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Incremental effective stress liquefaction modelling of sands McIntyre, Jay Duncan 1995

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I N C R E M E N T A L E F F E C T I V E STRESS LIQUEFACTION M O D E L L I N G OF SANDS by JAY D U N C A N McINTYRE B . A . S c , The University of British Columbia, 1992 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTERS OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Civil Engineering We accept this thesis as conforming to^be r e q u i r e d s t a n d a r d T H E UNIVERSITY OF BRITISH COLUMBIA April, 1996 ® Jay Duncan Mclntyre, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date V 7 DE-6 (2/88) 11 ABSTRACT A one dimensional incremental effective stress liquefaction model is presented. Two versions of the model have been developed - a static model to simulate the behaviour of a sand element to laboratory cyclic simple shear tests, and a dynamic model to compute the response of sand under harmonic or earthquake loading. For undrained cyclic loading, neither model incorporates strain softening response typical for loose sands, but have been calibrated to predict the response type known as cyclic mobility. The static version of the model is calibrated to match characteristic laboratory cyclic simple shear test results. Comparisons between laboratory results and modelled data are presented. The dynamic version of the model incorporates the key components of the static model, and can be used to predict the response of a simplified soil profile under dynamic loading. In particular, earthquake acceleration time history records may be used as input. The dynamic model was applied to simulate the field case history recorded at the Wildlife Site in California during the 1987 Superstition Hills earthquake. The recorded downhole acceleration time history was used as input for the dynamic model and the predicted response, in terms of surface acceleration, relative displacement, and porewater pressure, compared with the measured values. The predicted acceleration and displacement time histories are in good agreement with the measured values in terms of both the amplitude and characteristic frequency of response. The predicted porewater pressures are not in good agreement with the recorded values. The predicted porewater pressure rise is much faster than the measured data. The slower response may be due to limitations in the compliance of the measuring system and to the possibility that liquefaction did not occur simultaneously at all points in the liquefied layer. iii TABLE OF CONTENTS ABSTRACT ii T A B L E OF CONTENTS iii LIST OF FIGURES v A C K N O W L E D G E M E N T S viii Chapter 1: Introduction 1 Chapter 2: Observed Behaviour of Sand 2.1 Monotonic Loading 3 2.1.1 Drained Monotonic Loading Behaviour 3 2.1.2 Undrained Monotonic Loading Behaviour 6 2.2 Cyclic Loading 9 2.2.1 Drained Cyclic Loading Behaviour 9 2.2.2 Undrained Cyclic Loading Behaviour 12 13 2.3 Summary 20 Chapter 3: Incremental Stress-Strain Model 3.1 Approach to the Problem 22 3.2 Linear Elastic Isotropic Model 22 3.2 Shear-Volume Coupling 23 3.2.1 Martin, Finn and Seed (1975) Model 23 3.2.2 Byrne (1991) Model 25 3.3 Proposed Shear-Volume Coupling Equation 29 3.4 Shear Stress-Strain Law 31 3.5 Calculation of Porewater Pressure Response 33 3.6 Summary of Static and Dynamic Models 34 3.6.1 Static Model 35 3.6.2' Dynamic Model 36 3.7 Key Input Parameters 37 3.7.1 General 37 3.7.2 Maximum Shear Modulus, Gmax 38 3.7.3 Undrained Strength, Su 39 3.7.4 Failure Ratio, Rf 41 3.7.5 Constrained Modulus, M . _ 41 3.7.6 Constant Volume Friction Angle, (|)cv 42 3.7.7 Coefficient of At-Rest Earth Pressure, k,, 43 iv Chapter 4: Model Calibration 4.1 General 45 4.2 Comparison with Laboratory and Empirical Data 45 4.2.1 Calibration to Seed's Liquefaction Chart 45 4.2.2 Prediction of Laboratory Test Data 47 Chapter 5: Verification of Model with Field Event 5.1 General 55 5.2 Background 55 5.3 Site Description 55 5.4 Instrumentation 58 5.5 Recorded Site Response 58 5.6 Controversy Regarding Piezometer Response 63 5.7 Interpretation of Recorded Site Response 66 5.8 Analysis Procedure 67 5.9 Results of Analysis 67 Chapter 6: Summary and Conclusions 72 REFERENCES 74 APPENDIX 1 Input Parameters 77 APPENDIX 2 Predicted Overall Response to Undrained Cyclic Loading 80 APPENDIX 3 Numerical Integration Technique 85 APPENDIX 4 Print-out of Static and Dynamic Programs 88 LIST O F FIGURES Figure Page 2.1 Stress-strain and volume change response of granular skeleton to 4 monotonic simple shear loading. 2.2 Relationship between stress ratio and strain increment ratio for 5 Touyoura sand (after Matsuoka, 1974). 2.3 Characteristic response of saturated sands in undrained static 7 compression (after Chern, 1985). 2.4 Compaction versus shear-strain history (after Youd, 1972). 10 2.5 Effect of confining pressure sample settlement in 10 cycles (after 11 Silver and Seed, 1971). 2.6 Shear stress ratio versus dilatancy rate in constant load cyclic 12 drained simple shear tests on Ottawa sand (after Lee, 1991). 2.7 Cyclic loading behaviour of contractive sand - true liquefaction 14 and limited liquefaction (after Chern, 1984). 2.8 Cyclic loading behaviour of dilative sand - cyclic mobility (after 16 Chern, 1984). 2.9 Cyclic mobility of dilative sand under simple shear loading (after 18 Sivathayalan, 1994). 2.10 Post-liquefaction monotonic response to simple shear loading (after 19 Sivathayalan, 1994). 3.1 Volumetric strains from constant amplitude cyclic simple shear 24 tests (after Martin et al., 1975). 3.2 Incremental volumetric strain curves (after Martin et al., 1975). 25 3.3 Alternative volumetric strain curves (after Byrne, 1991). 26 3.4 Normalised incremental volumetric strain curves (after Byrne, 1991). 27 vi Figure Page 3.5a Relationship between volumetric strain and shear strain for dry sands. 28 3.5b Relationship between volumetric strain, shear strain, and penetration 28 resistance for dry sands (after Tokimatsu and Seed, 1987). 3.6 Schematic of volumetric strain accumulation due to cyclic shear 30 strains. 3.7 Simplified volumetric strain accumulation over one half cycle or 31 shear strain. 3.8 Shear stress-strain model for unload-reload. 32 3.9 KC T versus effective overburden pressure. 40 3.10 Modified hyperbolic stress-strain formulation (after Byrne and 43 Mclntyre, 1994). 4.1 A comparison of the relationship between cyclic stress ratio and 46 (Ni)6o values for clean sand. 4.2 A comparison of laboratory and modelled volumetric strain curves 48 for constant cyclic shear strain amplitude tests. 4.3 A comparison of the modelled and observed relationship between 50 volumetric strain ratio and number of cycles for dry sands. 4.4 Contractive deformation during cyclic simple shear loading (after 52 Sivathayalan, 1994). 4.5 Predicted response of loose Fraser River sand to cyclic simple shear 54 loading. 5.1 Location of Wildlife Site and epicenters of recent earthquakes in 56 Imperial Valley (after Holzer et al., 1989). 5.2 Plan and cross section of Wildlife Site showing stratigraphy and 57 location of accelerometers and piezometers (after Bennet et al., 1984). 5.3 Cross section of USGS piezometer tip used at Wildlife Site (after 59 Youd and Holzer, 1994). vii Figure Page 5.4 Acceleration time histories - Wildlife Site, 1987 Superstition Hills 60 earthquake. 5.5 Displacement time histories - Wildlife Site, 1987 Superstition Hills 61 earthquake. 5.6 Surface acceleration vs. relative displacement - Wildlife Site, 1987 62 Superstition Hills earthquake. 5.7 Change in soil stiffness during selected cycles - Wildlife Site, 1987 63 Superstition Hills earthquake. 5.8 Measured pore pressure time histories, in terms of pore pressure 64 ratio - Wildlife Site, 1987 Superstition Hills earthquake. 5.9 Comparison of measured and predicted time histories - Wildlife Site, 68 1987 Superstition Hills earthquake. 5.10 Predicted dynamic response at Wildlife Site for the 1987 Superstition 70 Hills earthquake. 5.11 Comparison between measured and predicted pore pressure ratios 71 - Wildlife Site, 1987 Superstition Hills earthquake. A l . l Characteristic cyclic shear stress-strain and effective stress path 83 response for static model. Vlll A C K N O W L E D G M E N T S The author would like to express his appreciation to Dr. Peter Byrne for his assistance and direction throughout the course of this post-graduate work. The comments and editorial remarks made by Dr. Y . Va id are also sincerely appreciated. I would like to thank Dr. Mi l ton Hsu for his guidance and financial assistance during the infancy of this project. Finally, I would like to thank my wife for her encouragement and understanding throughout this entire program. 1 Chapter 1: Introduction A primary concern for earth structures comprised of granular soils is the horizontal and vertical displacements that may result from cyclic loading. The magnitude and direction of these movements is dependent on material factors such as the soil type and density, as well as the magnitude and duration of the cyclic load. Soil structures can experience cyclic loading from many load sources. Some common sources are seismic waves from earthquakes, dynamic ice loads, blast explosions, water waves, and oscillating winds. Cyclic shear loading results in the tendency for granular soils to undergo volumetric compaction, whether the soil is initially loose or dense. If the pore spaces in the soil are filled with an incompressible liquid that cannot escape during the period of cyclic loading, then this tendency for volume change results in the transfer of normal load from the particle skeleton to the pore fluid. This load transfer causes an increase in the porewater pressure and a corresponding reduction in effective stress. As the effective stress decreases, so does the strength and stiffness of the soil, resulting in increased displacements. With sufficient cycles, the effective stress of the soil may reduce to zero and the soil will behave as a viscous liquid. Generally, this liquefied state is transient, and with continued shearing the soil will dilate and regain a portion of its original strength. The transient state of liquefaction is a major concern to engineers as a soil structure may undergo intolerable deformations, rendering it unserviceable or unstable. Because of the potentially damaging effects of the liquefaction phenomenon, the modelling and prediction of liquefaction events has become a major topic of interest to many geotechnical researchers. In the following chapter a summary of the response of granular soil to cyclic loading will be made. Following this, a relatively simple incremental elastic-plastic model will be proposed and verified by comparison with observed laboratory results. As a further step in the validation process, the model will be used in the dynamic mode to predict the response of the Wildlife site - an instrumented site in California where liquefaction occurred during earthquake in 1987. 3 Chapter 2: Observed Behaviour of Sand The stress-strain behaviour of sand is often complex. When the voids in the particle skeleton are filled with water the complexity of the response may increase, especially if the water is unable to drain during the loading sequence. Laboratory experimentation on a large variety of sands has shown that their behaviour is non-linear, inelastic, anisotropic, stress level dependent, and stress path dependent. It is not an easy task to accurately model such a complex material. In order to properly predict the response of sand in-situ, the analytical model must first capture the observed element behaviour from standard laboratory tests. In the following sections, the drained and undrained response of sand to cyclic loading will be reviewed, but first the response of sands to monotonic loading will be briefly discussed. 2.1 M o n o t o n i c L o a d i n g 2.1.1 Drained Monotonic Loading Behaviour The characteristic response of drained or dry sand under monotonic simple shear loading is shown in Fig. 2.1. During shearing of a dense sand, significant energy is required to overcome the interlocking of individual particles. The characteristic stress-strain curve for a dense sand shows a peak stress at relatively low strain (see Fig. 2.1(a)). Thereafter, as the particle interlocking is overcome, the shear stress necessary for additional strain decreases. The reduction in the degree of interlocking results in an increase in the volume of the specimen, as shown in Fig 2.1(b). Eventually. the sample becomes loose enough to allow particles to move around one another without any further net volume change, and the shearing stress reaches an ultimate constant value. For loose sands there is no significant particle interlocking to overcome, and the shear stress increases gradually to an ultimate value without a prior peak. The increase in stress is accompanied by a decrease in volume as shown in Fig. 2.1(b). Thus the loose sample contracts when sheared, while the dense sample initially contracts but then dilates with 4 further straining. When sufficiently strained, both samples achieve a state of constant volume (critical state). Under the same vertical stress, the ultimate shear stress for loose and dense specimens are essentially equal as indicated by Fig 2.1(a), and is given by xu l t = a'-tan<t>cv, where <|>cv is the constant volume friction angle. b Pi CO CO & Dense .• tan(j)c a) Shear Response Shear Strain, 7 CO S3 'S3 -fcs O B CONTRACTIVE - 0 - _ _ Loose -O b) Shear Induced Volumetric Strain CO <D £3 CD c) Contractive - Dilative States Fig. 2.1: Stress-strain and volume change response of the granular skeleton to monotonic simple shear loading. 5 The volumetric response of the loose and dense samples can be further explained by referring to Fig. 2.1(c) which shows the stress paths for both samples. This plot shows that the stress path of the dense sample crosses over the constant volume friction angle line. Fig. 2.1(b) and Fig. 2.1(c), therefore, indicate that beneath the § c v line sands are contractive, while if the stress state is above the (|)cv line, sands exhibit dilative behaviour. In his study of the stress-strain relationships on the mobilised plane, Matsuoka (1974) performed drained triaxial compression and extension tests, as well as plane strain tests, on both Toyoura sand and Sagami River sand. A plot of the experimental results is shown in Fig. 2.2. Matsuoka demonstrated there is a linear relationship between stress ratio, T M P / a M P , a. a -0.3-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 LEGEND» T E S T e T Y P E (kPo) O TC 0 . 