ANALYSIS AND DESIGN OF EARTH STRUCTURES TO RESIST SEISMIC SOIL LIQUEFACTIONPeter M. Byrne, Civil Engineering Department, The University of British Columbia, Vancouver, British Columbia, CanadaErnest Naesgaard, Civil Engineering Department, The University of British Columbia, Vancouver, British Columbia, CanadaMahmood Seid-Karbasi, Civil Engineering Department, The University of British Columbia, Vancouver, British Columbia, Canada

ABSTRACTThe National Building Code of Canada, 2005 has increased the seismic loading from a 10% chance of exceedance to a 2% chance of exceedance in 50 years event. This has increased the design accelerations by a factor of about 2 in Greater Vancouver as well as many other areas of Canada. For the design earthquakes scenario the risk of soil liquefaction and resulting damage to existing building and lifeline structures has greatly increased. This paper examines liquefaction induced damage to civil engineering structures in past earthquakes as well as the characteristic liquefaction response observed in laboratory tests. State-of-Practice as well as State-of-Art procedures for analysis and design of building and structures supported on potentially liquefiable soils are examined. Reliable prediction of seismic displacements of foundations, and design to limit displacements to prevent collapse of the supported structure is the challenge. State-of-practice analysis is based on a total stress approach in which it is assumed that the liquefiable soil will remain undrained at the in situ void ratio. It is further assumed that this is a conservative assumption as drainage during and after shaking will lead to lower void ratios and stiffer and stronger material. State-of-art coupled stress-flow effective stress analyses together with field experience and laboratory model tests show that this is an unconservative assumption in layered materials, and has led to a number of failures. Liquefaction generates excess pore pressures that drain upwards. If the liquefiable stratum contains silt or clay sub-layers that block the flow, this can result in a temporary expansion of the sand directly beneath the silt, and lead to a greatly reduced strength resulting in large displacements or flow slides. These concepts are discussed in relation to developing better analysis and design procedures for remediation of liquefaction effects.

RÉSUMÉLe Code National de Bâtiment de Canada, 2005 a augmenté le chargement sismique d'un événement avec 10% de chance d’être excédé à un 2% de chances en 50 ans. Ceci a augmenté les accélérations de conception par un facteur d'environ 2 dans le Grand Vancouver de même que dans beaucoup de secteurs au Canada, et a fortement augmenté le risque de liquéfaction de sol et les dommages résultants au bâtiment existants et les structures de ligne de sauvetage en cas du plus haut tremblement de terre de conception. Cet article examine les dommages induits par liquéfaction aux structures d'ingénierie civile dans les tremblements de terre passés de même que la réponse caractéristique de liquéfaction observée dans les tests de laboratoire. L’Etat de Pratique de même que des procédures d’Etat d'Art pour l'analyse et la conception de bâtiment et des structures soutenues sur les sols potentiellement liquéfiables sont examinées. La prédiction fiable de déplacements sismiques de fondations, et la conception pour limiter des déplacements pour empêcher l'effondrement de la structure soutenue est le défi. L’analyse d’Etat de Pratique est basée sur une approche de tension totale dans laquelle il est supposé que le sol liquéfiable restera non drainé au rapport de vides in situ. On suppose encore que ceci est une prétention conservatrice car drainage pendant et après la secousse mènera à des rapports de vides plus bas et le matériel sera plus raide et plus fort. Analyses d’Etat d’Art de tension effectives à écoulement d’effort couplé ensemble avec l'expérience de champ et les tests de modèle de laboratoire montrent que ceci est une supposition non conservatrice dans des matériaux à couches, et a mené à un nombre de ruptures. La liquéfaction produit des pressions de pore excessives qui drainent vers l'haut. Si la couche liquéfiable contient des sous-couches de silts ou argile qui bloquent le flux, ceci peut avoir pour résultat une expansion temporaire du sable directement en dessous du silt, et mener à une forte réduction de la résistance ayant pour résultat de grands déplacements ou écoulements de pentes. Ces concepts sont discutés par rapport à mieux développer les procédures d'analyse et conception pour le redressement des effets de liquéfaction.

