vc=0.12; τst/σ'vc =0.0 20 10 0 -15 -10 -5 0 -10 5 10 15 (c) -20 -30 Shear Strain, γ (%) Figure 7-4 Comparison of undrained simple shear test L2 laboratory test to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response. 107 Chapter 7– Model Calibration, Validation and Application Example 1 ∆u/σVC' 0.8 0.6 0.4 Series1 (a) 0.2 UBCSAND1v02 0 0 5 10 15 20 NO OF CYCLES 30 25 30 σ'vc=100kPa; Drc=40% τcyc/σ'vc=0.065; τst/σ'vc =0.1 Lab data UBCSAND1v02 Shear Stress, τ (kPa) 20 10 0 0 25 50 100 75 125 Vertical Effective Stress, σ'v (kPa) -10 (b) -20 -30 30 Shear Stress, τ (kPa) σ'vc=100kPa; Drc=40% τcyc/σ'vc=0.065; τst/σ'vc =0.1 20 10 0 -15 -10 (c) -5 0 5 10 15 -10 -20 Lab data UBCSAND1v02 -30 Shear Strain, γ (%) Figure 7-5 Comparison of Undrained simple shear test L12 laboratory test with initial static bias to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response – using same calibration parameters as those used without static bias. 108 Chapter 7– Model Calibration, Validation and Application Example 1 (a) ∆u/σVC' 0.8 0.6 0.4 Series1 0.2 UBCSAND1v02 0 0 5 10 15 20 NO OF CYCLES 30 25 30 σ'vc=100kPa; Drc=40% τcyc/σ'vc=0.065; τst/σ'vc =0.1 Lab data UBCSAND1v02 Shear Stress, τ (kPa) 20 10 0 0 25 50 125 Vertical Effective Stress, σ'v (kPa) -10 (b) -20 -30 30 σ'vc=100kPa; Drc=40% τcyc/σ'vc=0.065; τst/σ'vc =0.1 Shear Stress, τ (kPa) 100 75 20 10 0 -15 -10 -5 0 5 10 15 -10 (c) -20 Lab data UBCSAND1v02 -30 Shear Strain, γ (%) Figure 7-6 Comparison of undrained simple shear test L12 laboratory test with initial static bias to UBCSAND simulation (a) pore pressure ratio vs. number of cycles, (b) stress path, and (c) stress strain response – using additional hfac7 calibration parameter to control pulldown prior to triggering. 109 Chapter 7– Model Calibration, Validation and Application Example 0.16 Lab sig'vo=50 kPa, DR=38% 0.15 UBCSAND sig'vo= 50 kPa, DR=38% Cyclic Stress Ratio (CSR) 0.14 Lab sig'vo=100 kPa, DR=40% UBCSAND sig'vo=100 kPa, DR=40% 0.13 Lab sig'vo=200 kPa, DR=44% 0.12 UBCSAND sig'vo=200 kPa, DR=44% 0.11 0.1 0.09 0.08 0.07 0.06 0 5 10 15 20 25 30 35 Number of cycles to Liquefaction, N LIQ Figure 7-7 Comparison of cycles to liquefaction between undrained laboratory simple shear tests and UBCSAND simulations. 110 Chapter 7– Model Calibration, Validation and Application Example 60 Shear Stress, τ (kPa) σ'vc=100kPa; Drc=80% τcyc/σ'vc=0.35; τst/σ'vc =0.0 (a) 40 20 0 -15 -10 -5 0 5 10 15 -20 -40 Point of γ=3.75% (i.e. Assumed triggering point of liquefaction for comparison purposes) -60 Shear Strain, γ (%) (b) Figure 7-8 Undrained dense sand simple shear stress vs. strain comparison of D3 laboratory test to UBCSAND simulation (a) Laboratory test, and (b) UBCSAND simulation. 111 Chapter 7– Model Calibration, Validation and Application Example 60 σ'vc=100kPa; Drc=80% τcyc/σ'vc=0.35; τst/σ'vc =0.0 Shear Stress, τ (kPa) 40 20 0 0 25 50 75 100 125 -20 -40 Point of γ=3.75% (i.e. Assumed triggering point of liquefaction for comparison purposes) -60 Vertical Effective Stress, σ'v (kPa) 60 σ'vc=100kPa; Drc=80% τcyc/σ'vc=0.35; τst/σ'vc =0.0 Shear Stress, τ (kPa) 40 UBCSAND1v02 20 0 0 25 50 75 100 125 -20 -40 -60 Vertical Effective Stress, σ'v (kPa) Figure 7-9 Undrained dense sand simple shear stress path comparison of D3 laboratory test (a) to UBCSAND simulation (b). 112 Chapter 7– Model Calibration, Validation and Application Example 1 ∆u/σVC' 0.8 0.6 0.4 0.2 Lab data 0 0 5 10 NO OF CYCLES 15 20 1 ∆u/σVC' 0.8 0.6 0.4 UBCSAND1v02 0.2 0 0 5 10 15 20 NO OF CYCLES Figure 7-10 Undrained dense sand simple shear pore pressure ratio comparison of D3 laboratory test (a) to UBCSAND simulation (b). 113 Chapter 7– Model Calibration, Validation and Application Example Figure 7-11 Empirical liquefaction triggering chart with overlain UBCSAND simulations (from Beaty 2009; Beaty and Byrne 2011). 114 Chapter 7– Model Calibration, Validation and Application Example ↑ Figure 7-12 Volumetric strain (εv) within an infinite slope column with a low permeability crust over loose sand. Figure 7-13 UBCSAND numerical model of infinite slope (1D) column with low permeability barrier cap. At time x-x' zone 'I' has expanded to the critical state, dilation goes to zero, and flow failure is initiated. 115 Chapter 7– Model Calibration, Validation and Application Example Figure 7-14 Section through Massey Tunnel in Fraser River. Figure 7-15 Typical transverse FLAC model grid used from Massey Tunnel. Scale: the tunnel is 24m wide - vertical and horizontal scale are the same. Figure 7-16 Typical distorted mesh with displacement vectors. Scale: the tunnel is 24m wide vertical and horizontal scale are the same. 116 Chapter 7– Model Calibration, Validation and Application Example Figure 7-17 Typical distorted mesh with displacement vectors for level ground condition. Scale: the tunnel is 24m wide - vertical and horizontal scale are the same. Figure 7-18 Layout of Massey project centrifuge model showing location of instrumentation Model 2 (Legend: a3 = accelerometer; P12 = piezometer; L v and L h = LVDT). Dimensions in centimeters. Model 1 and Model 3 are similar. 117 Chapter 7– Model Calibration, Validation and Application Example Figure 7-19 Model #1 accelerometer A10 (dark line is FLAC class A prediction and light line is from centrifuge test). Figure 7-20 Comparison of class A numerical prediction and centrifuge results for Model #2 with 10m wide densification. 118 Chapter 7– Model Calibration, Validation and Application Example Figure 7-21 Comparison of pore pressure in piezometer P5 in Model #3 (dark colour is FLAC class A prediction and lighter colour is centrifuge test result). Figure 7-22 Typical grid and input time history used for UBC Liquefaction Initiative C-CORE centrifuge tests (shown in prototype scale). 119 Chapter 7– Model Calibration, Validation and Application Example Figure 7-23 Comparison of Centrifuge tests and numerical results for profile with low permeability silt barrier (COSTA-C): (a) displaced grid, (b) Horizontal displacement of sliding block over barrier (for the centrifuge data the solid line is measured and the dashed line is corrected to better match final displacements), (c) vertical displacement at crest, (d) centrifuge and numerical surface profiles, (e) calculated lateral displacement contours in metres, (f) and (g) pore pressure time histories at P3 and P6, (h) acceleration time history at A6. 120 Chapter 7– Model Calibration, Validation and Application Example Figure 7-24 (a) Initial and displaced profile of centrifuge test CT5 with three drainage slots (b) numerical analysis of same. 121 Chapter 7– Model Calibration, Validation and Application Example Figure 7-25 Comparison of vertical displacement near crest with (CT5) and without (COSTA-C) drainage slots. Post-shaking flow initiated in the COSTA-C test at approximately 70s. Figure 7-26 Kokusho shake table model at start of test (Kokusho 2003). 122 Chapter 7– Model Calibration, Validation and Application Example Figure 7-27 Kokusho shake table test profile at end-of-shaking. Figure 7-28 Post-failure profile of Kokusho shake table test. Failure occurred a short time after all shaking had stopped. 123 Chapter 7– Model Calibration, Validation and Application Example Figure 7-29 UBCSAND/FLAC model grid of Kokusho shake table test. Figure 7-30 Original boundary of model and displaced shape of FLAC model grid. 124 Chapter 7– Model Calibration, Validation and Application Example (a) (b) Figure 7-31 Contours of horizontal displacement (a) with barrier layers in place and flow on and (b) with barriers but no flow. 125 Chapter 7– Model Calibration, Validation and Application Example Figure 7-32 Calculated displacement time histories of shake table model with and without barrier and with and without flow. 126 Chapter 7– Model Calibration, Validation and Application Example Figure 7-33 Lower San Fernando Dam which failed into the reservoir approximately 30s after end of earthquake shaking. Blue areas are soil that was mixed and deemed to have liquefied. Lower figure is reconstructed profile which shows location of liquefied soil before failure. (Seed et al. 1973). 127 Chapter 7– Model Calibration, Validation and Application Example Figure 7-34 Grid, locations of low permeability barriers, (N1) 60 and shear strength input parameters used in Lower San Fernando numerical example. 128 Chapter 7– Model Calibration, Validation and Application Example Figure 7-35 Profile showing zones deemed to have liquefied at end of strong shaking (R u > 0.7). Figure 7-36 Displacement time histories of point near upstream toe of dam for both 3a coupled effective stress analysis (ESA) and 3b total stress analysis (TSA) alternative. Both analyses stopped due to excessive element distortion. 129 Chapter 7– Model Calibration, Validation and Application Example Figure 7-37 (a) Horizontal velocity (m/s) and (b) horizontal displacement (m) at end of analyses (40s) for coupled effective stress analysis. 130 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests 8 P-WAVE MEASUREMENTS AS AN SATURATION IN CENTRIFUGE TESTS 8.1 Introduction INDICATOR OF It has been shown that the triggering of liquefaction in sandy soils is sensitive to the compressibility or saturation of the pore fluid (Chaney 1978; Kokusho 2000; Tsukamoto et al. 2002; Yang 2002; Yang et al. 2004; Yang 2005). Cycles to liquefaction increase with increasing pore fluid compressibility or decreasing saturation. Pure water is relatively incompressible (Fluid bulk modulus ≈ 2 x 10 Pa); however, air is highly compressible and 9 the inclusion of small bubbles of air within the pore fluid (saturation < 100%) greatly increases the compressibility of the fluid. Measuring the fluid compressibility or saturation directly is often difficult. However, in the laboratory the compressibility and degree of saturation of the soil can be correlated with the Skempton ‘B’ (Skempton 1954) as indicated in Equation (8-1) (Yang 2002). 1 B= [8-1] 1+ n where: B Kb K + n b (1 − Sr ) Kw Pa = Skempton B = ∆u/∆σ m in undrained triaxial tests ∆u = change in pore pressure from an increment in mean normal pressure ∆σm n = porosity K b = bulk modulus of soil skeleton (individual soil grains are assumed incompressible) K w = bulk modulus of pore fluid P a = atmospheric pressure S r = saturation (as a fraction) The B-value is easily measured in undrained triaxial tests. However, in centrifuge and shake table tests, measuring the B-value and related degree of saturation is problematic. It has been proposed by Yang and Sato (1998), Kokusho (2000), Tsukamoto et al. (2002), and Yang (2002) that the P-wave or compression wave velocity (Vp) within the soil medium can be correlated to the degree of saturation and B-value as indicated by Equation (8-2) (Yang 2002) and illustrated in Figure 8-1. 