89 100 X TE 0 . 6 4 300 A PS 0 . 6 6 =200 + TC 0 . 6 8 196 O TE 0 . 6 8 196 • TC 0 . 6 8 396 V TE 0 . 6 8 3 9 6 WHERE < TC = T r i o x i o l Compression TE = T r i o x i o l Ex tens ion PS = Plo in Stroin Fig. 2.2: Relationship between' stress ratio and strain increment ratio for Toyoura sand (after Matsuoka, 1974). 6 and the strain increment ratio, A s M P / A y M P , where A s M P is the increment of volumetric strain and A y M P is the increment of shear strain. The 'MP' subscript indicates the stresses and the resulting strain increments are on the mobilised plane. The mobilised plane refers to the plane in the sand specimen on which the ratio between the mobilised shear stress and the normal stress, T M P / a M P , is a maximum. This plane forms an angle of 4 5 ° + with the major principle stress plane, where <|>m is the mobilised angle of internal friction. Matsuoka showed that regardless of the test type, sand density, or the confining stress, the linear relationship shown in Fig. 2.2 is valid. Data points lying on the left side of the vertical axis in Fig. 2.2 ( A E M P / A y M P = 0) represent sand in a contractive state, while those on the right side of the vertical axis represent dilative states. The further a particular data point is located from this vertical axis, the more dilative or contractive the sand is at that point. Thus a typical dense sample of sand would travel up the diagonal line from left to right and cross over the vertical axis, while looser samples would never enter the dilative range. 2.1.2 Undrained Monotonic Loading Behaviour The characteristic undrained response of saturated sand under monotonic loading in the triaxial test is shown in Fig. 2.3. Three different stress-strain response types are shown in Fig. 2.3(a). Type 1 and type 2 responses exhibit contractive or strain softening behaviour while type 3 response corresponds to dilative or strain hardening behaviour. Type 1 and type 2 responses are generally associated with loose sand; however, these response types may be observed for denser sands at high confining stresses (Vaid and Chern, 1985). Type 3 response is generally exhibited by denser sands. The effective stress ratio at the point of peak shear stress (Fig. 2.3(a)) has been termed the Critical Stress Ratio (CSR) by Vaid and Chern (1985). The CSR represents the Normal Effective Stress Fig . 2.3: Characteristic response of saturated sands in undrained static compression (after Chern, 1985). 8 onset of strain softening response. Kuerbis (1989), Chern (1985) and Chung (1985) reported that the CSR is unique for a given sand in triaxial compression. In both type 1 and type 2 responses the strain softening commences after the CSR is achieved. Type 1 behaviour has been called liquefaction (Vaid and Chern, 1985; Castro, 1969) or steady-state liquefaction (Kuerbis, 1989) as the sand ultimately reaches a state at which unlimited shear strain may take place under conditions of constant shear and confining stress. This strength is called the steady-state or residual strength. Type 2 response is known as limited liquefaction (Castro, 1969) as a limited amount of strain softening takes place followed by strain hardening. The onset of contractive behaviour occurs at the intersection of the CSR line (Fig. 2.3(c)). The arrest of contractive behaviour occurs at the point where deviator stress is a minimum and where pore pressure reaches a maximum. This point that represents the transition from strain softening to strain hardening behaviour is termed the phase transformation state (Ishihara et al., 1975) and is indicated by a sudden turn in the stress path (Fig. 2.3(c)). Prior to reaching the phase transformation state, positive porewater pressures develop. With further shearing, dilation causes a reduction in pore pressure. After crossing the phase transformation line, the stress path follows the undrained failure envelope as shown in Fig. 2.3(c).^  "Chern (1985) showed that the friction angle, ^pj/ss, at phase transformation or steady state is unique for a given sand under triaxial compression. For a given sand, this angle equals the constant volume friction angle, <|)cv, under drained conditions (Chern, 1985; Negussey et al., 1986). The characteristic stress-strain and porewater pressure behaviour for type 3 response is shown in Fig. 2.3(a). Note that the deviator stress never decreases at any stage of the shearing process. The pore pressure response in Fig. 2.3(b) shows an initial pore pressure increase as the sample tends to contract at the beginning of shearing. However, dilative behaviour is dominant and is indicated by a drop in porewater pressure. A state of phase transformation can be defined for sands exhibiting type 3 response, where the dilative behaviour commences. This will be recognisable for mildly dilative sands by the sudden turn 9 in the effective stress path, as shown in Fig. 2.3(c). However, for highly dilative sands the sudden turnaround may not be observed, and the evidence of phase transformation will only be shown in the pore pressure response (Luong, 1980). 2.2 Cyclic Loading 2.2.1 Drained Cyclic Loading Behaviour When a dry or drained sample of sand is loaded cyclically, it will undergo volumetric compaction whether it is initially loose or dense (Martin et al., 1975). Several researchers have investigated the behaviour of dry sand under cyclic loading. Youd (1972) carried out low strain amplitude cyclic strain controlled tests on dry and saturated drained samples of Ottawa sand using a simple shear apparatus. Fig. 2.4 shows the results of one such test performed on dry sand. The plot shows that for each strain excursion the sample initially contracts but then dilates. However, the net result for each cycle is a finite contraction (reduction in void ratio). After a large number of strain cycles, the sample reaches a constant maximum density. Silver and Seed (1971) observed that in cyclic triaxial tests the magnitude of volumetric compaction before reaching this constant volume state depends on the initial density of the sand sample. This observation is shown in Fig. 2.5. From this figure it is also apparent that the volumetric strain behaviour of cyclically loaded sand is essentially unaffected by vertical stress level. This same observation was made by Youd (1972) who concluded that the volume change behaviour of sands under cyclic loading is not significantly influenced by the value of vertical stress but depends on the shear strain amplitude. Data presented by Lee (1991) indicates that a similar relationship between stress ratio, t /a y , and the dilatancy rate or strain increment ratio, ^ ^ ° / ^ » exists for cyclically loaded sand as does for sand loaded monotonically. Results from drained cyclic simple shear tests on Ottawa sand are shown in Fig. 2.6. Note in this figure dilatancy rate is equivalent to 10 0 . 5 6 0 0 . 3 4 0 0 . 4 2 0 0 4 0 0 - 0 . 0 2 -0 .01 OJOO (0.51) (025) ( 0 0 0 ) Sheor displacement, t, in inches (millimeters) •o.oi *oaz (0.25) (Oil) Fig. 2.4: Compaction versus shear-strain history (after Youd, 1972). strain increment ratio and the shear strain increment, dy, is always positive regardless of deformation direction. Fig. 2.6 indicates the relationship between stress ratio and strain increment ratio is similar for Ottawa sand during the first quarter cycle of the test (i.e. the loading portion) as reported by Matsuoka (1974) during monotonic triaxial and plane strain tests on Toyoura and Sagami River sands (Fig. 2.2). That is, the data points travel from left to right up the diagonal line, from a stress ratio of zero to a maximum x/a. Fig. 2.6 shows that the loose sample, with a relative density of 7%, does not cross over the vertical axis to the dilative side. As the cyclic test continues and the sample is unloaded, the sand must 11 2.0 1.5 1 1 0 c • u 0.5 . D , -45 ptrc«nl i 1 i r u i n t Till ITT • OT (lb per iq ft). o 500 • o 2000 A 4000 i i i i 11 it 0 0.01 0.1 1.0 10.0 2.9 z o r -u < a . O u O v— U i a z < 1.5 1.0 0.5 1 1 1 1 1 1 11 ' Df-60 percent Cycle 10 1—i i i 11 I I 1 1—f 1 1 Ml fj- (lb per sq o 500 o 2000 A 4000 n i A C i 111 A / O.Ot 0.1 1.0 10.0 - 2.0 < 1 5 1.0 0.5 ! D r=80 percent Cycle 10 tr(/b per iq ft)] O 500 • a 2000 A 4000 i • i i 1111 ni A J O - T T T I 7 ' 1 1 1 I I I ! 0.01 0.1 1.0 , 0 ° CYCLIC SHEAR STRAIN - tf^ (percent ) Fig. 2.5: Effect of confining pressure on sample settlement in 10 cycles (after Silver and Seed, 1971). contract. Thus, the denser samples, which had entered the dilative range in the first quarter cycle, immediately cross back into the contractive zone. Upon unloading the initial data points are far to the left of the vertical axis, indicating that the sand is highly contractive at this time. As the unloading continues, the stress ratio decreases and eventually becomes 12 negative. As shearing continues the sand becomes less contractive and travel from left to right down the diagonal line. As before, those sands which are sufficiently dense will cross over the vertical axis into the dilative range, while loose samples will not. Again, when unloading takes place, the dense sands flip from the dilative range into the contractive range. -0.8H 1 1 1 1 1 1 1 1 -0.8 -0.6 -OA -0.2 0.0 0.2 0.4 0.6 0.8 DILATACY RATE , (-^f») Fig. 2.6: Shear stress ratio versus dilatancy rate in constant load cyclic drained simple shear tests on Ottawa sand (after Lee, 1991). As explained above, the application of cyclic loading to drained or dry sand results in a progressive decrease in volume. However, if the sand is saturated, and drainage is unable to occur during the loading sequence, the tendency for volumetric strain will result in a transfer of the confining stress to the pore fluid resulting in a pore pressure rise. This undrained cyclic response will now be discussed in detail. 13 2.2.2 Undrained Cyclic Loading Behaviour During undrained cyclic shearing, a sand specimen may 'fail' or undergo excessive deformation at a shear stress that is lower than its undrained shear strength during monotonic loading. This is especially true when the direction of the shear stress reverses in each loading cycle (Lambe and Whitman, 1969). This phenomena occurs because the excess porewater pressure does not return to zero after unloading, but rather accumulates with cycles. As the pore pressure accumulates, the effective stress is reduced, thereby reducing the shearing resistance of the soil. The increase in pore pressure is caused by the tendency of the soil particles to rearrange when sheared. If the soil was in a drained condition, the cyclic loading would result in a decrease in volume. However, during undrained loading, the overall volume change is zero. This volumetric constraint results in the generation of porewater pressures. As in the monotonic loading case, three different types of response can be observed for cyclic loading. The first two types of response can be grouped together as they both exhibit contractive, or strain softening behaviour. Type 1 response is characterised by large unidirectional deformations due to strain softening behaviour. In general, the sand loses a large portion of its shear resistance and deforms continuously under constant stress (Fig. 2.7(a)). This type of response is termed liquefaction, true liquefaction, or steady state liquefaction (Chern, 1984), and the stress level at which continued strain occurs has been termed steady state or residual strength of the sand. An example of type 2 response, known as limited liquefaction, is shown in Fig. 2.7(b). The stress-strain behaviour is characterised by a limited duration of strain softening behaviour associated with considerable straining and pore pressure development. This strain softening response is triggered when the critical stress ratio is mobilised. Contractive behaviour continues until the phase transformation line is reached. 14 Fig . 2.7: Cycl ic loading behaviour of contractive sand liquefaction (after Chern, 1984). - true liquefaction and limited 15 Following the occurrence of limited liquefaction, continued cyclic loading results in additional strain development due to cyclic mobility (type 3 response). This response is initiated when the stress path reaches the phase transformation line (Fig. 2.7(c)). At this point the path makes a sudden turn as the sand begins to dilate. This dilation phase is indicated in Fig. 2.7(b) by a gradual increase in deviator stress. Upon unloading the stiffness increases to a magnitude comparable to that prior to strain softening, and the corresponding deformations are minimal. Despite the stiff response, the unloading phase is characterised by the generation of large pore pressures, and the effective stresses reduces to essentially zero. As the cycle continues and the sample is loaded in extension, large straining takes place in the opposite direction to the first strain excursion at approximately zero deviator stress. This straining takes place while the effective stress remains near zero. With sufficient straining, the sample will eventually dilate and travel up the undrained failure envelope. This corresponds to an increase in deviator stress on the extension side (Fig. 2.7(b)). Once again, upon unloading the stiffness is high and the pore pressure generation is large, causing the effective stress to once more temporarily reduce to near to zero. Repetition of this cyclic loading and unloading sequence results in a progressive increase in deformation. The accumulation of strain with loading cycles is shown in Fig. 2.7(d). It is important to note that the deformation due to limited liquefaction always occurs before the temporary states of zero effective stress (Vaid and Chern, 1985). Another example of type 3 response is shown in Fig. 2.8. Note that at no point during the cyclic loading sequence does the sand exhibit strain softening behaviour. In this case, significant deformations are only a result of the temporary zero effective stress states that occur as a result of continued unloading and reloading after the phase transformation line has been intersected. This type of response was first called cyclic mobility by Castro (1969). As indicated by Fig. 2.8(b), a progressive decrease in effective stress takes place due to the loading cycles. After sufficient cycles, the phase transformation line is reached, and the stress path experiences a sudden turn. Upon unloading, pore pressure generation increases and the 16 effective stress decreases substantially, but does not necessarily go to zero. In the case shown in Fig. 2.8(b), the phase transformation line is crossed once more before the effective stress is driven to zero. At this point, considerable deformations may develop; however, as the loading cycle continues, the sand dilates and quickly regains strength. Strain accumulation with cycles of loading for cyclic mobility is shown in Fig. 2.8(c). Significant PT Cyclic Mobility Fig. 2.8: Cyclic loading behaviour of dilative sand - cyclic mobility (after Chern, 1984). 