1. INTRODUCTION

Earthquakes have caused severe damage to civil engineering structures, particularly where soil liquefaction was observed. Liquefaction can occur in granular soils whose pore spaces are filled with water, and involves

generation of high pore water pressures and large reductions in soil shear stiffness and strength that has led to very large shear deformations. In addition, the bulk stiffness of the soil skeleton greatly reduces upon liquefaction and can result in large post-liquefaction

settlements as pore water pressures dissipate; resulting in further damage.

The detrimental effects of liquefaction on urban infrastructures came into prominence during the 1964 Magnitude 7.5 PGA 0.2 g, Niigata earthquake. At Niigata, liquefaction caused apartment buildings to overturn, manholes and wood pilings to float out of the ground, bridge decks to fall off their supports and roadways and embankments to crack and spread (Figures 1 and 2). Liquefaction induced damage was also very severe during the Kobe earthquake of 1995, M7.1 with PGA 0.5 g. Damage to port facilities was particularly severe due to very large lateral displacements of the order of 3 m from soil liquefaction (Figures 3 to 7).

Soil liquefaction resulted in major damage in Turkey during the M7.5 earthquake event with PGA 0.5 g in 1999. Here liquefaction of silty soils (which previously had been deemed to be resistant to liquefaction) gave rise to large movements (Figures 8 and 9). Figure 10 shows an attempt to numerically simulate the overturned building of Figure 9.

Probably the best documented case histories of liquefaction effects on earth structures are the San Fernando dams in California, during the earthquake of 1971. The crest of the upper dam moved 1.5 m downstream, while the lower dam had a flow slide on its upstream side which removed the crest of the dam. The slide occurred shortly after severe shaking had ceased. The earthquake was a magnitude, M6.6 event with PeakGround Acceleration, PGA = 0.5 g. A plan view of the San Fernando dams is shown in Figure 11. A view of the upper dam is shown in Figure 12. The failed section of the lower dam is shown in Figures 13 and the failure mode is depicted in Figure 14.

The flow failure of the upstream slope of the Lower San Fernando dam was not expected at the time and resulted in a detailed study of the dam and the possible causes of the failure. The sands present at the site had relative densities of about 55%, and laboratory testing carried out on undisturbed samples after the failure showed high shear strengths, more than sufficient to prevent a flow slide. It was argued at the time that minor disturbance caused compaction of the samples and led to the high measured strength. Recent research suggests that localized expansion of the sand layers occurred after shaking due to redistribution of water associated with generated excess pore pressures. The expansion of the sand in turn caused its strength to drop and accounts for the failure. This hypothesis is supported by the observations that the slide occurred some time after the end of strong shaking; allowing time for redistribution of water and related expansion to take place. The expanded zones are very localized and difficult to detect. In addition, upon dissipation of excess pore pressure the expanded zones will compact. During the failure process the fine and coarse layers of the hydraulically placed dam also mixed. This mixing further reduces the strength of the liquefied soils (Byrne, 1989; Baziar and Dobry, 1995; and Naesgaard and Byrne, 2005).

Figure 1. Collapse of building in Niigata earthquake, Japan, 1964, M=7.5

Figure 2. Collapse of Showa Bridge in Niigata earthquake, Japan, 1964, M=7.5

Figure 3. Large lateral spreads damaged many port structures in Kobe 1995 earthquake

Figure 4. Bridge collapse due to foundation lateral spreading in Kobe 1995 earthquake

Figure 5. Bridge span collapse due to foundation spreading in 1995 Kobe earthquake.

Figure 6. Damaged port facilities due to foundation failure in 1995 Kobe earthquake.

Figure 7. Soil movements at bridge foundation, 1995 Kobe earthquake. (Photos in Fig. 1 to 7 fromhttp://cee.uiuc.edu/sstl/education/liquefaction)

Figure 8. Subsidence and lateral spreads in 1999 Turkey earthquake.

Figure 9. Building over-turning collapse due to foundation failure, Turkey 1999 earthquake.

Figure 10. Model of building over-turning collapse, Turkey 1999 earthquake.

Upper SanFernando Dam

Lower SanFernando Dam

Upper SanFernando Dam

Lower SanFernando Dam

Upper SanFernando Dam

Lower SanFernando Dam

Figure 11. Aerial view of San Fernando dams.

Figure 12. Upper San Fernando dam after 1971 earthquake.