131 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests 1/ 2 4G / 3 + K b /( 1 − B ) V p = ρ [8-2] G = shear modulus and ρ = density where: Data from Tsukamoto et. al. (2002), and Takahashi, et. al. (2006) showed a correlation between laboratory measurements and V p calculated from Equations (8-1) and (8-2) while Ishihara et al. (1998) and Tamura et al. (2002) showed that some laboratory Pwave B-value measurements deviated from the relationship given by Equation (8-2). V p has been used to locate zones of reduced saturation in the field (Sato and Yang 2000; Kokusho 2000) and has been correlated to Skempton ‘B’ and to cyclic resistance to liquefaction in laboratory tests (Tsukamoto et al. 2002; Ishihara et al. 1998) (Figure 8-2). In a recent UBC Liquefaction Initiative centrifuge test program carried out to study liquefaction-induced slope failure mechanisms, there was concern that the soil within the model was not fully saturated and that this may affect liquefaction triggering and model behaviour. The centrifuge model was prepared by air-pluviation of the fine to medium sand through a funnel with the same width as the model and with the sand drop height adjusted to achieve the desired relative density. The centrifuge test model was then inundated by: placing the specimen under a vacuum, flushing it with carbon dioxide, and slowly introducing a viscous pore fluid. A viscous pore fluid with a viscosity of 35 times the viscosity of water was used to improve pore fluid flow scaling when subjected to the 70 g centrifuge acceleration. P-wave velocity (V p ) measurements were made after inundation and following spin-up. However, the results were noted to be erratic. In order to investigate these anomalies in a fundamental manner, a laboratory test program was devised that would simulate the centrifuge saturation procedure and allow the verification of saturation by measurements of the Skempton B, and V p . Shear wave velocity (V s ) was also measured. Numerical analyses were conducted using the program FLAC (ITASCA 2002) in fullycoupled mechanical strain - groundwater mode in order to gain insight into the behaviour observed. This Chapter presents the results of the laboratory program along with those from numerical analyses seeking explanations for the erratic behaviour observed during centrifuge modeling. 132 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests 8.2 Laboratory Element Testing Program The laboratory element testing program was designed to simulate the sand placement and saturation procedures used in the previously mentioned centrifuge tests. The sand material (Fraser River Sand) and pore fluid for element testing were also similar to those used in the centrifuge tests. Figure 8-3 illustrates the laboratory setup. As indicated in Table 8-1, the testing involved measurement of B-value, V s , and V p on two triaxial specimens (Tests A and B) of 75mm diameter and 156mm height, under manifold testing conditions (i.e., with and without pore fluid, with and without cell water, with varying mean normal effective stress levels, and before and after 4- to 6-day saturation periods). All mean normal effective stress (σ΄ m ) variations on the triaxial specimens were exerted as hydrostatic loadings. Bender elements, as supplied by GDS Instruments Ltd. (2002), were located in the centre of the top and bottom platens; one set being the transmitter, and the other the receiver. The bender elements could be excited either in bending to generate shear waves or axially to generate a compression wave. Excitation frequencies were in the range of 3,500-7,000 Hz. The Fraser River sand used for the tests is a fine to medium grained sand with D50 of 0.26mm, D10 of 0.18mm and minimum and maximum void ratios of 0.62 and 0.94, respectively. The pore fluid used for the tests consisted of a 1.75% by weight methyl cellulose (methocel) water solution. The methocel solution has a viscosity of approximately 35 times that of water and is used in centrifuge tests to achieve better pore fluid flow scaling. The solution was pre-mixed using warm water. It was then cooled and de-aired by placing it under a 50 kPa vacuum for more than 8 hours. Specimen preparation and testing procedure were as follows: (i) The test specimens were prepared by air pluviation of the Fraser River sand through a funnel with an opening of approximately 2mm by 12mm. The sand was allowed to free fall 10 to 15mm as the funnel was manually moved back and forth over the specimen. The “as-placed” relative density of the specimens was approximately 20%. (ii) The specimen was then placed in the triaxial cell. Prior to the introduction of pore fluid, isotropic confining pressures ranging from 15 to 50 kPa were applied by various combinations of cell pressure and pore vacuum. Both air and water were 133 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests used to provide cell pressure. P-wave and shear wave velocities were measured for each load case. (iii) The specimen was then placed under a pore vacuum of 40 to 50 kPa and purged with carbon dioxide (CO 2 ). (iv) Methocel solution (pore fluid) was introduced at the bottom of the specimen with 1.5m head under a nominal cell pressure and 50 kPa pore vacuum, as illustrated in Figure 8-3. This was continued until clear methocel solution was exiting at the top of specimen (2.5 to 4 hours). Following this, the specimen was cycled with alternating 50 kPa vacuum and the methocel at atmospheric pressure. P-wave and shear wave velocities were measured at various stages during the inundation process. (v) V s , V p and B were measured at various cell and pore pressure increments. (vi) The specimen for Test A was then held under a constant back-pressure of 226 kPa for 4 days to provide an opportunity for improved saturation; similarly, the specimen for Test B was held under a constant back-pressure of 156 kPa for six days. (vii) After this holding period (for improved saturation), V s , V p and B were again measured at various cell and pore pressure increments. As noted in Table 8-1, cell pressure was supplied using air or water. Air was used for portions of the test to confirm that the P-wave travel path was not through the cell fluid in lieu of the specimen as desired. Results from the laboratory tests are summarized in Table 81. As may be noted, there was no noticeable difference in the P-wave velocity when air was used in the cell instead of water. Figure 8-4 shows a plot of V s with respect to the mean effective stress σ΄ m giving a best fit correlation as follows: V s = 1.7 x P a (σ΄ m / P a )0.23 [8-3] V p for unsaturated dry specimen conditions varied between 1.90 and 1.96 V s . This corresponds to a small strain dry sand Poisson’s ratio of 0.32 and a Bulk Modulus (K b ) to Shear modulus (G) ratio (= K b /G) of 2.4. 134 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests Figures 8-5(a) and 8-5(b) show typical V p and V s time history traces for a B of 0.48 and 0.95. Figure 8-6 shows the measured V p and V s as a function of Skempton B. Note that the arrival time does not change significantly with a change in B once the specimen is inundated with pore fluid and that the measured V p is indicative of a near fully saturated specimen, even for B less than 0.5. This is contrary to the findings of Tsukamoto et al. (2002), Yang (2002), and Kokusho (2000) and correlation from Equation (8-2). As shown in Figure 8-9, the measurements also show greater disparity from the correlation inferred from Equation (8-2) than the data measured by Ishihara et al. (1998) and Tamura et al. (2002). As expected, V s is essentially independent of B. 8.3 Numerical Modeling The experimentally observed V p vs. B correlation, as illustrated in Figure 8-6 is contrary to the expected behaviour inferred from Equation (8-2). A numerical analysis using the program FLAC (Cundall and Board 1988, ITASCA 2002) in coupled mechanical strain – groundwater flow mode was conducted to provide insight to the behaviour observed. FLAC is a finite difference numerical program which solves problems dynamically in accordance with the equations of motion using an explicit approach involving very small time steps. The formulation in FLAC is done within a framework of Biot theory with Darcy flow in a porous medium. The approximately 75mm diameter by 150mm high specimen was modeled in plane strain as a rectangular grid of 35 by 75 elements (Figure 8-7). The soil was modeled with an elastic skeleton with soil and pore fluid properties as indicated in Table 8-2. The permeability of the soil with the methocel pore fluid is 1/35th of the permeability of the same soil with water. The numerical modeling was carried out in two ways. In the first type of analysis, the pore fluid was assumed homogeneous and fluid modulus (K w ) was kept at a constant value at all nodes. This value was changed for each FLAC run to simulate varying degrees of saturation. This analysis is deemed to be representative of the air being uniformly disseminated within the pore fluid with many small bubbles and has been designated homogeneous-partial-saturation (HPS). Figure 8-8(a) illustrates the HPS conditions. The HPS type analyses produced the V p vs. B relationship shown as squares in Figure 8-9 and is in close agreement with Equation (8-2), as illustrated by the solid line in Figure 8-9. 135 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests In the other type of analysis, the pore fluid was modeled with scattered pockets of air bubbles. This has been designated non-homogeneous-partial-saturation (NHPS), and is illustrated in Figure 8-8(b). For the NHPS case, the air bubbles are modeled individually as a few pockets of large bubbles scattered throughout the grid. In the FLAC model, the air bubbles were modeled by reducing the fluid modulus (K w ) to a low value expected within the air bubbles (K w = P a + u) where P a = atmospheric pressure and u = pore pressure) at selected nodes scattered throughout the grid (Figure 8-10). The fluid modulus at other nodes was set to a high value of that representative of near fully saturated water (2.2x109 Pa). The V p vs B correlation from the NHPS type analyses is presented in Figure 8-9 (circle symbols) and shows a similar behaviour to that observed in the laboratory tests (triangle symbols). The P-wave time histories from the numerical analyses were similar to those observed from the bender elements in the laboratory. A major portion of the numerical modeling was conducted in plane strain instead of the axisymmetric modeling commonly used to simulate triaxial tests, in order to facilitate all elements being the same size or volume. This was important in the NHPS analyses as it allowed easy control of air bubble size and spacing. The HPS case was modeled both in plane strain and axisymmetric and the results were similar. The intent of the numerical modeling was not to give an exact replication of the laboratory model but to provide insight into the observed behaviour. 8.4 Discussion It is postulated that in the laboratory test there were scattered zones of bubbles surrounded by zones with either no or minimal air bubbles (non-homogeneous-partial-saturation (NHPS)). This will result in a stiff matrix with local soft pockets. The first arrival P-wave will be that which routes through the stiffer zones. This behaviour was simulated in the NHPS numerical model which showed that scattered discrete bubbles did not reduce V p even though they dramatically reduced the Skempton B. However, if there are numerous small bubbles within each pore space (homogeneous-partial-saturation (HPS) then the stiffness of the fluid is also homogeneous and the P-wave velocity will be a function of Skempton B, with saturation and fluid stiffness in accordance with Equations 8-1 and 8-2. 