17 amounts of deformation may result if numerous momentary zero effective stress states are experienced. Fig. 2.9 shows the cyclic mobility response of sand to cyclic simple shear loading. The response is entirely dilative and significant strains occur only after the stress path crosses the phase transformation line. Note that strains of up to 3.75% are developed without an excursion through a zero effective stress state. Thus, the prior attainment of a transient state of zero effective stress may not be a necessary condition for liquefaction, if liquefaction is defined by an arbitrary strain level. This type of behaviour has been noted in cyclic triaxial studies (Thomas, 1992). It is important to note that in cyclic triaxial tests, the shear stress level at the critical stress ratio (CSR) and the phase transformation (<))cv) lines is considerably smaller in triaxial extension than in compression. As a result, contractive deformation during cyclic loading is always triggered in the extension stage. However, in cyclic simple shear loading, there is symmetric response to horizontal shear stress about the vertical axis. Thus, strain development is symmetric and contractive behaviour can be triggered in either direction. These differences are apparent in Fig. 2.8 and Fig. 2.9. Sivathayalan (1994) studied the post-liquefaction monotonic response of Fraser River sands under simple shear loading. Fig. 2.10 shows the stress-strain and stress path response of a loose sand (D r = 28%) in post-liquefaction monotonic loading. Following triggering of contractive deformation the sand develops large strain (up to about -10%). Unloading of the sand results in large porewater pressure development and very little strain recovery. During monotonic loading, the sand initially deforms at essentially zero stiffness. The stiffness eventually increases with straining, but the rate of increase is initially slow. These test results indicate that looser sands will undergo larger deformation before regaining substantial strength after liquefaction. Fig. 2.9: Cyclic mobility of dilative sand under simple shear loading (after Sivathayal; 1994). 19 o o_ V) W 9 • 4 -J C (/) Cyclic Loading (Last Cycle) Post Cyclic Monotonic Loading Shear Strain 7, % 25 0 25 50 t 75 100 Normal Stress cr», kPa Fig . 2.10: Post-liquefaction monotonic response to simple shear loading (after Sivathayalan, 1994). 20 2.3 Summary Under drained monotonic loading, after sufficient straining, both loose and dense samples of a given sand will achieve a state of constant volume. The ultimate shear stress for both loose and dense specimens will be essentially equal for sands tested under the same vertical or confining stress. This shear stress, Tu1„ is given by xult = a'tan(j)cv, where <j)cv is the constant volume friction angle. The stress path for dense sands will cross over the <|>cv line, which separates the contractive and dilative stress states. Loose sands exhibit purely contractive behaviour, and do not cross over the <|>cv line. When loaded cyclically, drained sands undergo volumetric compaction whether the sample is originally loose or dense. After many strain cycles, the specimen will reach a constant maximum density. The magnitude of volume change before the state of constant volume is reached depends on the initial density of the sand. The overall volumetric strains are non-recoverable, and can be called plastic strains. The volumetric strain behaviour of drained sand under cyclic loading is essentially unaffected by the level of vertical stress, and depends only on the amplitude of the shear strain. Undrained sand, when monotonically loaded, may exhibit one of three stress-strain response types depending on the sand density, confining stress and loading conditions. The first type of response is termed liquefaction or steady-state liquefaction, as the sand undergoes unlimited strain under constant shear stress. A second type of response is possible, where the sand sample exhibits strain softening but eventually strain hardens. This response type is known as limited liquefaction. The phase transformation state represents the boundary between strain softening and strain hardening behaviour. For a given sand the phase transformation angle equals the constant volume friction angle under drained loading conditions. The third stress-strain response type exhibits only strain hardening behaviour. Sand which responds to undrained loading in this manner is considered dilative. 21 The response of undrained sands to cyclic loading is similar to the monotonic response. Generally the stress-strain response is initially stiff, with only small strains developing as the sample is cyclically loaded. The tendency for volumetric contraction results in the generation of porewater pressures and the corresponding reduction in effective stress. If the sand is sufficiently loose, with continued loading cycles the stress path will approach the phase transformation line. At this point, three response types are possible. As for monotonic loading, the first response type is called liquefaction, and is characterised by large unidirectional deformations. The second possible type of response is limited liquefaction, where only a limited duration of strain softening behaviour takes place, associated with considerable straining and pore pressure development. The third type of stress-strain response is cyclic mobility. In this case deformations are only the result of temporary zero effective stress states, and at no point during the cyclic loading does the sand exhibit strain softening behaviour. As mentioned above, there is a tendency for volumetric contraction during the shearing of sands. When a sand sample is saturated and undrained, the volume of the sample cannot change. The tendency for volume change results in an increase in porewater pressure. However, it is important to note that the same undrained response described above can be obtained in a drained sand if the volume of the sample is intentionally fixed during shearing. Thus, the drained "skeleton" response of sands is the key to understanding the overall response to cyclic and monotonic loading. 22 Chapter 3: Incremental Stress-Strain Model 3.1 Approach to the Problem In the preceding chapter, the drained and undrained response of sand to both monotonic and cyclic loading has been reviewed. As outlined in chapter 2, the behaviour of sand under cyclic loading is complex. For any analysis procedure to provide reliable and useful results, the model must capture this complex behaviour. A common approach in the analysis of soil structures is to model the soil as a collection of soil elements. To adequately model the effects of dynamic loading the analysis procedure must take into account the stress, strain and porewater pressure response of each soil element. Once the element behaviour is captured, it may be incorporated into a finite element or finite difference code which can compute the response of the entire soil structure system. The essence of the problem, therefore, is formulating an element stress-strain model that captures the observed laboratory element behaviour discussed in the preceding sections. In the undrained case, the model must match the laboratory results in terms of pre-liquefaction and post-liquefaction response. In the following sections, such a model will be presented. However, first a simplified approach to modelling the shear stress-strain response of sands will be reviewed. 3.2 Linear Elastic Isotropic Model The linear elastic isotropic simple shear model is a simplified representation of the stress-strain response in soils. The element stress-strain equations for the model are as follows: (1) 23 where G t is the tangent shear modulus, and M is the constrained modulus. Both of these moduli will be discussed in sections to follow. Whether in incremental format or not, this model completely uncouples volumetric strain and shear strain. In other words, a volumetric strain can only result from a change in vertical effective stress, and a shear strain can only result from a change in shear stress. However, from discussions in Chapter 2 it is known that volumetric strains are induced from shearing (see Fig. 2.1(b)). Therefore, this formulation is not an accurate representation of the true response of sands to shear loading. To better model the shear stress-strain behaviour, the formulation must incorporate shear-volume coupling. This would require the addition of a shear volume coupling term, D t , into the stress-strain formulation as shown below. The topic of shear-volume coupling will now be discussed in detail. In particular the formulation of the shear-volume coupling term, D t , will be investigated. 3.2 Shear-Volume Coupling A key component in the response of granular material to both monotonic and cyclic loading is the coupling that exists between shear and volumetric strains. It is this shear-volume coupling that induces the porewater pressure rise when drainage is prevented during loading of the sand skeleton. 3.2.1 Martin. Finn and Seed (1975) Model: The first shear-volume coupling model was presented by Martin et al. (1975). This model was developed from constant cyclic shear strain amplitude test data using a simple shear apparatus. The results from constant amplitude simple shear tests are shown in Fig. (2) 24 3.1 . The plot shows the accumulated volumetric strain versus number of cycles for crystal silica sand at a relative density of 45%. It is apparent from Fig. 3.1 that the accumulated volumetric strain increases with cycles for all strain levels, but at an ever decreasing rate. CYCLES Fig. 3.1: Volumetric strains from constant amplitude cyclic simple shear tests (after Martin etal., 1975.) Martin et al. (1975) plotted the same data in terms of volumetric strain increments per cycle as shown in Fig. 3.2. This figure indicates the volumetric strain for a given cycle is a function of the accumulated volumetric strain and the level of cyclic shear strain amplitude. Based on the data shown in these two figures, Martin et al. (1975) developed the following four parameter equation linking shear strains to volumetric strains: A E V = C , ( Y - C 2 £ V ) + ' 1 (3) y + c 4 s v where: Ae v = the net volumetric strain during a load cycle under drained conditions; E v = the accumulated volumetric strain, in percent, from previous loading cycles; Y = the shear strain amplitude, in percent, for the cycle in question; and, 25 Cj = four material constants determined using data plotted from constant strain amplitude cyclic tests. CYCLIC SHEAR STRAIN AMPLITUDE, y% Fig. 3.2: Incremental volumetric strain curves (after Martin et al., 1975). Eqn (3) is relatively complex, and is based on laboratory data for one type of sand at a single relative density. In addition to this, Byrne (1991) stated that Eqn (3) is not always stable. Byrne proposed an alternative two parameter equation that provides good agreement with laboratory measurements over a range of densities. 3.2.2 Bvrne (1991) Model: The parameters for the Byrne (1991) model can be obtained from cyclic load test results, or can be estimated from standard penetration test resistance values. As with the Martin et al. (1975) model, this model can be used to predict plastic volume changes and settlements under dry or drained conditions, along with pore pressure changes in saturated undrained sands. In developing his two parameter model, Byrne plotted the Martin et al. (1975) data in a different form as shown in Fig. 3.3. In this figure the volumetric strain per cycle, Ae v, is 26 w < »-«/) o £C »-UJ 2 _) O > 0.2 < u - J u > z UJ 5 UJ o 0.1 h Fig. 3.3: VOLUMETRIC STRAIN,€ v % Alternative volumetric strain curves (from Byrne, 1991). plotted versus the accumulated volumetric strain, ev, for three different strain levels. By dividing both axes by shear strain the curves of Fig. 3.3 collapse to a single curve as shown in Fig. 3.4. Byrne (1991) represented this curve by the following relation: = C,EXP where for the data shown, Q = 0.8 and C 2 = 0.5. In more general terms, the constant C- can be expressed as follows: (E ~) v v / cycle 1 y (4) (5) 27 However, as there is considerably more data on (s v ) c y c l e l 5 as a function of relative density, Byrne preferred to represent d as follows: ( £ v ) c y c l e l 5 c, = 5y (6) 1.0 MFS Doto points from y • 0 . 1 , 0 . 2 , 0 . 3 % « 6 8 Fig. 3.4: Normalised incremental volumetric strains (from Byrne, 1991.) Byrne proposed that the parameter Q controls the amount of volume change while C 2 controls the shape of the accumulated volume with number of cycles. It is noteworthy that this shape is the same for all relative densities (Byrne, 1991), and thus the parameter C 2 is simply a constant fraction of Ci as follows: 28 Therefore, the incremental shear-volume coupling Eqn. (4) involves only one constant, Q , which depends on the density of the sand. Tokimatsu and Seed (1987) presented accumulated volumetric strain data after 15 cycles for a range of relative densities and cyclic shear strains. They also related the volumetric strain data from cyclic tests to normalised standard penetration resistance values. Both of these plots are presented in Fig. 3.5. Fig. 3.5a is a reproduction of data that was originally presented by Silver and Seed (1971). The dashed lines in Fig. 3.5a represent Fig. 3.5: (a) Relationship between volumetric strain and shear strain for dry sands; (b) Relationship between volumetric strain, shear strain and penetration resistance for dry sands (after Tokimatsu and Seed, 1987). Silver and Seed's interpretation of the laboratory data. Based on these relations, Byrne (1991) developed the following equation relating the parameter Q to relative density: Cj = 7600(Dr) (8) 29 in which D r is the relative density in percent. Using the lines in Fig. 3.5b, Byrne also related Ci to (NOeo values by the following equation. C,=«-T[(N,)„r (9) The soil parameter C 2 is calculated as before from Eqn. (7). The volumetric strain equation (Eqn. (4)) can be used to compute volumetric strains arising from any one-dimensional series of strain pulses. For a random series of strain cycles Byrne (1991) proposed that Eqn. (4) be modified to compute volumetric strains for each half-cycle as follows: ( A £ v ) 1 / 2 C y c l e = 0.5y C l EXP 2u (10) Although computing volumetric strains after each half-cycle considerably increases the resolution of the analysis procedure, the optimal approach would be to calculate volumetric response incrementally. This would allow for a coupled incremental analysis. 