Damage to earth structures from seismic liquefaction arises from soil movements and can be categorized as follows:

a) Flow slides and bearing failures with very large movements that occur during or after shaking when the post-liquefaction strength drops below the static driving shear stress.

b) Lateral Spreads & shear Induced foundation displacements that occur intermittently (and progressively increase) during earthquake shaking when the combined static and inertial driving forces exceed the soil strength. However the strength is greater than the static driving shear stress and movements stop when shaking ceases.

c) Post-liquefaction settlements due to consolidation arise from dissipation of excess pore water pressures associated with liquefaction. High excess pore pressure and liquefaction disrupt the soil fabric making it much more compressible. This has two effects: 1) it reduces the coefficient of consolidation and thus slows down the rate of dissipation of excess porewater pressure after liquefaction, and 2) it greatly increases the amount of settlement arising from dissipation of excess porewater pressure. This settlement is in addition to shear induced settlements.

Figure 13. Failure of Lower San Fernando dam due to liquefaction in 1971 earthquake.

Figure 14. Rebuilt model of Lower San Fernando dam Failure mode (Seed et al., 1973).

The seismic loading that buildings and structures are required to withstand in Canada are prescribed by the National Building Code of Canada, NBCC. Over the past 30 years, the seismic loading has increased dramatically from a 100 year event, to a 475 year event (10% chance of exceedance in 50 years), to a 2475 year event (2% chance of exceedance in 50 years) in NBCC 2005. In Greater Vancouver, the design peak ground acceleration, PGA has gone from about 0.10 g to 0.46 g in that 30 year period, a factor in excess of 4. Many older buildings had built-in structural “over-strength”, so that if located on firm ground the buildings may still be adequate under the increased seismic loading of the new building code. However, older buildings located on liquefiable soils are not likely to meet the new code requirements as liquefaction potential is directly related to acceleration level, and if the new code earthquake were to occur could be severely damaged. Foundations for new buildings will require a higher level of treatment to prevent liquefaction occurrence, and/or curtail resulting displacements under NBCC, 2005. The increased seismic design levels in the building code make it important that we fully understand the liquefaction phenomenon and develop new and improved analysis procedures for predicting liquefaction response. Such analyses if verified against case histories will allow a reliable assessment of liquefaction response.

This presentation will focus on understanding the liquefaction response, and discuss analysis procedures that allow rational design of foundation treatment to reduce displacements to tolerable levels.

2. SOIL RESPONSE TO SEISMIC LOADING

An earthquake applies time histories of acceleration at the base of a soil structure as shown in Figure 15. The horizontal accelerations induce oscillating horizontal dynamic shear stresses, xy as shown in Figure 15b. The effects of such cyclic shear stresses on soil behaviour can be assessed by applying them to elements of the soil under simple shear conditions and observing the response.

Such tests show that cyclic loading causes a granular soil to contract. If contraction is prevented or curtailed by the presence of a low compressibility fluid such as water in the pores that cannot escape, it induces excess pore pressure and a consequent reduction in stiffness and strength. If the excess pore pressure rises to reduce the effective stress to zero, the soil has essentially zero stiffness and strength and is said to have liquefied. However, unless it is very loose, the soil dilates as it is loaded causing the pore pressure to drop, and the soil regains strength and stiffness. Since seismic loading occurs rapidly there is generally little time for drainage to occur, and testing to simulate field conditions is generally carried out under undrained or constant volume conditions.

In practice, for testing purposes, the random oscillating seismic stresses, τxy, are replaced with an equivalent uniform cyclic stress cyc. In addition, it has been found that liquefaction depends largely on Cyclic Stress Ratio, CSR = cyc/'vo, where 'vo is the initial vertical effective stress prior to the earthquake, and cyc is an equivalent uniform cyclic stress often set equal to 0.65 x max where max is the peak horizontal dynamic shear stress from the earthquake.