136 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests The NHPS saturation observed in the laboratory specimen may be partially related to the use of viscous methocel solution in lieu of water. The high viscosity may lead to the existence of a few local pockets of air bubbles rather than finely scattered small bubbles. Tests by others who were using water as the pore fluid (Tsukamoto et al., 2002) have indicated P-wave velocities in accordance with Equations 8-1 and 8-2. It is postulated that it may be possible to achieve better saturation of the specimen by inundating the specimen with de-aired water prior to introducing the methocel solution. A drawback of this procedure is that pockets of water may get trapped within the specimen and not become displaced by the methocel. It would be difficult to verify that all of the water had been displaced. Placing the specimen under a backpressure of greater than 150 kPa and holding it for several days was effective in increasing the Skempton B to levels indicative of good saturation (B > 0.85). The Skempton B remained high when the backpressure within the specimen was reduced to atmospheric pressure. It is suggested that this common laboratory procedure be considered as part of the process for saturating centrifuge models. 8.5 Chapter 8 Summary and Conclusions Centrifuge tests were carried out as part of the “UBC Liquefaction Research Initiative” (NSERC Grant 246394) for the purpose of validating the UBCSAND numerical model. During this work it was noted that the behaviour in the centrifuge did not match that predicted by the numerical models and it was suspected that the pore fluids in the centrifuge were not fully saturated, even though P-wave velocity (V p ) measurements were indicative of full saturation. To check why this may be the case, triaxial tests that simulated the centrifuge saturation procedure were carried out. The triaxial cells had bender elements in the top and bottom platens that allowed measurement of shear wave velocity (V s ) and P-wave velocity (V p ) during the testing. The results from the testing showed that when the measured Skempton B in the triaxial sample was high (representative of full saturation), the corresponding V p was also high. However, a high V p did not always correspond to a high Skempton B. It is postulated that the behaviour in the later case is due to the air bubbles giving the low B-value being widely spaced, thus, allowing the P-waves to find travel paths through the medium that bypassed the bubbles (thus giving the high velocity). From this, two types of partial saturation have been defined: homogeneous-partial-saturation (HPS) 137 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests whereby the air bubbles are numerous, very small and included within each void space throughout the specimen, and non-homogeneous-partial-saturation (NHPS) whereby the air bubbles are much larger and are only present within some pore spaces. For HPS, the V p will be a function of the Skempton B and average saturation (S r ) av , whereas, for NHPS, V p is independent of Skempton B and average saturation. With NHPS, high V p values indicative of high saturation are possible, even though the B value is relatively low. NHPS conditions are postulated to be more prevalent in centrifuge tests where viscous solutions, such as methocel, are often used in lieu of water. In addition to the laboratory test, the behaviour was also numerically simulated using the program FLAC. This later simulation provided confidence in the postulated explanation for the behaviour. When the triaxial specimen was placed under backpressure for several days this resulted in air going into solution and increased saturation and related Skempton B values. This procedure is standard practice in the saturation of laboratory specimens, and it is proposed that the procedure also be considered for saturating centrifuge test specimens. 138 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests TABLES Chapter 8 P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests 139 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests Table 8-1 Initial Cell Pressure (P i ) (kPa) ----50 99 152 198 298 358 398 355 508 450 --------------30 54 106 157 219 197 160 253 280 323 106 158 208 130 ------(1) (2) Summary of Laboratory Test Conditions and Results Initial Pore Pressure (u i ) (kPa) Final Cell Pressure (P f ) (kPa) Final Pore Pressure (u f ) (kPa) Mean Shear Effective P-wave Wave Pore Skempton Stress Velocity Velocity fluid (2) (1) 'B' (σ' m ) f (V p ) (V s ) (kPa) (m/s (m/s) TEST A 0 -15 15 240 123 air ----15 -10 25 252 142 air ----TEST A SPECIMEN PURGED WITH CO2 10 99 27 72 0.34 (0.36) 1420 159 meth. 27 152 44 108 0.32 (0.36) 1420 177 meth. 44 198 60 138 0.35 (0.36) 1562 184 meth. 60 298 109 189 0.49 (0.50) 1562 200 meth. 109 362 146 216 0.58 (0.59) 1562 208 meth. 113 398 137 261 0.59 (0.61) 1562 226 meth. 137 360 120 240 0.44 (0.45) 1562 210 meth. BACK PRESSURE OF 221 kPa ON TEST A SPECIMEN FOR FOUR DAYS 221.06 403.4 261.1 142 0.82 (0.85) 1562 195 meth. 353 561 400 162 0.89 (0.92) 1562 203 meth. 295 368 226 142 0.84 (0.87) 1562 195 meth. TEST B 0 -21 21 255 130 air ----0 -30 30 268 138 air ----0 -40 40 278 147 air ----0 -50 50 288 154 air ----0 -20 20 243 124 air ----47.6 0 47.6 288 150 air ----TEST B SPECIMEN PURGED WITH CO2 0 -50 50 288 150 C0 2 ----3 40 7 32 0.45 (0.47) 1415 129 meth. 30 103 54 50 0.47 (0.48) 1415 117 meth. 66 155 91 64 0.52 (0.55) 1557 148 meth. 95 211 124 87 0.54 (0.57) 1557 157 meth. 145 198 135 63 0.49 (0.50) 1557 142 meth. 134 174 120 54 0.60 (0.63) 1557 142 meth. 116 93 73 20 0.63 (0.65) 1557 109 meth. 140 283 153 130 0.43 (0.46) 1557 183 meth. BACK PRESSURE OF 154 kPa ON TEST B SPECIMEN FOR SIX DAYS 152 324 189 134 0.86 (0.98) 1557 188 meth. 189 364 225 139 0.88 (0.99) 1415 188 meth. 67 158 114 44 0.90 (0.96) 1557 136 meth. 115 208 161 47 0.93 (0.99) 1730 142 meth. 161 135 95 40 0.91 (0.98) 1557 135 meth. 91 79 46 33 0.89 (0.95) 1557 127 meth. 77 4 73 1416 157 meth. ----73 -20 93 1415 173 meth. ----0 -20 20 1557 117 meth. ----- Cell Fluid air water water water water water water water water water water water air air air air air air air air air air air air air air water water water water air air air air air air B-value in brackets is corrected for compliance of pore pressure measuring system meth. = 1.75% methylcellulose in water (methocel) 140 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests Table 8-2 Soil and Fluid Properties used in Numerical Analyses Description Symbol Value Units Confining pressure (isotropic) σ' m 100 kPa Dry density ρd 1450 kg/m3 Shear modulus G 5.5x104 Bulk modulus of soil skeleton Fluid modulus (100% saturation) kPa Kb 5 1.34x10 kPa Kw 6 kPa -3 2.2 x10 Permeability K 1.3 x10 cm/s Porosity N 0.45 --- 141 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests FIGURES Chapter 8 P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests 142 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests 1 (a) Skempton 'B' 0.8 0.6 0.4 0.2 0 0.9 0.92 0.94 0.96 0.98 1 0.8 1 Degree of Saturation (Sr) P-wave velocity (m/s) 2000 (b) 1500 1000 500 0 0 Figure 8-1 0.2 0.4 0.6 Skempton 'B' Variation of (a) Skempton ‘B’ with Degree of Saturation (S r ) and (b) P-wave velocity (V p ) with Skempton ‘B’ for Fraser River Sand, with D r =20% K b =92 MPa, G=56 MPa, and porosity = 0.45 in accordance with Equations (1) and (2). CSR(partial saturation) CSR(fully saturation) 2.5 2 1.5 1 0.5 0 Figure 8-2 500 1000 1500 P-wave velocity (Vp) (m/s) 2000 Normalized cyclic strength ratio (CSR) of sand compared to P-wave velocity (V p ) as measured in laboratory tests conducted with Toyoura sand. Developed using data from Tsukamoto et al. (2002) and Ishihara et al. (1998). CSR is that which causes liquefaction (double amplitude strain of 5%) in 20 cycles (for Relative Density of 4070%). 143 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests TRIAXIAL CELL FILLED WITH AIR OR WATER METAL PLATENS SAMPLE DIGITAL OSCILLOSCOPE BENDER ELEMENTS AMPLIFIER RUBBER MEMBRANE Legend R M VACUUM Pressure PT transducer TRIAXIAL CELL AIR AIR 1.5m Regulator R valve methocel R R SAND M AIR R PT PT WATER R Figure 8-3 C02 Laboratory test layout. Shear Wave Velocity (m/s) 250 200 150 100 50 Dry Saturated 0 0 50 100 150 200 Mean Effective Stress (kPa) Figure 8-4 Variation of shear wave velocity (V s ) with effective mean normal confining pressure (σ’ m ) as measured in laboratory tests. 144 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests 6 Input source motion at bottom of sample from shear wave Amplitude P-wave arrival 0 Measured wave at top of sample (a) -6 6 Input source motion at bottom of sample from shear wave Amplitude P-wave arrival 0 Measured wave at top of sample (b) -6 0 0.5 1 1.5 2 Time (ms) Figure 8-5 Typical P-wave time histories as measured in the laboratory. Traces (a) is for B = 0.48 with a calculated V p of 1415 m/s and trace (b) is for B = 0.95 with a calculated V p of 1557 m/s. 145 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests Wave Velocity (m/s) 2000 1500 Vp 1000 500 Dry Vs 0 0 0.2 0.4 0.6 0.8 1 Skempton 'B' Figure 8-6 Measured shear wave (V s ) and P-wave (V p ) versus Skempton ‘B’. Figure 8-7 35 by 75 element mesh used for FLAC analyses. 146 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests air bubbles (b) (a) Figure 8-8 (a) Represents homogeneous-partial-saturation (HPS) where the air bubbles are small and scattered throughout the void spaces. This is numerically modeled by reducing the fluid modulus (K w ) at all the nodes in the model. (b) Represents non-homogeneouspartial-saturation (NHPS) where there are only a few large bubbles at select locations. This is numerically modeled by reducing the fluid modulus only at select nodes within the model. 2500 FLAC - non homogeneous (NHPS) FLAC - homogeneous (HPS) Laboratory Equation (2) Tamura Dr=65% Ishihara Dr=60% P-wave velocity (V p) (m/s) 2000 1500 1000 500 0 0 0.2 0.4 0.6 0.8 1 Skempton B Figure 8-9 Comparing V p versus Skempton B from both HPS and NHPS FLAC analyses to that observed in laboratory tests and that calculated from Equation (2). Data from Ishihara et al. (1998) and Tamura et al. (2002) that deviated from the correlation from Equation (2) are also shown. 