3.3 Proposed Shear-Volume Coupling Equation In order to perform a coupled incremental analysis, the increment of volumetric strain, dsv, as a function of the shear strain increment, dy, is required for each time step. To accomplish this, the relationship between cyclic shear strain and accumulated volumetric strain must be adequately defined. Fig. 3.6 illustrates the characteristic shear-volume response of dry or drained sand to cyclic shear strain loading. As discussed in Chapter 2, for each strain excursion, the sand initially contracts but will eventually dilate; however, the net result is a finite increase in density (i.e. contraction) for each cycle. The relative amounts of dilation and contraction within a particular cycle is a function of the sand type and the its 30 initial density. Byrne developed a simple incremental formulation by assuming that the volumetric strain accumulates linearly with shear strain during any half-cycle, as shown in Fig. 3.7. With this assumption, Eqn. (10) becomes dev = 025dyC 1EXP ^ 2b v \ y J (11) (Aev) cycle (Asv) k cycle Fig. 3.6: Schematic of volumetric strain accumulation due to cyclic shear strains. where dev is the increment of volumetric strain associated with the shear strain increment, dy, and y is the largest strain in the current 1/2 cycle. The terms associated with dY can be grouped into a single shear-volume coupling term, D t . Then Eqn. (11) takes on the following form. dev = dY • D t (12) 31 Shear Strain, y ( A E v ) i / 2 C y c l e de v (&£\)V2 cycle dy 2y Fig. 3.7: Volumetric strain as a function of shear strain increment. From Eqn. (12) it is readily apparent that volumetric strain is directly related to shear strain. Therefore, we must know the shear strain behaviour of a granular material before we can predict its volumetric response. An appropriate shear stress-strain relationship is required to link applied stresses to material deformation. 3.4 Shear Stress-Strain Law The model incorporates an incremental shear stress-strain law based on the work of Duncan and Chang (1970). The stress-strain curves are assumed to be hyperbolic as first proposed by Kondner and Zelasko (1963). The formulation results in tangent moduli that vary with both stress level and material density. A tangent shear modulus, Gj, is used and is defined at any stress state by: i2 (13) G . = 41 = G t dy where: G n max 1 - - ^ R , . T f f . the maximum shear modulus that occurs at zero shear strain; the current shear stress; 32 xf = the failure shear stress; and R f = the failure ratio, to be discussed in a later section. To model stress and strain reversals associated with cyclic loading the stress-strain formulation is modified as follows: * -|2 G t - Gmax 1 -Rf (14) where: G m a x = the shear modulus immediately upon unloading; T* = (xA + T); tf* = (Ta + Tf); xA = the shear stress at the reversal point, as shown in Fig. 3.8. The stress level at the reversal point, T a , becomes the reference strain from which the incremental shear modulus is subsequently calculated. T G t - G m a x ( l - *Rf) Fig. 3.8: Shear Stress-Strain Model for Unload-Reload. 33 The shear-volume coupling equation and the stress-strain law for the model have now been discussed. By combining these two components, the undrained response of a soil element may be predicted. The following section describes the method used by the model to calculate the porewater pressure response. 3.5 Calculation of Porewater Pressure Response Under cyclic loading there is a tendency for granular material to undergo volumetric contraction regardless of whether the material is loose or dense. If drainage of the porewater is prevented during loading, a volumetric constraint is imposed on the soil skeleton. The tendency for volumetric contraction results in a transfer of the normal loads from the soil skeleton to the pore fluid, resulting in an increase in porewater pressure. The predicted skeleton response under undrained conditions is calculated using the same skeleton stress-strain model as described in the previous section, but with the addition of the volumetric constraint imposed by the pore fluid. If the porewater and solids are assumed to be incompressible, the overall volumetric strain during an undrained loading increment will be zero. However, grain slip is known to occur within the soil skeleton during undrained loading, meaning that the plastic and elastic components of the volumetric strain increment must be equal and opposite. d s y = d s v e + d e y P =0 (15) The elastic component of the volumetric strain can be calculated from da (16) M Rearranging Eqn. (16) and substituting from Eqn. (15) we get da = M • d s y e = - (17) 34 Since da' = da - du, and if we assume that for level ground conditions da = 0, then the rise in porewater pressure du = - da'. Therefore we can directly calculate the increment of excess porewater pressure as du = M - d s v p = M - D f d y . (18) Thus, after the plastic volumetric strain is calculated from Eqn. (9) the excess porewater pressure increment can be calculated. The total excess porewater pressure is determined by summing all the pore pressure increments. As the model uses an effective stress analysis, the pore pressure is used to modify the incremental shear modulus. 3 . 6 Summary of Static and Dynamic Models Two versions of the model have been developed - a static version and a dynamic version. The static version is designed to simulate the behaviour of a soil element in laboratory cyclic simple shear tests. The dynamic version computes the response of sand under harmonic or earthquake loading. The primary difference between the static and dynamic versions is the source of the shear stress and the mechanism by which it is transferred to the soil. From this point onward the two versions will be considered as separate models, and will be referred to as the static model and the dynamic model, or alternately as the static and dynamic programs. The static model was developed to confirm the model could simulate observed laboratory cyclic simple shear data. Once this was verified, the dynamic model was developed, and incorporates the key components of the static model, including the same shear-volume coupling equation, the same shear stress-strain law, and the same procedure for calculating the excess porewater pressure increment. For undrained cyclic loading conditions, both the static and dynamic models have been calibrated to predict the response type identified as cyclic mobility in Chapter 2. That is, at no point during the cyclic loading sequence does the model predict strain softening 35 behaviour. Significant deformations are only the result of temporary zero effective stress states that occur as a result of continued unloading and reloading after the phase transformation line has been intersected. The characteristic stress-strain and effective stress pattern adopted in the model is discussed in detail in Appendix 2. The reader is reminded at this point that both the static and dynamic models have only one degree of freedom. The purpose of the model is to capture the element shear stress-strain and porewater pressure response of the soil. Once this is accomplished the model can be incorporated into a finite element or finite difference code, where it would govern the response of each soil element within a larger soil structure. Both the static and dynamic versions of the model will be discussed in the following sections. 3.6.1 Static Model The static model is designed to simulate behaviour under laboratory cyclic simple shear tests on a soil element under either drained or undrained conditions. The simulated cyclic simple shear tests can be either load controlled or strain controlled. Depending on what type of test is chosen, an increment of shear stress or shear strain is applied. If a load controlled test is chosen, an increment of shear stress, dx, is applied and the resulting increment of shear strain, dy, is calculated by dy = (19) where G t is the tangent shear modulus defined by Eqn. (14). Once the shear strain increment is computed, the plastic volumetric strain increment, ds v p, is calculated using Eqn. (11). If the test is undrained, the overall volume change in the soil is zero. The excess pore pressure increment, du, is then computed by Eqn. (18) as discussed in Section 3.5. After the calculation of the porewater pressure increment is made, 36 the new vertical effective stress, a v', is computed and the new mean normal effective stress is calculated as follows: 1 + r O 2 ) (20) where k„ is the at-rest coefficient of earth pressure (discussed in Section 3.7.7). The maximum shear modulus is then updated by ^max — ^max - ^ 7 (21) V°"moV and the tangent shear modulus, to be used in the next load increment, is calculated as before: G t = G max * 1 * Rf T f 2 (14) If a strain controlled test is specified, the model follows the above analysis procedure except that an increment of strain is applied and the resulting increment of shear stress is computed via the tangent shear modulus. See Appendix 2 for a more detailed, step-by-step description of the predicted overall response of saturated sands to undrained cyclic shearing. 3.6.2 Dynamic Model The dynamic model contains many of the key components of the static model, including the same shear-volume coupling equation, the same shear stress-strain law, and the same procedure for calculating the excess porewater pressure increment. In its present format, the dynamic model computes the shear stress-strain and porewater pressure response of an idealised soil profile subjected to dynamic ground motions. The analysis procedure is carried out in the time domain. The idealised soil profile consists of a surficial non-liquefiable crust underlain by a liquefiable sand layer. The elevation of the 37 groundwater table is specified by the user. This soil profile is simplified into a single degree of freedom lumped-mass system. The mass of the system is taken as the mass of the surficial crust plus the mass of one-half the thickness of the liquefiable layer. The program was formulated in this manner so that the validity of the model could be assessed by comparison with recorded field events. The dynamic program accepts accelerations as the input motion and computes the response of the simplified system. The program will accept either harmonic input accelerations, or the user can input an actual earthquake acceleration time-history. If harmonic accelerations are specified, then the user must input the peak ground acceleration, the period of the input motion, the number of cycles desired, and the magnitude of the time step. With this information, the model will calculate the input acceleration for each time increment. If an acceleration time-history is chosen, then during each time increment the computer reads the acceleration file and stores the acceleration values into an array. Regardless of how the input acceleration is obtained, the program will next calculate the system response at each time step using a integration technique (see Appendix 3). Once the system displacement is computed, the shear strain increment is calculated by dividing the increment of system displacement by the thickness of the liquefiable layer. As for the static program, the shear strain increment is computed, followed by the calculation of the plastic volumetric strain increment. Next the porewater pressure increment is determined as discussed in section 3.5, followed by the calculation of the new effective stress value. Once the effective stress is updated, the tangent shear modulus, G t , to be used in the next time increment, is computed. 3.7 Input Parameters 3.7.1 General Prior to analysis, both the static and dynamic models require specific properties to be input or calculated. The required input values are generally different for the two programs. 38 In the following sections, six soil parameters that are common to both the static and dynamic programs will be discussed. These six parameters are the maximum shear modulus, G m a x , the undrained shear strength, Su, the failure ratio, R f, the constrained rebound modulus, M , the constant volume friction angle, <|>cv, and the coefficient of earth pressure at-rest, k0. These parameters are either specified directly by the user, or else are calculated during the analysis using empirical correlations. For further information on input parameters, see Appendix 1. 3.7.2 Maximum Shear Modulus - G m a x : The tangent shear modulus, G t , relates shear stresses to shear strains. The maximum shear modulus, G m a x , is a required input parameter for the analysis procedure and represents the stiffness of the soil prior to significant straining. Using an empirical equation developed by Seed and Idriss (1970), the model determines the maximum shear modulus for sands as follows: G m a x = 2 2 ( K 2 ) m a x R o- - u 1/2 (22) where: K 2 = a modulus parameter that depends on the relative density or (N,) 6 0 value of the sand; P a = the atmospheric pressure; o m = the mean normal stress; and u = the present level of porewater pressure. The modulus parameter, K 2 , used in the model is related to ( N ^ values and has the form: K 2 = 20*(N,) 6 0 1 / 3 . This formulation was suggested by Seed et. al. (1986) and allows for analysis of existing databases and case histories where laboratory test data may not be available. When relative density values are known for a sand, a different modulus parameter equation may be used, K 2 = 15 + 0.6*D r, (after Seed et al., 1986). 39 As noted, Eqn. (19) is empirical, being based on collected laboratory data and field experience. These correlations provide an initial estimation of G m a x that is acceptable for some analyses. However, superior methods for determining G m a x exist, including in-situ shear wave velocity measurements, and laboratory bender element tests. If the analysis requires a more accurate value then alternative in-situ or laboratory tests should be conducted and the resulting value of G m a x can be entered as input to the model. 3.7.3 Undrained Shear Strength - su A required input parameter for the analysis is the peak undrained shear strength of the sand. This strength is the maximum shear stress that can be developed in the soil during undrained loading. The model will continually check whether the current stress level exceeds the prescribed undrained strength. If it does, the stress is set equal to the undrained strength. When a direct laboratory or in-situ measurement of the sand's undrained strength exists, this value can be entered into the model as input. In the absence of such data, the undrained strength will be estimated using a correlation with Standard Penetration Test (SPT) normalised blowcounts. The correlation between s u/a v o' and (N^o used in the static and dynamic models is similar to that developed by Stark and Mesri (1992). The correlation has the same shape as Seed's liquefaction curve (Seed et al., 1984). For sands with normalised SPT blowcounts less than or equal to 20, the model multiplies the (N.) 6 0 value by a constant, as follows: A - = 0.015x(N1)6 0_c s (23) where (N^o.^ is the clean sand equivalent, normalised SPT blowcount, and su is the peak undrained strength of the sand. For sands with blowcounts above 20, the s u/a v o' value increases exponentially. The undrained strength is then computed by multiplying the s u/a v o' value by the effective overburden stress and a correction factor, ICj. 40 The KC T correction factor accounts for the influence of overburden pressure on the cyclic resistance of the soil. For soil elements at depth with effective overburden stress exceeding 1 ton per square foot (tsf), thOe K a factor is commonly applied to account for the reduction in liquefaction resistance or cyclic resistance ratio (CRR). The correction factor is defined as the ratio of the CRR at the confining stress of concern to that at the reference stress of 1 tsf. The KC T factor used in the model is based on the results of laboratory tests conducted at the University of British Columbia, as discussed by Pillai and Byrne (1994). The KC T factor decreases with increasing confining stress (as shown in Fig. 3.9) and is expressed as follows: 0.4 0.2 o.o I 1 1 1 1 1 0 500 1000 1500 Efective Confining Stress (kPa) Fig. 3.9: K versus effective overburden pressure. 41 3.7.4 Failure Ratio (Rf parameter) The R f parameter, or failure ratio, is a unitless value used in the shear stress-strain law (Eqn. (14)). It is the ratio of the failure stress level to the ultimate strength of the material, as determined from the best-fit hyperbola. The failure ratio defines the strain level, yf, at which the failure stress occurs. The failure ratio influences the shape of the shear stress-strain response curve and is used to modify the hyperbola to fit the observed data. Fig. 3.10 shows how the failure ratio influences the shear stress-strain response. A R f value of zero specifies a linear elastic-plastic material with a Yf = V G m a x . An R f = 1 specifies a strain hardening material with y f equal to infinity. Laboratory results indicate that for most sands, the failure ratio lies between 0.5 and 0.9 (Duncan and Chang, 1970). The failure ratio can be input directly into the computer code, or, alternatively, can be calculated by empirical relations dependent on relative density and correlated to the SPT (N])^ blowcount value. For materials with blowcounts greater than 10 the proposed relation is as follows: R f = 0.8 - ( ( N | ) ^ " 1 0 ) * 0.125 . (25) For (N^go blowcounts less than 10, the equation to calculate the failure ratio changes as shown below. R f = 0.8 + ( 1 0" ( 1o' ) m ) * 0.2 (26) Thus, the higher the SPT blowcount, the lower the failure ratio and the smaller the failure strain. 3.7.5 Constrained Modulus, M The constrained rebound modulus, M , is the ratio of the of axial stress to axial strain for confined compression. The modulus is typically modelled by the following formulation: 42 M = K M * P a (27) where: K, •M = the constrained modulus number; P. a = the atmospheric pressure, and; a' v = the current vertical effective stress. K M should be determined from unload-reload tests. When this data is unavailable, empirical correlations may be used. Although some studies show that K M does not vary with density, the proposed model assumes a linear variation of K M with (N^Q values. In terms of SPT blowcounts, the value of K M varies from 500 to 2000 over a (Nj)go range of 1 to 20. The higher the blowcount, the higher the K M value. This results in 3.7.6 Constant Volume Friction Angle, <|>cv The constant volume friction angle is a required input parameter for the model and can be determined from laboratory testing. After considerable straining of a drained sand, both the stress and the void ratio achieve values that are independent of the initial density. At this condition the sand will strain without further volume change at a constant stress. This condition has been referred to as the constant volume, ultimate, residual, or critical condition. The shear stress (or deviator stress in the case of triaxial tests) that exists at this condition can be used to define the constant volume friction angle, <|>cv. As mentioned in Chapter 2, for a given sand the constant value friction angle under drained conditions is equal to the phase transformation/steady state angle, ^pj/Ss under undrained triaxial compression. The constant volume friction angle is a function of the soils mineralogy. For uniform fine to medium sands, (j)cv typically varies between 26 to 30 degrees, while for well graded sands, the constant volume friction angle generally lies between 30 to 34 degrees (Lambe and Whitman, 1969). 43 Y Fig. 3.10: Modified hyperbolic stress-strain formulation (after Byrne and Mclntyre, 1994). 3.7.7 Coefficient of At-Rest Earth Pressure, k^ , The coefficient of lateral at-rest earth pressure is the ratio between the horizontal and vertical stresses within a soil deposit. The value of changes during shearing, and this is accounted for in the model by the use of the following equation: 44 k 0 = k 0 0 + ( l - k 0 0 ) - — (28) °"vo where k<,0 is the at rest earth pressure coefficient at the beginning of loading, and r j v o is the initial overburden pressure. During undrained loading, when the pore pressure value equals the overburden pressure, the k,, value becomes unity. That is, the vertical and horizontal stresses are equal when the effective stress within the sand drops to zero. 45 Chapter 4: Model Calibration 4.1 General As discussed in Chapter 3, two stress-strain models have been developed - a static model and a dynamic model. The static model was developed to simulate the characteristic behaviour of sands in laboratory cyclic tests. The dynamic model contains the key components of the static model, and was developed to compute the response of an idealised soil profile under harmonic or earthquake loading. The static version was developed first. Once it was demonstrated that the static model could accurately predict laboratory data and established empirical correlations, the dynamic model was then developed. In Chapter 5 the validity of the dynamic model will be demonstrated through a comparison of predicted and measured field response from a documented liquefaction event. In the following sections of this chapter, the validity of the static model will be verified through comparison between predicted and observed laboratory cyclic shear test results. However, as a first step in the verification procedure, the static model will be used in a comparison with the empirical correlations developed by Seed et al. (1984), commonly referred to as Seed's liquefaction curve. 4.2 Comparison with Laboratory and Empirical Data 4.2.1 Calibration to Seed's Liquefaction Chart Seed and his colleagues (Seed et al., 1984) developed correlations between SPT normalised blowcount values and the cyclic stress ratio to cause liquefaction in 15 cycles of earthquake loading. The correlations are based on a database of field observations from sites that experienced earthquakes ranging in Richter magnitude from M=5.3 to M=8.0. The cyclic stress ratios from these events were adjusted to correspond to a magnitude of 7.5 by dividing them by correction factors proposed by Seed et al. The curve is presented in Fig. 4.1, and represents a lower bound that separates liquefied and non-liquefied response. 46 Also shown in Fig. 4.1 is the curve computed by the static model. The modelled curve was generated by fixing the number of loading cycles to 15 and then varying the cyclic stress ratio over a range of ( N ^ density states. The data points shown on Fig. 4.1 are the miriimum cyclic stress ratio values to cause liquefaction in 15 cycles for a particular sand density. 47 The modelled liquefaction relationship is very similar to the curve developed by Seed et al. (1984). In fact, the model was calibrated to match the response indicated by Seed's liquefaction curve. Only at high blow count values do the curves diverge. Seed's relation indicates that sands with an ( N i ) 6 0 value greater than 30 cannot liquefy. At that density, Seed's curve becomes vertical. However, the liquefaction resistance curve generated by the static model indicates that sands with SPT blow counts greater than 30 may still liquefy. Apparently, Seed et al. (1984) felt that a clean sand with a ( N ^ value greater than or equal to 30 could not liquefy under 15 cycles of loading from a magnitude 7.5 earthquake. The choice of limiting liquefaction response at a blowcount of 30 appears arbitrary, especially since the majority of sites within the database had SPT blowcounts under 30. In fact, only 4 data points out of more than 60 on the liquefaction resistance curve correspond to sites with (Ni)6o values greater than 30. Additionally, the database of earthquakes used to develop the curve consisted mainly of events with cyclic stress ratios less than 0.30. In fact, only 2 events had x a v /a 0 ' values greater than 0.40. It is conceivable that sands with SPT blowcounts greater than 30 could liquefy if cyclic stress ratios greater than 0.45 were experienced. 4.2.2 Prediction of Laboratory Test Data Both cyclic simple shear and cyclic triaxial tests have been used to study and quantify the behaviour of sands under cyclic loading. The two types of testing yield slightly different results as the stress-strain conditions associated with the tests differ. Simple shear loading causes continuous rotation of principal stress axes about the vertical axis, whereas cyclic triaxial loading typically involves a jump in stress axis rotation of 90° when stress reversal occurs. Therefore, for simple shear tests, there is no extension or compression phase of the test. The choice of positive and negative strains is generally arbitrary. The horizontal plane is the plane of maximum shear stress in a simple shear test (Roscoe, 1970), and this generally is the best representation of the field condition. For this reason, the static model was calibrated to match the response observed in simple shear tests. 48 As discussed in Chapter 3, Martin, Finn and Seed (1975) performed constant amplitude simple shear tests on crystal silica sand. Data points from their study are shown in Fig. 4.2. The plot shows the accumulated volumetric strain versus the number of cycles for three different shear strain amplitudes. Also shown in Fig. 4.2 are the volumetric strain curves predicted by using the static model. Overall, the predicted accumulated volumetric strain 2.4 Number of Cycles Fig. 4.2: A Comparison of Laboratory and Modelled Volumetric Strain Curves for Constant Cyclic Shear Strain Amplitude Tests. curves are similar to the laboratory results for all three strain amplitude levels. In particular, the ultimate volumetric strain values (after 50 cycles) are in very good agreement with the test results. However, for the first 20 to 30 cycles, the static model predicts volumetric strains that are slightly less than the laboratory values. This discrepancy increases with increasing shear strain amplitude. Despite these small differences, the overall predicted 49 response matches well with the laboratory response reported by Martin, Finn and Seed (1975). It is not surprising that minor discrepancies exist between the predicted and observed volumetric strains, as the model is designed to simulate laboratory tests for a wide range of sand types. To fine tune the predicted response for a specific type of sand, material-specific parameters could be incorporated into the model. Fig. 4.3 compares the observed and predicted relationship between volumetric strain ratio and number of cycles for dry sands. The volumetric strain ratio represents the volumetric strain value at any number of cycles divided by the strain level at 15 cycles. After performing a review of previous studies, Tokimatsu and Seed (1987) suggested that volumetric strain ratios for dry sands generally fall within the shaded zone shown in Fig. 4.3. The relationship predicted by the static model is shown in Fig. 4.3 by the dashed line. The predicted results of the static model are in clear agreement with the database summarised by Tokimatsu and Seed (1987). Fig. 4.4 shows the laboratory results of a cyclic undrained simple shear test performed on Fraser River sand with a relative density of 28%. For a detailed description of the sample preparation procedure and the testing apparatus, see Sivathayalan (1994). A cyclic shear stress of 12 kPa was applied to the specimen under a vertical effective consolidation stress of 100 kPa. Limited liquefaction was triggered during the eighth cycle of loading. Fig. 4.4a shows the variation of strain for each loading cycle. The shear strains are less than 0.5% for the first seven cycles, although they increase slightly for each cycle. During the eighth cycle the sample undergoes a sudden strain excursion of 10%, indicating a significant change in the stress-strain response. Fig. 4.4b shows the relationship between shear stress and shear strain for the same test. This figure clearly shows the initial stiff response of the sample under the cyclic stresses. The stress-strain curve is near-vertical, with an essentially elastic response. During the eighth loading cycle the stress-strain curve suddenly flattens out before the peak stress is 50 reached. The sample then strains with essentially zero incremental stiffness to approximately the -10% strain level. Upon unloading, the stiffness increases to a level comparable to that during the initial loading cycles. Once the shear stress level changes (crosses over the strain axis), the stress-strain curve once more flattens and the sand strains with a very low shear modulus. Number of Cycles 10 20 30 \ . \ T \ s \ S ^ \ \ T/RGF.ND: - - Static Model Observed (after Tokimatsu and Seed, 1987) liiiifwPsiiv Fig. 4.3: A Comparison of Modelled and Observed Relationship between Volumetric Strain Ratio and Number of Cycles for Dry Sands. 