147 Chapter 8– P-Wave Measurements as an Indicator of Saturation in Centrifuge Tests Figure 8-10 Typical variation of fluid modulus within NHPS FLAC grid. Small squares represent locations with low fluid modulus representative of air bubbles, whereas all other elements have a high fluid modulus of 2.2x109 Pa. 148 Chapter 9 – Discussion and Conclusions 9 DISCUSSION AND CONCLUSIONS 9.1 Summary and Conclusions Seismic design of major civil structures (bridges, dams and embankments) is moving increasingly towards using performance design methodologies. A key aspect of performance design is to determine the earthquake induced ground and structure movements. To do this, designers are increasingly resorting to the use of dynamic numerical analyses. The research presented in this dissertation involves development of these numerical design tools and procedures for use in engineering practice for estimating the earthquake induced ground deformations of soil that may potentially liquefy. The developed design methodology uses effective stress and total stress soil constitutive models, UBCSAND and UBCHYST, developed at the University of British Columbia (UBC). There are several assumptions and approximations in these analysis procedures and due to this, a hybrid procedure that combines the coupled effective stress analysis with a post-shaking static analysis check using empirical soil strengths that have been back-calculated from past earthquake case histories has been developed. In Chapter 2, an overview of items contributing to liquefaction and potential flow sliding are given. These are behaviours that are important for the proposed numerical model to capture and important for the proposed users of the model to understand. It is noted that sands have contractive tendencies when sheared at stress ratio less than the phase transformation stress ratio, have dilative tendencies sheared at higher stress ratio, but always contractive tendencies on unloading. The net response of cyclic shear loading of saturated sands is nearly always a tendency for contraction. This in turn generates pore water pressures, potential liquefaction of the sands (low effective stress and very low shear moduli), and sets up a hydraulic gradient within the deposit. The gradients cause water flow (generally upward or toward a free face) out of some zones and into others. Those zones losing water generally contract and those gaining pore water expand. When there is a low permeability barrier above the sand layer the inflow of pore water into the soil directly below the barrier results in localization of both the expansion and shear strain. As the sand expands, its potential shear-induced-dilative-tendencies decrease, and with sufficient 149 Chapter 9 – Discussion and Conclusions expansion, the critical state, at which shear strain will induce no further expansion, is reached. This contraction and expansion can have a dramatic affect on both the effective stress and the shear strength of the sand. This is particularly the case for loose sand under low permeability barriers where expansion tends to be localized and strengths can go to very low values, or at the limit, zero when a water interlayer forms. Most of the concepts in Chapter 2 have been discussed by others in some manner in the past. However, the framework and emphasis of the discussion is important and is that of the author. Soil mixing, another mechanism that can lead to significant strength reduction of layered liquefiable soils and liquefied soils, is investigated in Chapter 3. Many natural cohesionless soils and man-made hydraulically placed soils are layered. When the fine and coarse layers mix, the resulting “residual” shear strength is much lower than that of the individual layers at the same void ratio. This mixing provides explanation for the 1970 flow liquefaction failure observed at the Mufulira Mine site and, at least partially, accounts for the low end-of-failure residual shear strength back-calculated for the Lower San Fernando Dam failure. Mixing is not a new concept, but is a mechanism that can both result in liquefaction and lower the undrained strength of liquefied soils. The author proposed the concept of cyclic mixing and developed rudimentary cyclic shear tests to investigate the concept. The rudimentary cyclic shear tests were carried out within a 12mm internal diameter vinyl tube. During the large strain cyclic shearing the movement of the soil particles was clearly visible. The ratio of the grain-size of the coarse and fine layers was shown to be a key factor in a layered soil’s susceptibility to mixing. A preliminary postulation by the author is that cohesionless soils may be susceptible to cyclic mixing when ((D 15 ) coarse layer / (D 85 ) fine layer ) > 3 to 4. It is also postulated that mixing will more readily occur when effective stresses and related inter-particle contact forces are low. Mixing is probably more common as a post-liquefaction mechanism and helps explain the low post-liquefaction strengths and large run-out distances observed in some instances. Within a post-liquefaction low effective stress – low strength environment both mechanical shear and turbulent flow may induce mixing. The occurrence of mixing induced by turbulent flow is a concept postulated by the author following examination of the shake table test videos by Kokusho (http//:www.civil.chuo-u.ac.jp/lab/doshitu/index.html). 150 Chapter 9 – Discussion and Conclusions The post-liquefaction residual strengths back-calculated from case histories are discussed in Chapter 4. These are important because they are used in current engineering practice and also because they provide the input parameters for the proposed hybrid numerical procedure that is the topic of this thesis. Experience has shown that strengths back-calculated from laboratory tests tend to give strengths that are too high. This is postulated to be due to the laboratory testing process not accounting for void redistribution and mixing effects. If the in-situ soil is truly believed to behave in an undrained manner then the strengths from laboratory tests should be representative. In the field, the soil is generally not truly “undrained” and the “residual” strength is dependent on many variables (soil density, soil fabric, stratigraphy, initial stresses, duration of shaking, permeability / groundwater flow regime, method of analyses used, etc.) and is as much a site property as it is a soil property. Correlating the “residual” strength or strength ratio with soil density or penetration resistance is a gross simplification and consequently there is much scatter and uncertainty in the data. Chapter 5 introduces the constitutive models UBCSAND and UBCHYST that are key parts (especially UBCSAND) of the proposed hybrid analysis procedure. UBCSAND, when run within the program FLAC is a coupled effective stress model that considers shear induced soil skeleton volume changes, their effect on pore water pressure, and the resultant pore water flow from one element to another. As indicated in Chapter 2, these are key behaviours needed to capture soil liquefaction and related soil strength changes. The UBCSAND program was developed by Dr. Byrne and his students. New features added to UBCSAND as part of this research include: • Elimination of numerical instability for certain cases with static bias; • Improvement in the simulation of cases with large static bias; • Improvement in the liquefaction behaviour by adding a dilation delay (period during which shear does not induce volume change) thus allowing large shear strains to develop when the normal effective stress and shear stress is near zero, in agreement with behaviour observed in laboratory test data (Figure 5-10), 151 Chapter 9 – Discussion and Conclusions • Introduction of a “dilation cut-off” which set shear induced dilation to zero when an element reached a pre-described critical volume or void ratio (analogous to the dilation angle going to zero when the soil reaches the critical state); and, • Improvements in the modelling of dense soils by making the phase transformation stress ratio a function of soil density and changing the correlation between peak friction angle and soil density. UBCHYST is a simple robust “total stress” constitutive model often used together with UBCSAND for the numerical modelling of soil structures subjected to earthquake shaking. UBCSAND is used for portions of the soil profile deemed susceptible to liquefaction and UBCHYST is used for those deemed not susceptible. The advantage of the UBCHYST model over a simpler Mohr Coulomb model is the non-linear hysteretic loops developed by varying the tangent shear modulus during loading and unloading. This more closely replicates the behaviour of real soil and reduces the Rayleigh damping requirements; thus increasing analysis speed. The initial algorithm for UBCHYST was developed by Professor Byrne while the author coded UBCHYST as a constitutive model in FLAC, made improvements regarding unload/reload loops, developed options for strength reduction as a function of strain or number of cycles, and developed documentation explaining the model. The proposed hybrid effective stress – total stress design approach is presented in Chapter 6. Dynamic numerical analyses using the UBCSAND constitutive model may be the state-of-the-art in predicting liquefaction induced deformations. However, there are still shortcomings, including: element size / localization effects, underestimation of postliquefaction consolidation, no out-of-plane (3D) shaking, etc. In addition to the modelling shortcomings, there is also the difficulty of characterizing the soil stratigraphy with accuracy at any given real site. Case history derived post-liquefaction soil strengths also have large uncertainties associated with them and they are used in current engineering practice largely because there are no alternatives. When considering all this, it is proposed that a good design should consider both approaches and the hybrid effective stress – total stress design approach described in this chapter is the result. A description of the procedure and step-by-step details on implementing it are given in Chapter 6. 152 Chapter 9 – Discussion and Conclusions Chapter 7 provides examples of both the calibration and validation of the UBCSAND model and the proposed hybrid analysis procedure. Laboratory tests are one of the few cases where the loading, stress and pore pressure histories can be controlled and are known. Section 7.2 shows comparisons between a single element with the UBCSAND (version UBCSAND1v02) model and cyclic simple shear tests by Sriskandakumar (2004). It is shown that the new version of UBCSAND with the proposed revisions gives a reasonable match of both loose and dense sand samples when there is no static bias. With a static bias the match is not as good; however, the behaviour mechanisms are matched. Section 7.3 describes a calibration of the model to empirical liquefaction triggering charts. The empirical liquefaction triggering charts have been developed from numerous case histories and their use is accepted current design practice. This section shows that UBCSAND can be calibrated to simulate the liquefaction triggering behaviour inferred by these field case history derived correlations. Calibrating to these charts is commonly done when using UBCSAND for commercial design projects. Section 7.4 describes some early, 2004, one dimensional (infinite slope) column tests that the author carried out to demonstrate that UBCSAND/FLAC could simulate the post-shaking flow failure phenomena. To the author’s knowledge this is the first numerical simulations of this type of behaviour. This simulation demonstrates: the ability of UBCSAND/FLAC to generate pore pressure gradients and flow, the mechanism of some elements losing pore water and decreasing in volume and some elements gaining pore water and increasing in volume, and the mechanism of volume increase and eventual strength loss that occurs below a low permeability barrier layer that overlies liquefiable sands. Section 7.5 describes some work carried out for the seismic retrofit design of the George Massey tunnel. This is one of the earlier commercial uses of UBCSAND and included a centrifuge test program for validation. The author supervised and was responsible for the analyses. The numerical simulations of the centrifuge tests were Class ‘A’ predictions (carried out before the centrifuge tests were carried out) and gave good agreement with the actual test performance. 