51 Fig. 4.4c shows the stress path for the sand sample during the simple shear test. The critical stress ratio (CSR) line and phase transformation (PT) line have been overlaid onto the plot of the test results. The effective stress path cycles toward the origin in a saw-tooth pattern from an initial effective stress state of 100 kPa. During the eighth loading cycle strain softening behaviour is initiated once the CSR line is crossed. Once the phase transformation line (equivalent to <|>cv line under drained conditions) is reached, the stress path turns sharply. At this point the sand begins to dilate and the shear stress increases slightly. This coincides with the flat portion of the stress-strain curve during the large strain excursion (Fig. 4.4b). Once the stress cycle reaches its peak, the sample unloads and large porewater pressures are generated, driving the effective stress state to the origin. At this point the effective stress in the sand is practically zero and the sample strains with very low shear stress at essentially zero stiffness. The static model was used to simulate the cyclic simple shear test shown in Fig. 4.4. The predicted response is shown in Fig. 4.5. These results were obtained by setting the same test conditions as for the laboratory test, except for the density of the sand sample. It was found that for a sand with a (j)cv = 30°, a minimum relative density of approximately 49% was required to sustain 8 cycles of 12 kPa loading. Therefore, the results shown in Fig. 4.5 are for a sample with a relative density of 49%. The overall response predicted by the static model is very similar to the recorded laboratory data. In particular the model captures the variation in the stiffness of the sample and the large porewater pressure generation once the stress state unloads from the <j)cv line. However, the model predicts slightly lower strain development prior to the large strain excursion in the eighth cycle, and a more sudden change from stiff to loose response. The static model also predicts a more gradual development of porewater pressure, and a smoother effective stress path after the phase transformation line (<|>cv line) is intersected. As discussed previously in this chapter, it is unrealistic to expect a liquefaction model to predict the response of a particular type of sand under a specific test condition. However, 52 Fig . 4.4: Contractive deformation during cyclic simple shear loading (after Sivathayalan, 1994). 53 the model must demonstrate the ability to capture the overall behaviour of a material. Once this is accomplished, material specific and test specific correlations may be developed to simulate the response of a sand under particular testing conditions. 54 10 as 2a (73 tzi 00 -10 < c = 100 kPa xCy=12kPa (a) 4 6 Number of Cycles 10 15 -15 r -10 -5 0 Shear Strain, (%) 10 40 5 2 0 £ U & £ -20 -40 200 50 100 150 Vertical Effective Stress, (kPa) Fig. 4.5: Predicted Response of Loose Fraser River Sand to Cyclic Simple Shear Loading. 55 Chapter 5: Verification of Model with Field Event 5.1 General As a further step in the validation of the model, the dynamic program was used to predict the response of the Wildlife site - an instrumented site where liquefaction occurred during an 1987 earthquake. 5.2 Background The Wildlife site is located in southern California in Imperial Valley. Evidence of liquefaction was observed at or near the site following numerous past earthquakes. In particular, liquefaction was documented at the site during the April 26, 1982 Westmorland earthquake (Mw=5.9). This consistent pattern of liquefaction was a contributing factor in the selection of the Wildlife site as the location for a field instrumentation array. Other reasons for choosing the Wildlife site were that it is located in one of the most seismically active regions of California, and being in a wildlife preserve, the instrumentation could be left undisturbed for numerous years. In 1982 the site was instrumented by the U.S. Geological Survey (USGS) using accelerometers and piezometers in an effort to record ground motions and porewater pressures during earthquakes. With these recorded site responses, a better understanding of the liquefaction process and the post-liquefaction behaviour could be obtained. In addition, the measurements could be used to verify analytical and empirical models. 5.3 Site Description The Wildlife site is located in the floodplain of the Alamo River approximately 36 km north of El Centro (Fig. 5.1). Although the site is on level ground, it is located in close proximity (about 20 m) to the river's western bank. As the river is incised to a depth of about 3.7 m, there exists the oppormnity for lateral spreading towards this free field (Holzer 56 et al. 1989). In-situ and laboratory investigations (Bennett et al. 1984, Haag 1985) have shown that the site stratigraphy consists of a surficial silt layer approximately 2.5 m thick underlain Fig. 5.1: Location of Wildlife Site and epicenters of recent earthquakes in Imperial Valley (after Holzer etal., 1989). by a 4.3 m thick layer of loose silty-sand. Underneath the sand layer from a depth of 6.8 m to 11.5 m is stiff to very stiff clay. The groundwater table fluctuates within the surficial silt layer and is located at an approximate depth of 2.0 m. A schematic of the soil stratigraphy is shown in Fig. 5.2. 57 O CPT6 O CPT CONE PENETRATION CONE PENETRATION qc RESISTANCE kg/cm» R, FRICTION RATIO % V P PIEZOMETER • SM- STRONG MOTION SEISMOMETER • OR OSCILLOGRAPHIC RECORDER X WATER TABLE 13r 14 L Fig. 5.2: Plan and cross section of Wildlife Site showing stratigraphy and location of accelerometers and piezometers (after Bennet et al., 1984). 58 5.4 Instrumentation The liquefaction array at the Wildlife site consists of two 3-component accelerometers and six electric piezometers. One accelerometer was mounted at the surface on a concrete slab supporting an instrument shed. The second accelerometer was installed in a cased hole beneath the liquefiable layer at a depth of 7.5 m. Five of the six piezometers were installed within the liquefiable sand layer. In addition to the above, an inclinometer casing was installed approximated 12 m to the west of the array to allow measurement of permanent horizontal displacements in the subsurface. The piezometer tips were designed and fabricated specifically for the array, as there were no commercially manufactured push-in electronic piezometers available at the time which met the researcher's specifications (Youd and Holzer, 1994). An illustration of a piezometer tip is shown in Fig. 5.3. The purpose of the piezometer tip is to allow groundwater to be in contact with the sensing diaphragm while preventing soil from entering the diaphragm chamber. With this accomplished, the piezometers would be capable of measuring dynamic porewater pressure fluctuations. The diaphragm cavity, shown in Fig. 5.3, is connected to the external surface of the piezometer tip by six 3 mm diameter holes sloped downward toward the cavity to prevent soil from entering the holes during pushing. The holes were not screened, but were left open to freely allow water movement. Prior to installation, the piezometer tips were carefully saturated. During installation the piezometers were lowered down pre-drilled holes and the pushed approximately 0.3 m into the underlying soil (Youd and Holzer, 1994). Further details about the instrumentation and the installation procedure are given by Youd and Wieczorek (1984). 5.5 Recorded Site Response In November, 1987 the Wildlife site was shaken by two earthquakes - the Elmore Ranch earthquake and the Superstition Hills earthquake. Both events triggered the 59 instrumentation at the site; however, only the Superstition Hills earthquake (M = 6.6) generated dynamic pore pressures. Subsequent site investigations showed evidence of liquefaction in the form of sand boils and small ground fissures (Zeghal and Elgamal, 1994). Water proof cable Potting Soldered connection Pore pressure transducer 3 mm dia. holes Diaphragm chamber Delrin shell Fig. 5.3: Cross section of USGS piezometer tip used at Wildlife Site (after Youd and Holzer, 1994). Relative displacements of 180 mm were measured by means of the inclinometer casing; however, within the array, surface displacements were considerably smaller. Holzer et al. (1989) report that the cumulative opening across cracks at the array was 126 mm. Fig. 5.4 shows the measured acceleration time histories for the North-South component of the Superstition Hills quake. Fig. 5.4a shows the surface time history while 60 the downhole time history is shown in Fig. 5.4b. Ground motions in the East-West direction were smaller, and were not analysed using the model. By performing an integration of the acceleration records, the surface and downhole displacement time histories were obtained, and are shown in Fig. 5.5a and Fig. 5.5b respectively. 200 1 100 o •M a u T3 o o < in 1 1=1 O i-i <u o o < -100 -200 0 20 40 60 a) Surface acceleration time history, N-S component 80 100 200 100 -100 -200 20 40 60 Time (s) 80 100 b) Downhole acceleration time history, N-S component Fig. 5.4: Acceleration time histories - Wildlife Site, 1987 Superstition Hills earthquake The relative displacements between the surface and the stiff base are primarily of interest, and these were obtained by subtracting the surface and downhole displacements records at each time step. The resulting relative displacement time history is shown in Fig. 5.5c. Note that the relative displacements were essentially zero for about the first fourteen seconds of shaking despite the fact that absolute displacements were measured both at the 61 surface and downhole. This indicates that up until fourteen seconds, the soil units above and below the liquefiable sand layer were essentially moving together. After fourteen seconds, however, relative displacements are observed, indicating the uncoupling of the soil units above and below the sand layer. 15 0 •15 15 0 0 r! -15 15 0 0 •15 0 20 20 20 , a) Surface displacement time history. . . A A V V -40 60 Time (s) 80 100 b) Downhole displacement time history. \ / 1 / \ / 1 / V v V V " V v""v/' " " • "vr V \/ ^ '<->'1 40 60 Time (s) 80 100 c) Relative displacement time history. .A. - J i 1 I A A A A A ^ / ^ ^ - A A A A ~ ~ -P V V V v 1/ \ / V v ^ V V v ' l / v v ^ ' l 40 60 Time (s) 80 100 Fig. 5.5: Displacement time histories - Wildlife Site, 1987 Superstition Hills earthquake. Knowing the relative displacement time history, it is useful to plot surface acceleration versus relative displacement as shown in Fig. 5.6. This plot is similar to a shear stress 62 versus shear strain plot, where the shear stress would simply be the surface acceleration multiplied by the soil mass and the strains would be the relative displacements divided by the thickness of the liquefied layer. Since neither the soil mass nor the thickness of the liquefied layer are known with certainty, presenting the data in this form introduces less error. 200 -15 -10 -5 0 5 10 15 Relative Displacement (cm) Fig. 5.6: Surface acceleration vs. relative displacement; Wildlife Site, 1987 Superstition Hills earthquake. By isolating brief segments of the data from Fig. 5.6, it is possible to see how the soil modulus changes with cycles. Fig. 5.7 shows four discrete cycles at different times during the earthquake. For about the first 14 seconds of shaking, the soil is stiff as shown in Fig. 5.7a. At about 16 seconds of shaking (Fig. 5.7b) significant degradation of modulus has occurred. After approximately 35 seconds of shaking, further modulus degradation has taken place (Fig. 5.7c) and the response is characterised by an zone of essentially zero stiffness followed by a stage of strain hardening and then an abrupt increase in modulus upon 63 unloading. The behaviour shown in Fig. 5.7c corresponds exactly with laboratory observations discussed in Chapter 2. This same behaviour is seen in Fig. 5.7d, except the base accelerations are considerably smaller in this stage of the earthquake. 200 J , 100 a ••g S 0 u <^  <u -100 c/3 -200 12.92s- 13.15s -12 -8 -4 0 4 ! Relative Displacement (cm) a) Initial stiff behaviour 12 200 1 100 o lertat 0 < -100 C/3 -200 34.13s-36.91s -12 -8 -4 0 4 ! Relative Displacement (cm) c) Peak relative displacements 12 200 1 100 1 0 o < -100 urfac CO -200 1 U O < { jj 15.59s- 16.89s V -12 -8 -4 0 4 8 Relative Displacement (cm) b) Modulus degradation at triggering. 12 200 100 0 -100 -200 52.98s - 56.06s -12 -8 -4 0 4 8 Relative Displacement (cm) d) Continued soft behaviour 12 Fig. 5.7: Change in soil stiffness during selected cycles - Wildlife Site, 1987 Superstition Hills earthquake. 5.6 Controversy Regarding Piezometer Response There has been considerable discussion concerning the unexpected dynamic pore pressure behaviour measured by the piezometers at the Wildlife site (Youd and Holzer, 1994; Youd et al., 1989; Hushmand et al., 1992a; 1992b). The measured pore pressure time histories are shown in Fig. 5.8. Piezometer P4 malfunctioned during the liquefaction event. The transducer for piezometer P5 apparently was improperly calibrated (Youd and Holzer, 1994) and this resulted in pore pressure ratios greater than unity during the event. The measurements show a sudden rise in pore pressures after about fourteen seconds of shaking. This coincides with the stage of the earthquake where the site experiences the peak 64 acceleration pulses. The point of contention between many geotechnical specialists is the unexpectedly slow rate of pore pressure accumulation after this initial increase. The pore pressure records indicate that 100 percent of the effective overburden pressure (a recognised condition for liquefaction) was not attained until about 80 seconds after the instruments were originally triggered. These measurements would seem to disagree with the observations made from Fig. 5.7. Some experts have concluded from this unexpectedly slow pore pressure rise that the piezometers at -0.21 ' 1 ' 1 1 1 0 20 40 60 80 100 Time (sec) Fig. 5.8: Measured pore pressure time histories, in terms of pore pressure ratio - Wildlife Site, 1987 Superstition Hills earthquake. the Wildlife array were not accurately measuring dynamic porewater pressures during the earthquake. Hushmand et al. (1992a) attributed the apparent delay in response to insufficient saturation of the piezometer tip or possibly to soil disturbance during instrument installation. 