153 Chapter 9 – Discussion and Conclusions Section 7.6 described some numerical modeling of centrifuge tests carried out by C-CORE in Newfoundland as part of the UBC Liquefaction Initiative. These are some of the earlier centrifuge tests to incorporate an embankment slope with a low permeability soil barrier overlying loose liquefiable sand, and possibly the first centrifuge test to demonstrate flow failure after end-of-shaking. This is a behaviour that has been observed in several field case histories. The author showed that the UBCSAND/FLAC numerical model could simulate the observed post-shaking failure mechanism when shear-induced dilation was curtailed upon the element volume exceeding a preset threshold. This is analogous to shear induced volume change being zero when the soil reaches the critical state. These centrifuge tests were suspected of not being fully saturated. This initiated the triaxial testing program and numerical analyses described in Chapter 8 of this dissertation. The shake table tests by Kokusho (2003) were important because they illustrated the void redistribution mechanism that explains the post-shaking failures so often observed in field case histories, and because they rekindled interest by the research and design community in the void redistribution mechanism. Section 7.7 describes numerical modelling by the author that demonstrated that the observed deformation and post-shaking mechanism could be simulated using the UBCSAND constitutive model within FLAC. The 1971 failure of the Lower San Fernando Dam in California is one of the best documented liquefaction-induced-failure case histories available. A key aspect of this case history is that it failed approximately 30 seconds after the end-of-shaking. Therefore, the failure was caused by delayed soil strength reduction and not by earthquake inertial forces. Many attempts by others to numerically model this failure have been able to approximately simulate the deformation pattern but, to the author’s knowledge, none have successfully simulated the post-shaking shear failure mechanism. Section 7.8 describes the authors work on numerical modeling of the Lower San Fernando Dam using the proposed hybrid procedure. It is shown that the post-failure mechanism can be simulated by the UBCSAND coupled effective stress model. It is also shown that carrying out the proposed post-failure static analysis check, using the Idriss and Boulanger, 2008 post-liquefaction shear strength correlations, within the zones that liquefied during the strong shaking portion of the effective stress analysis, also results in flow failure. 154 Chapter 9 – Discussion and Conclusions The author was able to simulate post-shaking flow failure in the analyses described in Sections 7.4 (infinite slope column with low-permeability barrier), 7.6 (C-Core centrifuge tests), 7.7 (Kokusho shake table simulation), and 7.8 (Lower San Fernando Dam failure). In each of these cases, when the same analyses are repeated with flow not permitted (undrained condition) or repeated without the low permeability layers, then post-shaking failure did not occur. This highlights the importance of being able to simulating pore water flow and void redistribution mechanisms when numerically modelling potentially liquefiable earth structures subjected to earthquake shaking. Chapter 8 is on a slightly different theme then the rest of the thesis. However, the observations and conclusions derived are believed to be of technical interest and, therefore, were included in the main body of the dissertation. Centrifuge tests were carried out as part of the “UBC Liquefaction Research Initiative” (NSERC Grant 246394) for the purpose of validating the UBCSAND numerical model. During this work it was noted that the behaviour in the centrifuge did not match that predicted by the numerical models and it was suspected that the pore fluids in the centrifuge were not fully saturated, even though P-wave velocity (V p ) measurements were indicative of full saturation. To check why this may be the case, triaxial tests, that simulated the centrifuge saturation procedure, were carried out. The triaxial cells had bender elements in the top and bottom platens and thus shear wave velocity (V s ) and P-wave velocity (V p ), and Skempton B could be measured during the testing. The results from the testing showed that when the measured Skempton B in the triaxial sample was high (representative of full saturation), the corresponding V p was also high. However, a high V p did not always correspond to a high Skempton B. It is postulated that the behaviour, in the later case, is due to the air bubbles which give the low Skempton B, being widely spaced, thus, allowing the P-waves to find travel paths through the medium that bypass the bubbles (thus giving the high velocity). From this, two types of partial saturation have been defined: homogeneous-partial-saturation (HPS) whereby the air bubbles are numerous, very small and included within each void space throughout the specimen, and non-homogeneouspartial-saturation (NHPS) whereby the air bubbles are much larger and are only present within some pore spaces. For HPS, the V p will be a function of the Skempton B and average saturation (S r ) av , whereas, for NHPS, V p is independent of Skempton B and average saturation. With NHPS, high V p values indicative of high saturation are possible, even 155 Chapter 9 – Discussion and Conclusions though the B value may be relatively low. NHPS conditions are postulated to be more prevalent in centrifuge tests where viscous solutions, such as methocel are often used in lieu of water. In addition to the laboratory test, the behaviour was also numerically simulated using the program FLAC. This later simulation provided confidence in the postulated explanation for the behaviour. When the triaxial specimen was placed under backpressure for several days this resulted in air going into solution and increased saturation and related Skempton B values. This procedure is standard practice in the saturation of laboratory specimens, and it is proposed that the procedure also be considered for saturating centrifuge test specimens. 9.2 Recommendations 9.2.1 Recommendations based on the Work carried out for this Dissertation (1) The design process outlined in this dissertation is a viable design tool for use in engineering practice and its use is recommended. By carrying out the numerical modelling significant insight into the design problems can be obtained. In addition, the calculated displacements are likely more reliable than those obtained using simplified methods based on Newmark-type procedures (Newmark 1965) or those obtained using empirical correlations such as those by Youd et al. 2002). (2) For design work, the model can be calibrated to published liquefaction triggering and K α correlations (charts); however, the post-liquefaction triggering response, and response to a static shear bias, is important and is not adequately captured by these charts. For these aspects, the use of simple shear laboratory tests is recommended. It is suggested that tests with similar stress paths, to what will be experienced in the design problem, be carried out. (3) On important large projects, consideration should be given to carrying out physical model tests to calibrate and validate the numerical model. Centrifuge testing and large shake table tests may be useful for this purpose. (4) When carrying out centrifuge tests, measured P-wave velocities may not correlate well with saturations. Placing a specimen under back pressure for several days results in air going into solution, which increases saturation and related B values. 156 Chapter 9 – Discussion and Conclusions This procedure is standard practice in the saturation of laboratory specimens, and it is recommended that the procedure also be considered for saturating centrifuge test specimens. 9.2.2 Recommendations for Future Research UBCSAND (1) Further work is required for improving the model in conditions with large static bias. Much of the deformation during earthquake shaking is related to “marching” due to the static bias. More laboratory tests with stress paths that simulate that which occurs in the field are required to assist with model development. (2) In many instances, the volumetric strain due to post-liquefaction consolidation may be much larger than currently calculated by UBCSAND, especially in level ground conditions. This could be important as with the correct post-liquefaction volumetric strain, more water would be available for re-distributing, and there would be more potential for water films, etc. A version of UBCSAND with improved post- liquefaction consolidation behaviour should be developed. Introducing a volumetric ‘cap’ or yield envelope may be required to do this. An option in FLAC to have a rigid (infinitely high) fluid modulus may also help. (3) In this dissertation, analyses were carried out that showed that the UBCSAND constitutive model could capture the post-shaking flow failure observed in the 1971 flow failure of the Lower San Fernando Dam. It is suggested that analyses using the same methodology should be carried out for the Upper San Fernando Dam where flow failure did not occur during the same earthquake with similar soil conditions. (4) The model is a compromise - a balance between simplicity, practicality, robustness and ability to capture field behaviour. This should be considered when implementing changes. Mixing The importance of mixing on post-liquefaction flow failure run-out distances and residual strength is believed to be underestimated. Further study of the mixing phenomena should be undertaken. This could include more detailed assessment of which grain-size 157 Chapter 9 – Discussion and Conclusions variations, shear strain values, number of cycles, etc. would induce mixing and which would not. Also, it is observed that much mixing occurs between layers of non-plastic materials when there is liquefaction and extensive pore water void redistribution – perhaps development of special laboratory apparatus (or shaking table tests) are needed to adequately study mixing. Use of Large Shaking Tables Large-sized shaking tables provide opportunity to study pore water void redistribution phenomena and the mixing between layers. Tests similar to those by Kokusho (Kokusho 1999, 2000, 2003; and Kokusho and Kabasawa 2003) could be repeated on a much larger “real-life” scale with detailed measurements (using video recording or photographs, if clear sides are used) of void changes etc. Extension of Analyses Tools into three dimensions Most real soil-structure interaction design problems (bridge piers, building foundations, etc.) have a strong three dimensional component and the computing power necessary for these analyses is currently available. Making the UBCSAND/UBCHYST type design tools three dimensional, is a logical goal. Residual or Post-liquefaction Strength of Soils for Design Current empirical case-history based procedures are inaccurate and improved correlations and methodologies are needed. Variables other than those in the current correlations may be important. Static bias and stratigraphy are expected to have a significant effect. The use of coupled effective stress numerical analyses to back-analyze case histories and generic design sections might prove helpful in developing new design correlations or methodologies. Use of particulate models (such as, ITASCA’s PFC3D program) Current research using PFC3D to simulate laboratory tests suggests that the use of particle models can capture the behaviour observed in laboratory triaxial and simple shear tests (Pinheiro et al. 2008; Dabeet 2010) and provide useful assistance in understanding soil behaviour and liquefaction phenomena. Improvement in particulate codes and in computing power are areas where further development is required. Using available codes to further 158 Chapter 9 – Discussion and Conclusions develop understanding in soil behaviour and to develop engineering design tools is another suggested area of research. Liquefaction Triggering An improved understanding of the effect of static bias on triggering is required. Current practice is to set K α (a factor to allow for static bias) to unity which implies that there is no effect. However, both laboratory testing and field experience indicate that there can be a quantifiable effect. A simple procedure may not be possible and numerical models may be required to capture the behaviour, or else numerical models may be able to assist in developing an improved simplified design procedure. 159 References REFERENCES Adalier K., Elgamal A., Meneses J., and Baez J.I. 2003. Stone column as liquefaction countermeasure in non-plastic silty soils. 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Earthquake Engineering, Tenth World Conf., Balkema, Rotterdam, ISBN 9054100605. 168 Appendix A APPENDIX A UBCSAND Flow Chart and Fish Code UBCSAND1v02 FLAC Fish Code NOTE: Determining the appropriateness and accuracy of this routine for any purpose is sole responsibility of end user. Routine is provided to specific organizations by author and is not transferrable outside of this organization. Please refer new users and potential bugs to primary author at pmb@civil.ubc.ca. ****************************************************************************** * FISH version of UBCSAND MODEL from * * Mohr-Coulomb model with * * strain hardening/softening * * Effective stress stress approach * * primary and secondary plastic hardener * * * * Revisions * * NOV 14 2001 pmb Change to post trigger plastic modulus and * * crossover counter m_count4,m_ocr * * DEC 27 2001 pmb m_triax = 1 to simulate comp ext tests * * Feb 6 2002 pmb Modified plastic hardeners and basic relationship * * between plastic and elastic moduli * * Feb 13 2002 pmb Change to anisotropy (only for first time loading) * * Sep 12 2002 mhb Change m_count4 to $gplim & $ratlim * * modified $hard1 for m_n160 of 5 to 10 * * modified m_dt at low $sig * * reset 2ary yield surface if dilation * * introduced zart for averaging stress components * * limited maximum m_knew2 to m_knewp * * March 30 2008 pmb Added one-sided loading and tension changes * * Nov. 1 2008 PMB Changed start of Running section * * Added m_mohr = 1.0 to simulate mohr model * * Add m_hfac4 to allow reduction of dilation after triggering * * * January 8,2009 Modify one-sided loading * * January 13,2009 Modify for drained condition or fmod = 0, use * * m_hfac3 = 0 as a signal * * May 12,2009 Modify for silt. Change m_hfac3 and m_hfac4 * * July 24.2009 Modified $hard associated with m_hfac3 and $hard2 * 169 Appendix A * associated with m_hfac4 * * July26,2009 Changed default on $hard to 0.001. * * Mar31-May12,2010 UBCSAND1v02 - General clean-up plus added: * * -revised dilation cut-off option, * * -revised pull-down of yield envelope on unloading, * * -reduced default mean and vertical effective stress, * * -revised tension yield envelope default, * * -added zero dilation interval following passing * * through zero stress origin, * * -changed m_Pa to $Pa = atmospheric pressure, * * -changed m_phicv to m_phipt * * -changed m_ratcv to m_ratpt * * (pt stands for phase transformation * ****************************************************************************** set echo off def m_mss constitutive_model 99 f_prop m_kge m_ne m_kb m_me m_ocr m_triax f_prop m_kgp m_np m_phipt m_phif m_rf m_n160 f_prop m_g m_k m_coh m_ten m_ind f_prop m_csnp m_nphi m_npsi m_e1 m_e2 m_x1 m_sh2 f_prop m_anisofac m_$fac m_css m_knew m_knew1 m_knew2 f_prop m_ratio m_ratpt m_ratf m_gpsum m_ratcrs m_knewp f_prop m_dratmob m_ratmob m_dt m_flago m_ratmobold m_cross f_prop m_hfac1 m_hfac2 m_hfac3 m_hfac4 m_hfac5 m_hfac6 f_prop m_epsum m_epsum1 m_rtymax m_ratmax m_ncyc m_ncyc1 f_prop m_epsav m_epsum4 m_epsum4old m_ratmax0 m_ratmax1 m_sxyold f_prop m_ratioy m_rtmax m_gpstar m_mohr m_epsum5 m_epsum6 f_prop m_signal1 m_dilcut m_ratmax3 m_cnt m_epsum7 m_epsum8 float $sphi $spsi $s11i $s22i $s12i $s33i $sdif $s0 $rad float $s1 $s2 $s3 $dc2 $dss float $si $sii $psdif $fs $alams $ft $alamt $cs2 $si2 float $apex $epsav $tpsav $de1ps $de3ps $depm $eps $ept float $bisc $pdiv $anphi $tco $sig $hard1 $area $hard2 float $sd $sxy $dumsig $dumsd $dumsxy $epn $epsum $cross float $eps1 $epn1 $ratmax $hard $sy $dumsy $ratlim $gplim float $epsum5 $Pa ;---------------------------------------------Case_of mode ; ---------------------; Initialisation section ; ---------------------Case 1 ; --- data check --$m_err = 0 170 Appendix A if m_phif > 89.0 then $m_err = 1 end_if if m_coh < 0.0 then $m_err = 3 end_if if m_ten < 0.0 then $m_err = 4 end_if if $m_err # 0 then nerr = 126 error = 1 end_if ; ----FLAG TO SET UP INITIAL CONDITIONS THE FIRST TIME IT GOES THROUGH ;-----AND EACH RESTART if m_flago < 5.0 then ;AVOIDS CHANGES ON RESTART m_ratf = sin(m_phif * degrad) m_ratpt = sin(m_phipt * degrad) m_k = m_kb * $pa m_g = m_kge * $pa m_e1 = m_k + 4.0 * m_g / 3.0 m_e2 = m_k - 2.0 * m_g / 3.0 m_sh2 = 2.0 * m_g ; --- set tension to prism apex if larger than apex --$apex = m_ten if m_phif # 0.0 then $apex = m_coh / tan(m_phif * degrad) end_if m_ten = min($apex,m_ten) end_if if $ratlim = 0.0 then $ratlim = 0.01 ;used for crossovers end_if if $gplim = 0.0 then $gplim = 0.00005 ;used for crossovers end_if if m_n160 = 0.0 then m_n160 = 5.0 end_if Case 2 ; --------------; Running section ; --------------m_flago = m_flago +1.0 171 Appendix A if m_flago < 5.0 then ;FOR STARTUP m_ratmob= m_ratf ;Treats as Mohr model m_dt = m_ratpt - m_ratmob $sphi = m_ratmob ; makes it elastic on startup $spsi = -m_dt m_npsi = (1.0 + $spsi) / (1.0 - $spsi) m_nphi = (1.0 + $sphi) / (1.0 - $sphi) m_x1 = m_e1 - m_e2*m_npsi + (m_e1*m_npsi - m_e2)*m_nphi m_csnp = 2.0 * m_coh * sqrt(m_nphi) if abs(m_x1) < 1e-6 * (abs(m_e1) + abs(m_e2)) then $m_err = 5 nerr = 126 error = 1 end_if end_if zvisc = 1.0 if m_ind # 0.0 then m_ind = 2.0 end_if $anphi = m_nphi ; --- get new trial stresses from old, assuming elastic increments --$s11i = zs11 + (zde22 + zde33) * m_e2 + zde11 * m_e1 $s22i = zs22 + (zde11 + zde33) * m_e2 + zde22 * m_e1 $s12i = zs12 + zde12 * m_sh2 ; $s33i = zs33 + (zde11 + zde22) * m_e2 + zde33 * m_e1 ; $s33i = $s22i $s33i = .5*($s11i+$s22i) $sdif = $s11i - $s22i $s0 = 0.5 * ($s11i + $s22i) $rad = 0.5 * sqrt ($sdif*$sdif + 4.0 * $s12i*$s12i) ; ----principal stresses --$si = $s0 - $rad $sii = $s0 + $rad $psdif = $si - $sii ; --- determine case --; section ; if $s33i > $sii then ; --- s33 is major p.s. --; $icase = 3 ; $s1 = $si ; $s2 = $sii ; $s3 = $s33i ; exit section ; end_if ; if $s33i < $si then ;; --- s33 is minor p.s. --172 Appendix A ; $icase = 2 ; $s1 = $s33i ; $s2 = $si ; $s3 = $sii ; exit section ; end_if ; --- s33 is intermediate --$icase = 1 $s1 = $si $s2 = $s33i $s3 = $sii ; end_section section ; --- shear yield criterion --$fs = $s1 - $s3 * $anphi + m_csnp $alams = 0.0 ; --- tensile yield criterion --$ft = m_ten - $s3 $alamt = 0.0 ; --- tests for failure --if $ft < 0.0 then $bisc = sqrt(1.0 + $anphi * $anphi) + $anphi $pdiv = -$ft + ($s1 - $anphi * m_ten + m_csnp) * $bisc if $pdiv < 0.0 then ; --shear failure --$alams = $fs / m_x1 $s1 = $s1 - $alams * (m_e1 - m_e2 * m_npsi) $s2 = $s2 - $alams * m_e2 * (1.0 - m_npsi) $s3 = $s3 - $alams * (m_e2 - m_e1 * m_npsi) m_ind = 1.0 else ; --tension failure --$alamt = $ft / m_e1 $tco= $alamt * m_e2 $s1 = $s1 + $tco $s2 = $s2 + $tco $s3 = m_ten m_ind = 3.0 ; -----end_if else if $fs < 0.0 then ; --shear failure --$alams = $fs / m_x1 $s1 = $s1 - $alams * (m_e1 - m_e2 * m_npsi) $s2 = $s2 - $alams * m_e2 * (1.0 - m_npsi) 173 Appendix A $s3 = $s3 - $alams * (m_e2 - m_e1 * m_npsi) m_ind = 1.0 else ; --no failure --zs11 = $s11i zs22 = $s22i zs33 = $s33i zs12 = $s12i exit section end_if end_if ; --- direction cosines --if $psdif = 0.0 then $cs2 = 1.0 $si2 = 0.0 else $cs2 = $sdif / $psdif $si2 = 2.0 * $s12i / $psdif end_if ; --- resolve back to global axes --case_of $icase case 1 $dc2 = ($s1 - $s3) * $cs2 $dss = $s1 + $s3 zs11 = 0.5 * ($dss + $dc2) zs22 = 0.5 * ($dss - $dc2) zs12 = 0.5 * ($s1 - $s3) * $si2 zs33 = $s2 case 2 $dc2 = ($s2 - $s3) * $cs2 $dss = $s2 + $s3 zs11 = 0.5 * ($dss + $dc2) zs22 = 0.5 * ($dss - $dc2) zs12 = 0.5 * ($s2 - $s3) * $si2 zs33 = $s1 case 3 $dc2 = ($s1 - $s2) *$cs2 $dss = $s1 + $s2 zs11 = 0.5 * ($dss + $dc2) zs22 = 0.5 * ($dss - $dc2) zs12 = 0.5 * ($s1 - $s2) * $si2 zs33 = $s3 end_case ; zvisc = 0.0 end_section 174 Appendix A ; ----------------------------------------------------------------------; -----UBC add on to account for change of elastic and plastic parameters ; ----------------------------------------------------------------------; -------- PLASTIC STRAINS --if m_ind = 1.0 then $de1ps = $alams $de3ps = -$alams * m_npsi $eps1 = abs($de1ps-$de3ps) $epn1 = -($de1ps+$de3ps) $eps = $eps + $eps1*zart ; always positive $epn = $epn + $epn1*zart end_if ;-------------- STRESSES $sig = -0.5*(zs11+zs22) $sd = -(zs11-zs22) / 2.0 $sxy = zs12 $sy = -zs22 $dumsig = $dumsig + $sig*zart $dumsd = $dumsd + $sd *zart $dumsxy = $dumsxy + $sxy*zart $dumsy = $dumsy + $sy *zart $area = $area + zart SECTION ; ---GET AVERAGE VALUES OF STRESSES AND STRAINS if zsub > 0.0 then ;zsub loop ;-----STRAINS $epsav = 0.0 $epsum = 0.0 $epsav = $eps / $area ;PLASTIC SHEAR STRAIN INCREMENT $epsum = $epn / $area ;PLASTIC VOLUMETRIC STRAIN INCREMENT $eps = 0.0 $epn = 0.0 ;---- STRESSES $sig = $dumsig/$area $sig = max($sig,0.0005*$pa) ;0.005 APR 5 EN DECREASE DEFAULT BY 10 $sd = $dumsd/$area/$sig $sxy = $dumsxy/$area/$sig $sy = $dumsy/$area $sy = max($sy,0.0005*$pa) ;0.005 APR 5 EN DECREASE DEFAULT BY 10 m_ratioy = $sxy/$sy * $sig if m_triax = 1.0 then ;To simulate triaxial or plane strain tests having sxy = 0 only m_ratioy = (1.0 - $sy/$sig) ;When sxy = 0.0, control by sxx-syy end_if m_ratio = sqrt($sd*$sd+$sxy*$sxy) m_ratio = min(m_ratio,m_ratf) 175 Appendix A if abs(m_ratioy)> m_ratf then m_ratioy = m_ratioy*m_ratf/abs(m_ratioy) ;EN Apr 06 2010 end_if $dumsd = 0.0 $dumsxy = 0.0 $dumsig = 0.0 $dumsy = 0.0 $area = 0.0 ;************************************ ; Resets yield loci and other factors if m_ratmax = 0. then m_ratmax3 = 0.0 ; maximum mobillized stress ratio when ; cycles do not have cross-over. Reset to ; zero each time there is a cross-over. m_ratmax1 = 0.0 m_ratmax0 = 0.0 m_ratmax = 1.0 m_ratmob = m_ratio m_epsum = 0.0 m_epsum1 = 0.0 m_epsum4 = 0.0 m_epsum5 = 0.0 m_epsum6 = 0.0 m_ncyc = 0.0 m_ncyc1 = 0.0 m_ratcrs = m_ratio m_cross = 10.0 m_signal1 = 0.0 end_if ;************************************** m_dratmob = 0.0 if $epsav > 0.0 then ; PLASTIC LOOP m_epsum = m_epsum + $epsum m_epsav = m_epsav + $epsav m_epsum7 = m_epsum7 + $epsav ; EN Apr 8 2010 cumulated plastic shear strain following cross-over if $epsum < 0.0 then ; DILATION m_epsum1 = m_epsum1 + $epsav ; Nov1 2008 accumulated Plastic shear ; strain associated with expansion. ; Set to ;zero at end of each half ; cycle m_epsum5 = m_epsum5 + $epsav ; May 11, 2009 accumulated Plastic ; shear strain associated with ; expansion. 