65 Their conclusion, that the piezometers were not truly measuring free-field pore pressures, was mainly based on comparison with pore pressure responses during previous earthquakes and in centrifuge model studies. Hushmand et al. (1992a) argued that the long, smooth rise of pore pressures measured by four of the gauges seemed uncharacteristic of earthquake loading, and showed little correlation with the acceleration time histories. Youd and Holzer (1994), however, claimed that the piezometer tips were carefully de-aired and that only pin-head sized air bubbles could have survived their saturation procedure. They stated that air bubbles of this size would have diffused into the groundwater within months of piezometer installation. Youd and Holzer also argued that the smooth pore pressure response could be attributed to the fact that the Wildlife site is underlain by thinly cross-bedded silty sands with a relatively low permeability compared to typical model tests. In addition, the Superstition Hills earthquake could be considered 'noisy' in comparison to the simulated earthquake motions often used in centrifuge tests. Youd and Holzer (1994) argued that these two factors may account for the uncharacteristically slow and smooth response recorded by the piezometers. Hushmand et al. (1992a) carried out a field investigation program at the Wildlife site in 1989 to examine the response of the USGS piezometers. The study compared the response of the USGS piezometers to a reference BAT piezometer that was installed adjacent to each of the four functioning piezometers. Using a pressure-generation well, both the BAT device and the USGS piezometers were tested by applying fluctuating and quasi-static pressure changes. Details about the testing apparatus, testing procedure, and the results are found in Hushmand et al. (1992a). Upon comparing the pore pressure results, the investigators concluded that only one of the functioning USGS piezometers, namely piezometer P5, was working correctly. Unfortunately, this testing procedure was not truly diagnostic, and there is a reasonable explanation to account for the discrepancies between the measurements of the BAT device and the USGS piezometers. Basically, the BAT piezometer is significantly different than the USGS devices. The piezometer tip is constructed from porous plastic and 66 has a 37 mm filter length, whereas the vent holes on the USGS piezometers are only in contact with about 5 mm of adjacent soil. This means the BAT piezometer can sense porewater pressure changes from a considerably larger thickness of sediment and therefore may measure pore pressures along a different path than measured by the USGS devices. As pressure waves travel faster along a path with higher permeability, the BAT piezometer would respond to dynamic pore pressures quicker than the USGS piezometers if the larger BAT tip intercepted a sand layer located above or below the USGS vent holes. With this in mind, it would seem reasonable to conclude that the USGS piezometers functioned properly during the Superstition Hills earthquake, but their design and the presence of silty lenses around the vent holes likely impeded their response time to pore pressure variations. 5.7 Interpretation of Recorded Site Response There is no question that liquefaction occurred at the Wildlife site during the Superstition Hills earthquake. The displacement time histories (Fig. 5.5) indicate that relative displacements between the layers above and below the liquefiable layer began after about 16 seconds of shaking. Fig. 5.7 shows that an approximate 500 fold reduction in soil modulus has occurred after roughly 35 seconds of shaking. The recorded porewater pressure response shows that pore pressures equalled the overburden pressure approximately 80 seconds after the instruments were triggered. All of these observations signify the occurrence of liquefaction, but each observation indicates that liquefaction took place at a different time. Since relative displacements began after about 16 seconds, one would conclude that liquefaction was triggered around this time at some discrete zone or layer between the downhole and surface accelerometers. As strong shaking continued, the zone of liquefaction gradually increased in size. After about 30 seconds, when strong shaking subsided, sufficient modulus degradation may have occurred in the non-liquefied parts of the sand layer to permit continued liquefaction under the reduced acceleration pulses. This gradual increase in size of 67 the liquefied zone over time is another possible explanation for the slow response of the USGS piezometers. 5.8 Analysis Procedure The dynamic analysis of the Wildlife site was performed using a single degree of freedom, lumped-mass and spring model. The lumped mass was taken to be the mass of the surficial silty crust along with the mass of half the thickness of the liquefiable layer. No viscous damping was introduced into the system, but the spring was assumed non-linear, and represented the stiffness of the liquefiable layer by incorporating the stress-strain model discussed in Chapter 3. It was assumed that each soil unit had a uniform density and that the entire sand layer liquefied simultaneously. See Appendix 1 for a listing of the assumed input parameters for the Wildlife Site analysis. The model prediction was made by applying the downhole acceleration time history as the base input motion and then computing and system accelerations, velocities and displacements using a numerical integration technique. At each time interval, incremental pore pressures, shear strains and moduli values were calculated. The computed response in terms of surface accelerations, relative displacements, and porewater pressures are compared in the next section. 5.9 Results of Analysis A comparison of the measured and predicted surface accelerations is presented in Fig. 5.9a. This figure shows there is reasonable agreement between the overall forms of the predicted and observed responses, especially before the 40 second mark. After 40 seconds, the predicted response is significantly smoother than the measured response, with several acceleration pulses not being picked up by the model. Fig. 5.9b compares the predicted and measured relative displacement time histories, where it is seen that the model accurately predicts the initiation of relative displacements between the two accelerometers. Fig 5.9b 68 shows that, initially, the predicted relative displacements are larger than the observed displacements. This is the case as the model assumes the entire sandy layer liquefies simultaneously, while in the field the liquefied zone would have increased in size more gradually. After roughly 30 seconds, the predicted relative displacements become smaller than the observed displacements, and by about 40 seconds a bias is observed in the predicted -200 0 20 40 Time (s) a) Measured and predicted acceleration time histories. 60 80 15 10 5 0 1 B M -5 O H 5 -10 -15 0 20 Relative Displacement A i l l 1 Wfv M I V ffyVV\)\J Measured Predicted 40 Time (s) b) Measured and predicted displacement time histories. 60 80 Fig. 5.9: Comparison of measured and predicted time histories - Wildlife Site, 1987 Superstition Hills earthquake. 69 displacement response in Fig. 5.9b. This bias develops as a result of the idealised soil system experiencing a significant acceleration pulse in one directions while the soil stiffness is practically zero. Although offset from the observed response, the predicted displacements closely match the same trends of the measured relative displacement time history. The predicted surface acceleration versus relative displacement is shown in Fig. 5.10a. As discussed earlier, this plot corresponds to a shear stress versus shear strain plot. Up until about 16 seconds, the predicted site response is very stiff, indicated by near vertical loops in the figure. At around 17 seconds, liquefaction is triggered resulting in rapid modulus degradation and very flat loops, as shown in Fig. 5.10a. The overall shape of the predicted acceleration versus relative displacement plot is in reasonably good agreement with the observed response, except that in reality the overall site stiffness degraded more gradually than in the model. This difference in site response is likely a result of the assumption that the whole sand layer liquefies at one time. The predicted effective stress path is shown in Fig. 5.10b. This plot shows a gradual reduction in effective stress with cycles from an initial stress state of o' v o= 66 kPa and x s t= 0. This decrease in effective stress occurred as the input accelerations caused cyclic shear stresses which in turn generated excess porewater pressures. Fig. 5.10b shows that the stress point reached the <|>cv line several times before the developed strain was sufficient to trigger a large porewater pressure rise. After intersecting the <|>cv line for the third time, the stress point was then driven to a state of zero effective stress upon unloading. Once this state was reached, subsequent loops of dilation up the <j)cv line and contraction upon unloading were predicted to occur along with the accompanying porewater pressure fluctuations. As discussed in Chapter 2, significant cyclic deformations are only realised after the stress point has unloaded and reached a momentary state of zero effective stress. A comparison between the measured and predicted porewater pressure ratios is shown in Fig. 5.11. As shown in the plot, the predicted pore pressure rise is considerably faster 70 200 .CO «3 1 ioo 4-o I -•S 0 0) o o (3 O O o 3 •100 t -200 -15 -10 -5 0 5 Relative Displacement (cm) a) Acceleration versus relative displacement. 10 15 200 100 + 0 -ioo + o < -200 N-S Component ii kill i f P — i — i — i — i — i — i — 0 10 60 20 30 40 50 Vertical Effective Stress (kPa) b) Acceleration versus vertical effective stress. Fig. 5.10: Predicted dynamic response of Wildlife Site for 1987 Superstition Hills earthquake. 70 71 than the observed rise. Another clear difference between the predicted and observed porewater pressure response is the greater number of oscillations in pore pressure ratio predicted by the model. The measured porewater pressure response shows several significant pore pressure ratio oscillations, but only after about 30 seconds of shaking. These oscillations are a result of dilative loops after liquefaction has been triggered. Therefore, if piezometer P5 is reading correctly, this would suggest that liquefaction did not occur at the location of the piezometer tip until a later time. Most likely, this piezometer was measuring porewater pressures that migrated from adjacent zones which liquefied at an earlier time during earthquake shaking. 1.0 0.8 + s A, 0.6 0.4 0.2 0 20 40 Time (s) 60 80 Fig. 5.11: Comparison between measured and predicted pore pressure ratios - Wildlife Site, 1987 Superstition Hills earthquake. 72 Chapter 6: Summary and Conclusions An incremental effective stress liquefaction model has been presented. Two versions of the model have been developed - a static program to simulate the behaviour of a sand element to laboratory cyclic simple shear tests, and a dynamic program to compute the response of sand under harmonic or earthquake loading. Both the static and dynamic version of the model have one degree of freedom. The model was designed to capture the element shear stress-strain and porewater pressure response of a soil under a range of conditions. The key components of the model are (1) the shear-volume coupling equation, (2) the shear stress-strain law and (3) the procedure for calculating the excess porewater pressure response during undrained loading. The empirical shear-volume coupling equation is used to calculate the incremental volumetric strain from the increment of shear strain by assuming a linear variation/accumulation of strain during any half-cycle of loading. The shear stress-strain behaviour in both loading and unloading is modelled by modified hyperbolas. The formulation results in tangent moduli that vary in both stress level and relative density. The incremental porewater pressure response is calculated as a result of the volumetric constraint that exists due to undrained loading conditions. The static version of the model is calibrated to match the characteristic laboratory cyclic simple shear and cyclic triaxial test results. The model was programmed to simulate the observed trends in these tests. The dynamic version.of the model incorporates the key components of the static program, and can be used to predict the response of a simplified soil profile under dynamic loading. In particular, earthquake acceleration time history records may be used as input. The dynamic version of the model was applied to simulate the field case history recorded at the Wildlife Site in California during the 1987 Superstition Hills earthquake. The recorded downhole acceleration time history was used as input for the dynamic model and the 73 predicted response, in terms of surface acceleration, relative displacement, and porewater pressure, compared with the measured values. The predicted and observed surface accelerations are in reasonable agreement in terms of both the amplitude and characteristic frequency of response. The predicted relative displacements are also in reasonable agreement with the recorded values. In particular, the relative displacement pattern around 17 seconds is in good agreement. The author regards this to be the time where liquefaction was initiated somewhere in the sand layer. The predicted acceleration versus relative displacement (stress versus strain) curves are in very good agreement and indicate that prior to t=17 seconds the stress-strain response is very stiff, whereas after this time a major reduction in stiffness by a factor of about 500 occurs. This indicates that liquefaction and essentially 100% porewater pressure rise was triggered at least in some zones at about t = 17 seconds. The predicted porewater pressures are not in good agreement with the recorded values. The predicted porewater pressure rise is much faster than the measured data. The slower response may be due to limitations in the compliance of the measuring system and to the possibility that liquefaction did not occur simultaneously at all points in the liquefied layer. 