176 Appendix A end_if m_ratmax0 = max(m_ratmax0,m_ratioy) m_ratmax1 = min(m_ratmax1,m_ratioy) ;----Evaluate anisotropy factor m_css=$cs2 if m_css>=0.0 then m_$fac = m_anisofac else m_$fac = m_anisofac + (m_anisofac - 1.0) * m_css end_if ;----PLASTIC SHEAR MODULUS m_knew = m_kgp/$sig * $pa*($sig/$pa)^m_np*m_hfac1 ;-------secondary yield: $hard1 = max(0.5, 0.1*m_n160) ;correction at low N160 $hard1 = min(1.0, $hard1) m_knew1 = m_knew*( 4. + m_ncyc1) *$hard1 * m_hfac2 ;-------primary yield: if m_ocr <= 2.0 then if m_ratioy > 0.0 then $ratmax = m_ratmax0 else $ratmax = abs(m_ratmax1) end_if if abs(m_ratioy) > 0.99*$ratmax then m_knew1 = 0.5*(m_knew+m_knew1)*m_$fac end_if end_if ;-------modify for stress ratio: m_gpstar = m_knew1 *(1.0-(m_ratio*m_rf/m_ratf))^2 ; Modify for dilation ;-------dilation "softener" to control post-liq: ;-------m_epsum4 is the accumulated shear strain associated with dilation during ; the previous stress pulse. $hard = 1.0 if m_hfac3 > 0.0 if m_epsum4 > 1e-6 then ; if m_signal1 = 0.0 ; added July 24,2009 $epsum5 = m_epsum5 else $epsum5 = m_epsum6 end_if 177 Appendix A $hard = max(0.001,exp(-$epsum5*110.0*m_hfac3)) ; $hard range ; 1 to 0.001, ; July 26,09. ;;; m_knew2 = m_knew * $hard ; add this line if want to plot m_knew2 m_gpstar = m_knew * $hard end_if end_if ;--- Raise Yield locus, m_ratmob m_dratmob = m_gpstar*$epsav m_dratmob = max(m_dratmob,0.0) m_ratmob = m_ratmob + m_dratmob m_ratmob = min(m_ratmob,m_ratf) ;current yield locus end_if ;------------------------------------ ;END OF PLASTIC LOOP ;---- Lower m_ratmob upon unloading************************************* if $epsav = 0.0 ; All sub zones elastic If m_ratio < 0.90 * m_ratmob m_ratmax3 = max(m_ratmax3,m_ratmob) ; EN Apr 4 2010 if m_ratmax3 < m_ratpt then ; EN Apr 4 2010 m_ratmob = 0.95*m_ratmob else if m_cnt < 1.0 then m_ratmob = max(0.95*m_ratmob,(m_ratpt*m_hfac6)) ; EN April 3 2010 end_if end_if end_if end_if ;---CROSSING AXIS RESETS PLASTIC PARAMETERS if m_ratioy*m_sxyold < 0.0 then ;crossover check ;------Crossover has occurred $cross = max(m_rtymax,m_rtmax-m_ratcrs) / $ratlim $cross = max($cross, m_gpsum/$gplim) m_ratcrs = m_ratio m_rtmax = m_ratio m_gpsum = 0.0 m_rtymax = 0.0 ;-------Previous half cycle is "large" 178 Appendix A if max(m_cross, $cross) > 1.0 then if m_cross # 99.0 then m_ncyc = m_ncyc + 0.5 m_ncyc1 = m_ncyc1 + 0.5 m_ratmobold = m_ratmob else m_ratmobold = max(m_ratmob,m_ratmobold) endif m_ratmob = m_ratio m_ratmax3 = 0.0 ; EN Apr 4 2010 m_epsum4old = m_epsum4 m_epsum4 = m_epsum1 ; preserves the prior dilation if m_epsum4 > 1e-6 then ; reset 2ary yield surface if dilation m_ratmax0 = 0.0 m_ratmax1 = 0.0 m_ncyc1 = 0.0 endif m_epsum1 = 0.0 if m_cnt < 1.0 then m_epsum8 = m_epsum7 ; comment 1000 m_epsum7 = 0.0 ;EN Apr 8 2010 end_if m_cross = 0.0 ;-------Previous half cycle is "small" else m_ratmob = m_ratio + 0.75*(max(m_ratmobold,m_ratio) - m_ratio) m_epsum1 = m_epsum4 m_epsum4 = m_epsum4old m_cross = 99. ;remember small half cycle endif else ;-------No crossover m_gpsum = m_gpsum + $epsav ;ignore initial crossover step (uses old parameters) m_rtymax = max(m_rtymax, abs(m_ratioy)) m_rtmax = max(m_rtmax, m_ratio) end_if ; End Crossing loop m_sxyold = m_ratioy 179 Appendix A ;---COMPUTE NEW PARAMETERS ACCORDING TO THE CURRENT MOBILIZED FRIC ANGLE ;---------- SET PLASTIC VALUES if m_mohr = 1.0 m_ratmob = m_ratf end_if m_dt = m_ratpt-m_ratmob m_dt = min(m_dt,0.5*m_ratpt) ; EN April 2010 if m_epsum4 > 1e-6 ;Dilation occurring if m_dt > 0.0 then m_dt=(m_ratpt-m_ratmob) else ; Reduce Dilation with plastic shear strain if m_signal1 = 1.0 $epsum5 = m_epsum5- m_epsum6 $hard2 = max(0.01, exp(-($epsum5)^.25*m_hfac4)) ; 0.01 to 0.1 m_dt=(m_ratpt-m_ratmob)*($hard2) end_if end_if end_if if $sig < 0.01*$pa then m_cnt = 99.0 if m_epsum7 < m_epsum8*m_hfac5 then ; if plastic strain since crossover < function cummulated plastic dilation strain then m_ratmob = m_ratpt else m_ratmob = m_ratf ; added March 23 2010 end_if m_dt=(m_ratpt-m_ratmob) if m_signal1 = 0.0 ; Added July 24,2009 m_epsum6 = m_epsum5 m_signal1 = 1.0 end_if else m_cnt = 0.0 end_if m_ratmob = max(m_ratmob,0.01) ; Dilation cut-off (m_dilcut controlled outside of constitive model) if m_dilcut = 1.0 then m_dt = max(m_dt,0.0) end_if 180 Appendix A ;---------- PLASTIC PARAMETERS $sphi = m_ratmob $spsi = -m_dt m_npsi = (1.0 + $spsi) / (1.0 - $spsi) m_nphi = (1.0 + $sphi) / (1.0 - $sphi) ; ---STRESS DEPENDENT ELASTIC MODULI m_k = m_kb * $pa * ($sig/$pa)^m_me m_g = m_kge * $pa * ($sig/$pa)^m_ne m_e1 = m_k + 4.0 * m_g / 3.0 m_e2 = m_k - 2.0 * m_g / 3.0 m_sh2 = 2.0 * m_g m_x1 = m_e1 - m_e2*m_npsi + (m_e1*m_npsi - m_e2)*m_nphi end_if ;--------------------------------------END OF ZSUB > 0 END_SECTION Case 3 ; ---------------------; Return maximum modulus ; ---------------------if m_g = 0.0 then m_k = m_kb * $pa m_g = m_kge * $pa end_if cm_max = (m_k + 4.0 * m_g / 3.0) sm_max = m_g Case 4 ; --------------------; Add thermal stresses ; --------------------ztsa = ztea * m_k ztsb = zteb * m_k ztsc = ztec * m_k ztsd = zted * m_k End_case end opt m_mss set echo=on 181 Appendix A Typical FLAC fish properties code for UBCSAND1v02 def properties loop i (1,izones) loop j (1,jzones) ;ELASTIC m_n160(i,j) = max(m_n160(i,j),1.0) m_kge(i,j) = 21.7*25.*m_n160(i,j)^.333 ; Max. Normalized shear Shear Mod m_kb(i,j) = m_kge(i,j)*0.916 ; Bulk mod = m_kge* (2(1+mu))/(3*(1-2*mu)) m_me(i,j) = 0.5 ;mu = 0.1 , factor = 0.916 m_ne(i,j) = 0.5 ;PLASTIC PROPERTIES m_kgp(i,j) = m_kge(i,j)* m_n160^2*.003 +100.0 ;shear Mod m_np(i,j) = 0.4 m_phipt(i,j)= min(33.0,40.0-m_n160(1,1)*0.75) m_phif(i,j) = 33.0 + (m_n160(1,1)/10.)^1.65 ;plastic modification factors m_hfac1(i,j) = 0.65 ;primary hardener, Controls no. of cycles to trigger liquefaction m_hfac2(i,j) = 0.85 ;Secondary hardener, Refines shape of pore pressure rise with ;cycles m_hfac3(i,j) = 1.0 ;dilation "hardener", Controls post-trigger response m_hfac4(i,j) = 0.6 ;reduces dilation after triggering, 0 = no reduction, 1 is large ;reduction m_hfac5(i,j) = 1.0 ;set length of zero dilation zone following going to origin m_hfac6(i,j) = 0.95 ;set amount of failure envelope pulldown below phipt on ;unloading ;failure ratio --same as in Hyperbolic model m_rf(i,j) = 1.0 - m_n160(i,j)/100. m_rf(i,j) = max(m_rf(i,j),.5) m_rf(i,j) = min(m_rf(i,j),.99) ;plastic anisotrophy ;m_anisofac(i,j) = .0166*m_n160(i,j) ;m_anisofac(i,j) = min(m_anisofac(i,j),1.0) ;m_anisofac(i,j) = max(m_anisofac(i,j),0.333) m_anisofac(i,j) = 1.0 ;m_anisofac ;Anisotrophy factor; 1 for isotropic, .333 for loose pluviated end_loop end_loop end 182 Appendix A Figure A-1 Flow diagram for UBCSAND1v02 illustrating how it fits within the FLAC (ITASCA 2008) Mohr Coulomb framework. 183 Appendix B APPENDIX B ANALYSIS OF A CONCRETE GRAVITY DAM OVER POTENTIAL LIQUIFIABLE SOIL ILLUSTRATING PROPOSED HYBRID PROCEDURE METHODOLOGY B.1 Introduction This example project is given to illustrate the design procedure that has been developed using UBCSAND and UBCHYST with a post-shaking total stress check for stability with empirical post-liquefaction residual strengths. The analyses were carried out for a concrete gravity dam over potentially liquefiable foundation soils. This is work from an actual project but is given here as an example analysis using the proposed hybrid effective stress – total stress procedure. The structure analyzed consisted of a concrete intake structure with downstream penstocks founded on fluvial, deltaic and glacial sediments. The analyses were carried out to assess the seismic performance of the concrete intake structure on a section normal to the axis of the dam. The intake structure is a concrete gravity dam approximately 13m high, 43m wide by 24m long, and has a sheet-pile and jet-grout installed bentonite-cement cut-off wall located under and adjacent to the structure. The dam is located in an area of high seismicity. Preliminary analyses were carried out using seven natural earthquake records that were linearly scaled so that the spectral values of the average response spectrum are similar to those of the uniform hazard response spectrum within the natural period of interest between 0.75s and 1.5s. Peak ground acceleration of the records varied from 0.52 to 0.74 g. B.2 Dynamic Numerical Analyses Two dimensional (2-D) non-linear dynamic numerical analyses were carried out using the finite difference program FLAC version 6.0 (ITASCA 2008). The analyses were carried out in ‘ground water mode’ and flow and pore pressure redistribution were allowed. 184 Appendix B Cohesionless (sandy) soils were modeled using the effective-stress constitutive model UBCSAND, while non-liquefiable cohesive (clay/silt) soils were modeled using the total stress constitutive model UBCHYST. In this context, ‘effective stress’ refers to constitutive models where shear strain, skeleton volume change, and pore pressure are coupled and directly included in the model. In the ‘total stress’ model, shear strain does not induce volume or related pore pressure change; however, pore pressure changes are indirectly accounted for by reducing stiffness and strength when pore pressure build-up or liquefaction is predicted. The concrete intake structure and reservoir water were included in the model. Figure B-1 shows a typical numerical model profile and material zoning. Descriptions of material properties are summarized in Table B-1. The model grid is illustrated in Figure B-2. In FLAC, the dynamic analyses were carried out in groundwater mode in a chronological manner simulating in-situ conditions as described below. The general procedure used for analyses included the following steps: • Set up model grid, material properties and pore water regime and bring to static equilibrium using Elastic and then Mohr Coulomb constitutive models. • Select representative elements for calibration of UBCSAND and UBCHYST. For each selected element note the representative material properties (shear wave velocity, vertical and lateral effective stress, and (N 1 ) 60 values) for use in the calibration. • Calibrate UBCSAND constitutive model by exercising a single element model. Calibration factors are adjusted so the response is in agreement with the Idriss and Boulanger (2008) liquefaction triggering chart and to match typical post-liquefaction response from laboratory tests. • Calibrate UBCHYST to give similar cyclic softening to that inferred from the project cyclic triaxial laboratory tests. • Enter calibration factors in the large model and bring to static equilibrium. • Turn on flow and change to dynamic configuration with large strain, multistepping, and nominal 1% Rayleigh damping and bring to equilibrium by 185 Appendix B running with input motion of zero. (When damping, model parameters, or FLAC configuration (i.e., static to dynamic mode) are changed in FLAC it often takes time for the model to return to equilibrium. Because of this, it has been found to be good practice to let the model come to equilibrium in dynamic mode (by running with zero input motion) prior to carrying out the dynamic analysis). • Firm ground (Pleistocene soil) outcropping time histories were converted to a format that could be used for input to the base of the FLAC numerical model. Two alternative procedures were used: 1. The half-space below the model was assumed to be flexible and input motions were given as stress time histories; or, 2. The half-space was assumed rigid and input motions were given as velocity time histories. • Set displacements to zero, apply earthquake time history at base, and solve past end of earthquake. • Check model resistance against static flow slide using empirically developed Idriss and Boulanger (2008) residual strengths (Figure 4-1(d)). • Calculate additional post-liquefaction consolidation settlement using equations by Ishihara and Yoshimine (as given in Idriss and Boulanger 2008). • B.3 Compile and summarize results of analyses. Results Typical end-of-earthquake shaking displacements are shown in Figures B-3 and B-4 for the horizontal and vertical directions respectively. Typical pore pressure ratio (R u ) at the end of earthquake shaking is shown in Figure B-5 and displacement time histories are shown on Figure B-6. As shown in Figure B-6, changing to the empirical residual soil strengths (based on back-analyses of case histories) in all zones that liquefied (R u > 0.7) did not cause any significant additional movements. 