74 R E F E R E N C E S Bennett, M . J . , Laughlin, P.V. , Sarmiento, J. And Youd, T . L . (1984). "Geotechnical Investigation of Liquefaction Sites, Imperial Valley, California," US Geological Survey Open File Report 84-242, pp. 1-103. Byrne, P . M . (1991). A Cyclic Shear Volume Coupling and Porewater Pressure Model for Sand," Second International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, Missouri, Report 1.24, Vol. 1, March, pp. 47-57. Byrne, P . M . and Mclntyre, J, .D. (1994). " Deformations in Granular Soils due to Cyclic Loading," A S C E Geotechnical Special Publication No. 40, Settlement '94 Conference, College Station, Texas. Castro, G. (1969). "Liquefaction of Sands," Ph.D. Thesis, Harvard University, Cambridge, Mass. Chern, J .C. (1985). "Undrained Response of Saturated Sands with Emphasis on Liquefaction and Cyclic Mobility", Ph.D. Thesis, The University of British Columbia, Vancouver. Chung, E . K . F . , (1985). "Effects of Stress Path and Pre Strain History on the Undrained Monotonic and Cyclic Loading Behaviour of Saturated Sand," M.A.Sc. Thesis, The University of British Columbia, Vancouver. Clough, R.W. and Penzien, J. (1975). Dynamics of Structures, McGraw-Hill Inc., p. 263. Duncan, J .M. and Chang, Y . Y . (1970). "Nonlinear Analysis of Stress and Strain in Soils," Journal of the Soil Mechanics and Foundations Division, A S C E , Vol. 96, No. SM5, September. Holzer, T . L . , Bennett, M.J . and Youd, T . L . (1989). "Lateral Spreading Field Experiments by the US Geological Survey," Proceedings, 2nd US-Japan Workshop on Liquefaction, Large Ground Deformation and Their Effects on Lifelines, Buffalo, New York, pp. 82-101. Hushmand, B. , Scott, R .F . , and Crouse, D.B. (1992b). "In-place calibration of USGS transducers at Wildlife liquefaction site, California, USA." Proc, 10th World Conf. On Earthquake Engineering, Madrid, Spain, pp. 1263-1268. Ishihara, K . , Tatsuoka, F. and Yasuda, S. (1975). "Undrained Deformation and Liquefaction under Cyclic Stresses, Soils and Foundations, Vol. 15, No. 1, pp. 29-44. 75 Kondner, R . L and Zelasko, J.S. (1963). "A Hyperbolic Stress-Strain Formulation for Sands," Proceedings, 2nd Pan-American Conference on Soil Mechanics and Foundation Engineering, Brazil, Vol. 1, 1963, pp. 289-324. Kuerbis, R . H . (1989). "Effect of Gradation and Fines Content on the Undrained Response of Sands", M.A.Sc. Thesis, The University of British Columbia, Vancouver. Lambe, T.W. and Whitman, R.V. (1969). Soil Mechanics. John Wiley and Sons, New York. Lee, C.-J . (1991). "Deformations of Sand Under Cyclic Simple Shear Loading," Second International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, Missouri, Report 1.12, Vol. 1, March, pp. 33-36. Luong, M.P. (1980). "Stress-Strain Aspects of Cohesionless Soils Under Cyclic and Transient Loading," Int. Symposium, Swansea, U.K. Martin, G.R. , Finn, W.D. Liam, and Seed, H.B. (1975). "Fundamentals of Liquefaction Under Cyclic Loading," Journal of the Geotechnical Engineering Division, A S C E , May, Vol. 101, No. GT5. Matsuoka, H . (1974). "Stress strain relationships of sands based on the mobilised plane", Soils and Foundations, Vol. 14, No. 2, pp. 47-61. Negussey, D . , Wijewickreme, W . K . D . , and Vaid, Y.P. (1986). "Constant Volume Friction Angle of Granular Materials", Soil Mechanics Series No. 94, Dept. of Civil Engineering, The University of British Columbia, Vancouver. Pillai, V.S. and Byrne, P .M. (1994). "Effect of Overburden Pressure on Liquefaction Resistance of Sand," Roscoe, K . H . (1970). Tenth Rankine Lecture, "The Influence of Strains in Soil Mechanics," Geotechnique, Vol. X X , No. 2, pp. 129-170. Seed, H.B. and Idriss, I .M. , (1970). "Soil Moduli and Damping Factors for Dynamic Response Analyses," Report No. EERC 70-10, University of California, Berkeley. Silver, M . L . , and Seed, H.B. (1971). "Volume Changes in Sand During Cyclic Loading," Journal of the Soil Mechanics and Foundations Division, A S C E , September, pp. 1171-1182. Sivathayalan, S. (1994). "Static, Cyclic and Post Liquefaction Simpler Shear Response of Sands," M.A.Sc. Thesis, The University of British Columbia, Vancouver. 76 Stark, T .D . and Mesri, G. (1992). "Undrained Shear Strength of Liquefied Sands for Stability Analysis," A S C E Journal of Geotechnical Engineering, Vol. 118, No. 11, November, pp. 1727-1747. Thilaklaratne, V. And Vucetic, M . (1989). "Liquefaction at the Wildlife Site - Effect on Soil Stiffness on Seismic Response," Proceedings, 4th International Conference on Soil Dynamics and Earthquake Engineering, Mexico City, pp. 37-52. Thomas, J. (1992). "Static, Cyclic and Post Liquefaction Undrained Behaviour of Fraser River Sand", M.A.Sc. Thesis, The University of British Columbia, Vancouver. Tokimatsu, K. and Seed, H.B. (1987). "Evaluation of Settlements in Sands Due to Earthquake Shaking," A S C E Journal of Geotechnical Engineering, Vol. 113, No. 8, August, pp. 861-878. Vaid, Y .P . , and Chern, J.C. (1985). "Cyclic and Monotonic Undrained Response of Saturated Sands," Advances in the Art of Testing Soils under Cyclic Conditions, A S C E Convention, Detroit, pp. 120-147. Youd, T . L . (1972) "Compaction of Sands by Repeated Shear Straining," Journal of the Soil Mechanics and Foundations Division, A S C E , July, pp. 709-725. Youd, T . L . and Wieczorek, G.F. (1984). "Liquefaction During the 1987 and Previous Earthquakes Near Wesmorland, California," US Geological Survey Open-File Report 84-680. Zeghal, M . And Elgamal, A .W. (1994). "Piezometer Performance at Wildlife Liquefaction Site California," Journal of Geotechnical Engineering, ASCE, Vol. 120, No. 6, June. 77 APPENDIX 1 78 APPENDIX 1 Input Parameters Static Model The static model simulates laboratory cyclic simple shear tests. The user can specify drained or undrained tests and strain or load controlled tests. Regardless of which test type is chosen, the user is requested to input the following soil and test properties prior to the commencement of the analysis: 1) the vertical effective stress, a / 2) the coefficient of lateral earth pressure, k 0 3) the constant volume friction angle of the sand, (j)CT 4) the relative density or an SPT (Nj ^ blowcount value for the sand. If a load controlled test is specified, the user must input the desired cyclic stress ratio, CSR. If a strain controlled test is requested, the maximum strain value is required. The analysis procedure is not carried out in the time domain. Depending on the type of test specified, a stress or strain increment (or decrement) is applied within the specified range. Dynamic Model The dynamic model computes the shear stress-strain and porewater pressure response of an idealised soil profile subjected to dynamic ground motions. The necessary input can be divided into three categories: geometric profile information, soil properties, and dynamic properties. The geometric input parameters include the crust thickness, the thickness of the suspected liquefiable layer, the depth to the groundwater table, and the slope of the ground (if any). Required soil properties include the unit weight of the soil, y, the constant volume friction angle, <|>cv, the coefficient of lateral earth pressure, kg, and the characteristic SPT (Nj) 6 0 blowcount for the liquefiable layer. Required dynamic system properties include the percentage of critical damping and the initial system displacement and system velocity. 79 After these parameters have been input, the program calculates other dynamic properties of the system including the natural frequency, the damped frequency, the period and the damping coefficient. For the liquefaction analysis of the Wildlife Site, the following input parameters were chosen: Crust thickness = 2.6 m Thickness of liquefied layer = 4.2 m Depth to water table = 2.0 m Ground slope = 0 % Average SPT normalised blowcount, (N^go = 12 Coefficient of at-rest lateral earth pressure, k<, = 0.5 Constant volume friction angle, ())cv = 33° Unit weight of water, yw = 9.8 kN/m 3 Percentage of critical damping, Ld = 0 % Initial system displacement, X 0 = 0 m/s Initial system velocity, V 0 = 0 m/s 80 APPENDIX 2 81 APPENDIX 2 Predicted Overall Response to Undrained Cyclic Loading Fig. A l . l illustrates the characteristic stress-strain and effective stress response pattern adopted in the model. The simulated test shown in Fig. A l . l is a load controlled cyclic simple shear test on a saturated, undrained sand. Note that at no time during the loading sequence is a strain softening response predicted. The model separates the overall response into five different phases, as discussed below. Phase 1: During phase 1, the stress-strain response is stiff and is characterised by small cyclic strains (Fig. A 1.1 (a)). The effective stress path response is shown in Fig. Al. l (b) , and during this phase the effective stress reduces with the stress state cycling towards the left in saw-tooth pattern. Phase 2: After sufficient cycles, the stress state may intercept the constant volume or phase transformation friction angle line (point A). If this occurs, the model assigns a much smaller shear modulus, resulting in significant plastic strains. This is represented in Fig. A l . l by the flat portion of the curve between points A and B. In terms of the stress path, phase 2 response is limited to the movement of the stress state up the constant volume friction angle line from point A to point B. Once the current stress cycle reaches its peak a loading reversal occurs. Although the conditions for liquefaction are not met in phase 2, liquefaction is triggered during this phase. Liquefaction is assumed to be triggered once a specified strain level is exceeded during phase 2. Strain limits from 0.2% to b) Effective stress path response Fig. A l . l : Characteristic Cyclic Shear Stress-Strain and Effective Stress Path Resp for Static Model. 0.5% have been used for the static model. If this strain limit is not exceeded during phase 2, upon unloading the model will switch back to phase 1 response. Phase 3: Once the current cycle of stress or strain reaches its peak, a loading reversal takes place. During this reversal (see Fig. A 1.1 (a)), the soil response is initially very stiff, and the soil unloads from point B at a shear modulus comparable to phase 1. Phase 3 response ends at point C, where the shear stress level reduces to near zero, or changes sign (crosses over the strain axis). Despite this initially stiff response, phase 3 is characterised by large porewater pressure generation. Consequently, the effective stress state moves toward the origin from point B (Fig. Al.l(b)). Phase 4: Upon unloading from point B, the shear stress level reduces. Once the stress level changes sign (crosses the strain axis) or reduces to a very low level, the model switches from phase 3 to phase 4. The shear modulus assigned during phase 4 is at least one thousand times smaller than the initial shear modulus during phase 1. Consequently, phase 4 is characterised by very loose response, as the sand undergoes large strains at essentially zero shear stress. This is illustrated in Fig. A 1.1(a) by the near horizontal line between points C and D. For all of phase 4, the effective stress state remains essentially at zero (point C in Fig. Al.l(b)). During phase 4, the sand specimen strains in the opposite direction than in phase 2. Eventually, with sufficient straining, the model will switch from phase 4 to phase 5 and the sample will begin to dilate and regain some of its original strength. Point D marks the end of phase 4, and represents a strain boundary -that is, the point beyond which the sample has not previously been strained. 84 Phase 5: As discussed above, point D marks the end of phase 4 and the beginning of phase 5. During phase 5, the sand sample dilates and regains some of its original strength. In terms of the shear stress-strain response, the shear stress level increases as the sample is strained beyond the previous strain boundary, as shown if Fig. A 1.1 (a). Note that the shear modulus increases with straining, but does not reach the same magnitude as at the beginning of loading. Fig. Al. l (b) shows the effective stress path, with the stress point travelling up the <j)'cvlfne from point D to point E . The effective stress increases as the stress point moves up the line and away from the origin. Once point E is reached, a loading reversal takes place and this marks the end of phase 5. As discussed above, liquefaction is triggered during phase 2. However, in terms of porewater pressure development, liquefaction does not occur until the stress state reaches point C for the first time. At this point, the effective confining stress is essentially zero, and the sand sample undergoes large deformations under very low shear stress. 85 APPENDIX 3 86 APPENDIX 3 Numerical Integration Technique (after Clough and Penzien, 1975) The dynamic model uses a numerical integration technique to compute the response of the idealised soil profile. In simple terms, the numerical integration technique determines the response of the system at a time in the near future using the present displacement, velocity and acceleration values. The basic assumption of the process is that the acceleration varies linearly during each time increment, while the properties of the system remain constant during the interval. The illustration below demonstrates the nature of the problem to be solved: AAAAA D— m F(t) kt+At t+At x The following equation of motion is valid at all times: mx + ex + kx = F(t) (1) If we assume that x varies linearly in time interval dt, the following equations are true: _ . At . . A t -Xt+At — x t + 2 x t "*~ 2 X'+AT . . At 2 - At 2 X . + A , = X , + X , A * + X , — + X , + A t - 7 -(2) (3) At time t+At Eqn (1) becomes: mx t + A t +cx t + A t +kx t + A t = F(t + At) (4) 87 Substituting for x t + A t and x t + A t from Eqns (2) and (3) into Eqn (4) and rearranging, the following solution is obtained: x t + A [=[H]{F-ca-kb} (5) where: At.. a = x. H x,, * 2 1 ' A At' b = xt + xtAt + xt —^-, and [H] = At , At2 m+c hk-2 6 The solution procedure can be summarised as follows: 1. Knowing xt, xt and xt, compute a, b and H. 2. Compute x t + A t from Eqn (5). 3. Compute x t + A t and x t + A t from Eqns (2) and (3). 4. Proceed with the next time step, knowing that k and c could change with x. 88 APPENDIX 4 89 O II • - CL •s* O »- z> • to O * *-» UJ | 1.25 o •o in z "v. :yc su; N-Z T3 II Tt ft) > g r Pa O :yc [Gmo > .SE o o z L! [Gmo > UJ t. * to 5 SUOP, o •o * Pa * a < z o a. o <o Ti her * z -* JOP o *•* Dr •v. •v. tO CO to * O \ « .O II D)»-•— z o -o o X i o + W w == * z m z * CM of * w HE OKI •- > c u o z 3< u z = t_> a o >«-f\l • « U O V O . 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