186 Appendix B B.4 Application Example Discussion and Conclusions Over 50 dynamic coupled effective stress analyses were carried out with various soil parameters and model profiles. Select analyses have been checked for post-liquefaction stability using empirically derived residual strengths within liquefied zones. Select analyses to estimate post-shaking consolidation settlement were also carried out. Due to the complexity of the problem and approximations in the methodology there will be significant uncertainty in the calculated displacements. However, the analyses give an approximate range of displacements and give considerable insight into potential behaviour and failure modes. Key findings of the analyses are as follows: (1) the intake structure only moves during earthquake shaking and stops moving at end of shaking (excepting post-shaking consolidation settlement); (2) the main movements are of a downward bearing failure type and a lateral shifting that is resulting from shear in relatively deep liquefied zones; (3) there is some tilting of the intake structure (upstream end settles more than downstream end); (4) there will be a relatively sharp differential vertical displacement (approximately. 0.1m to 0.7m) between the downstream toe of the intake structure and the ground surface a few metres further downstream; and, (5) depending on the earthquake record and model profile, the intake structure may move either upstream or downstream. In some analyses, reversing the direction of earthquake motion resulted in a change in the direction of net lateral movement. A post-liquefaction stability check in which the shear strength of liquefied zones were set to an empirically derived residual strength by Idriss and Boulanger (2008) did not result in any significant movements of the dam. Reducing the residual strength by a factor of 1.2 did not significantly change the result. In all the analyses, lateral movement of the dam centre was less than 0.6m (upstream or downstream) and shear strain settlement was less than 0.9m. Post-shaking consolidation settlements were all less than 0.2m. In analyses where the upper silt (Unit 3a and 3b in Figure B-1) was assumed to behave as loose sand, shear strain settlements were 187 Appendix B more than double and post-shaking consolidation settlements were more than fifty percent higher than when the silt was assumed to cyclically strain soften but not liquefy in a loose sand-like manner. Of the seven final earthquakes run, the Chi Chi TCU071-W earthquake record gives the highest settlements. For identical model and assumptions, the TCU071-W record gives settlements that are 50% higher than the average settlements (TCU071-W gives -0.75m, whereas, the average is -0.5m). Including vertical motion in addition to the horizontal base, excitation did not significantly change the resulting displacements. Applying the input motions as a velocity in lieu of as a stress history appeared to give slightly higher vertical settlements (approximately 10%). Calculated displacements were similar whether the water in the reservoir was modeled as an extremely weak soil or as applied pressures to the mud-line and dam. Extending the downstream length of the mesh by 100m resulted in a small (10%) decrease in settlement. In all analyses, lateral movements at the center of the intake structure were less than about 0.5m, vertical shear induced settlement less than 0.9m and post-shaking consolidation settlements of the intake structure less than 0.2m. Some analyses were carried out assuming that the upper silt soil units 3a and 3b would behave as loose sand and are deemed to give conservative upper-bound displacements. Considering this, best estimate centre of dam displacements, including post-shaking consolidation settlement, are deemed to be about 0.5m lateral movement (could be either upstream or downstream) and 0.5m settlement. On average, the upstream toe of the intake structure settled approximately 25% more and the downstream toe 33% less than the centre of the dam. Considering the various uncertainties, values in the range one-half to double the best estimate lateral displacement values and onehalf to one and one half (1.5x) the vertical settlements are suggested for the assessment of the intake structure. 188 Appendix B Table B-1 Summary of Soil Properties used in Example Analysis of Appendix B ELEVATION(1) SOIL LAYER 1b 2a 3a(M) 3a(S) 2b 3b(M) 3b(S) 2c 4 5 TILL J m W Notes: (1) (2) (3) TYPE SAND SAND/SILT SILT SILT SAND SILT SILT SAND/SILT SAND SILT SAND/SILT JET GROUT BOTTOM MUD WATER TOP (m) 133 127 122 122 119 117 117 112 106 99.5 79 BOT (m) 127 122 119 119 117 112 112 106 99.5 79 77 CTR(2) THICKNESS DEPTH (m) (m) 6 3 5 8.5 3 12.5 3 12.5 2 15 5 18.5 5 18.5 6 24 6.5 30.25 20.5 43.75 2 55 (N1)60-CS (3) DYN (blows/ft) 25 23 14 14 20 14 14 22 34 10 N/A N/A POROSITY DRY FRICTION (n) DENSITY (kg/m^3) 1632 1576 1545 1545 1633 1550 1550 1574 1639 1585 1650 1300 1243 1000 ANGLE(5) (degrees) 38 0 34.1 34.1 0 0 34.1 35.2 36.4 0 N/A 0 0 0.1 to 1.0 (4) RESID (blows/ft) 25 21 N/A 14 20 N/A 14 20 N/A N/A 0.39 0.44 0.39 0.39 0.44 0.41 0.41 0.41 0.39 0.41 0.41 0.51 0.55 PERMEABILITY COHESION (kPa) 0 0 145 0 0 145 0 0 0 145 N/A 100 0.5 0.1 to 1.0 Vs (m/s) 300 300 300 300 300 300 300 300 330 310 760 275 40 N/A k k(flac) (cm/s) (m s/kg) 0.00042 4.2E-10 1.6E-06 1.6E-12 0.000012 1.2E-11 0.000012 1.2E-11 1.6E-06 1.6E-12 1.6E-06 1.6E-12 1.6E-06 1.6E-12 1.6E-06 1.6E-12 0.000012 1.2E-11 1.6E-06 1.6E-12 1.6E-06 1.6E-12 1E-07 1E-13 1.6E-06 1.6E-12 ELEVATION TAKEN IN SECTION IMMEDIATELY DOWNSTREAM OF INTAKE STRUCTURE (at i =185 (x=20 m)) CTR DEPTH = Depth of center of layer in section by downstrean end of intake structure DYN = equivalent clean sand (N1)60-CS for use with UBCSAND in dynamic analyses (4) RESID = equivalent clean sand (N1)60-CS for use in post-shaking residual strength calculation (5) FRICTION ANGLE DURING DYNAMIC ANALYSIS = 33 + (N1)60/10 Figure B-1 Example Dam analysis model profile and soils types. Soil properties are summarized in Table B-1. 189 MODEL UBCSAND UBCSAND UBCHYST UBCSAND UBCSAND UBCHYST UBCSAND UBCSAND UBCSAND UBCHYST ELASTIC UBCHYST UBCHYST MOHR Appendix B Figure B-2 Example Dam analyses model grid details. Figure B-3 Example Dam analysis typical post-earthquake horizontal displacement contours. 190 Appendix B Figure B-4 Example Dam analyses showing typical post-earthquake vertical displacement contours. Figure B-5 Example Dam analysis showing typical post-earthquake pore pressure ratio contours. 191 Appendix B Figure B-6 Time histories showing displacement of center of intake structure and model base (top of till) for typical analysis (positive is downstream or upward). 192
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A hybrid effective stress – total stress procedure for analyzing soil embankments subjected to potential… Naesgaard, Ernest 2011
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Title | A hybrid effective stress – total stress procedure for analyzing soil embankments subjected to potential liquefaction and flow |
Creator |
Naesgaard, Ernest |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | Seismic design of major civil structures (bridges, dams and embankments) is moving increasingly towards using performance design methodologies which require determination of earthquake induced movements. Development of these numerical design tools and procedures for use in engineering practice for estimating the earthquake induced ground deformations of potentially liquefiable soil is the topic of this dissertation. Fully coupled effective stress numerical analyses procedures developed at the University of British Columbia (UBC) were used to simulate field and centrifuge test case histories. These analyses can offer considerable insight, but due to the complexity of the problem and variability of the parameters involved, there is considerable uncertainty. The author, therefore, recommends that the relatively new state-of-the-art effective stress analyses should be augmented by carrying out an additional analysis compatible with conventional design processes. This latter analysis uses published post-liquefaction “residual” soil strengths derived from back-analysis of field case histories by others. The developed design methodology uses the effective stress (UBCSAND) soil constitutive model for dynamic analyses, and empirical “residual” post-liquefaction soil strengths for a post-shaking total stress static analysis. In the proposed approach, the effective stress dynamic analysis is used to determine zones of liquefaction, to quantify earthquake induced deformations, and to provide overall insight. The post-shaking total stress static analysis, with “residual” strength parameters used in elements which liquefied, is carried out to capture the effects of complex stratigraphy and localization that may be missed by the effective stress model. Calibration and validation of the UBCSAND model was undertaken by comparing the model with field case histories and laboratory simple shear, shake table, and centrifuge tests. The measured response of some centrifuge tests being used for validation was indicative of the centrifuge model not being fully saturated. This was problematic as P-wave measurements within the centrifuge model suggested full saturation. A series of triaxial tests with P-wave measurements was carried out. These tests, and the numerical modeling of them, showed that high P-wave velocities were not always indicative of full saturation and they provided a logical explanation for the observed centrifuge response. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-04-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution 3.0 Unported |
DOI | 10.14288/1.0063041 |
URI | http://hdl.handle.net/2429/34004 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2011-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by/3.0/ |
AggregatedSourceRepository | DSpace |
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