Cyclic simple shear loading is depicted in Figure 16a. Typical shear stress-strain results for loose Fraser River sand tested in undrained (constant volume) conditions are shown Figure 16b, and stress path response in terms of shear stress vs. normal effective stress is shown in Figure 16c. From Figure 16b, it may be seen that the shear stress-strain response is stiff for a number of cycles in the pre-liquefaction stage, with shear strain less than 0.2%, followed by an abrupt change to a post liquefaction stage with very much softer response and strains of 10%. The stress path followed is shown in Figure 16c where it may be seen that starting from an initial vertical effective stress of 100 kPa, the effective normal stress drops with each cycle until the phase transformational line, or constant volume friction angle cv 33 is reached after 6 cycles. Once this has occurred, loading and unloading takes place close to the cv line, with loading involving an increase in effective stress and unloading involving a decrease in effective stress. This is illustrated in more detail in Figure 17 where it may be seen that prior to liquefaction the stress path is below the cv line and

xy

Time(b)

Acc.Time

Earthquake motion

´0 xy

(a)

Figure 15. Cyclic shear stress induced by base acceleration in an earth fill Embankment.

Figure 16. (a) cyclic simple shear loading, (b) shear stress-strain response, and (c) effective stress path (tests data from Sriskandakumar, 2004).

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0

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0 25 50 75 100 125

Vertical Effective Stress, 'v (kPa)

Shea

r S

tres

s, t

(kP

a)

'vc=100kPa; Drc=40%

cyc/'vc=0.10; st/'vc =0.0

Point of =3.75%(i.e. Assumed triggering point of liquefaction)

(c)

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Shear Strain, (%)

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, t(k

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cyc/'vc=0.10; st/'vc =0.0

Point of =3.75%(i.e. Assumed triggering point of liquefaction for comparison purposes)

(b)

(a)

´0

cyc

cyc

cyc

cyc

f

pt = cv

´

pt = cv

Figure 17. Effective stress path following liquefaction.

contraction occurs upon cyclic loading. Once the cv line is reached, further loading causes dilation as the stress point moves up just above the cv line. Upon unloading, the stress point drops slightly below the cv line and the soil contracts, driving the stress point back to the origin or zero effective stress (liquefied) state. Upon reloading the soil dilates, moving up just above the cv line gaining strength (de-liquefies). In subsequent unloading and reloading cycles, the pattern repeats itself. Note that although the soil liquefies during cyclic loading, it maintains its critical state strength at large strain. The post-liquefaction strength and stiffness depends very much on its density. Higher densities will have much higher post liquefaction stiffness and critical state strengths.

The test results shown above are for an applied CSR = 0.1, and triggered liquefaction in 6 cycles. If a higher CSR is applied the test element liquefies in less cycles, and if a lower CSR is applied it takes more cycles to liquefy as shown in Figure 18 in terms of CSR versus number of cycles to liquefaction. The CSR, that causes liquefaction in 15 cycles, is commonly taken as a reference and is referred to as the cyclic resistance ratio, CRR. For the (dry pluviated) loose Fraser sand tested CRR= 0.09. The

number 15 derives from a magnitude, M7.5 earthquakes which causes about 15 equivalent cycles. If the design earthquake is different than M7.5 then corrections for the appropriate number of cycles are made.

In practice, elements in the soil structure will likely have a static bias that will cause shear strain to accumulate in the direction of the bias as shown in Figure 19. In this case the bias was sufficient to prevent a change in sign of the shear stress (no crossover), large strains can accumulate once the stress point reaches the cv line and the stress point essentially moves up and down the line as loading and unloading occurs.

The cyclic stress-strain response of dense sand, Dr = 80% is shown in Figure 20. it may be seen that there is a gradual accumulation of shear strain with number of cycles leading to a gradual decay in stiffness with no loss in strength. The strain increase is less than 0.02% on the first cycles, and 10% after about 20 cycles.

The cyclic response of a soft low plasticity silt is shown in Figure 21. It may be seen that the shear strain increases gradually with strain and there is no abrupt change in shear strain. The soft silt behaviour is similar to that of

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Shear Strain, (%)

Shea

r S

tres

s, t

(kP

a)

'vc=100kPa; Drc=40%

cyc/'vc=0.065; st/'vc =0.1

Point of =3.75%(i.e. Assumed triggering point of liquefaction)

Figure 19. Stress-strain response for Fraser River sand with shear stress bias (Sriskandakumar, 2004).

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'vc=100kPa; Drc=80%

cyc/'vc=0.30; st/'vc =0.0

Point of =3.75% (i.e. Assumed triggering point of liquefaction)

Figure 20. Stress-strain response for dense Fraser River sand, Dr= 80% (Sriskandakumar, 2004).

Figure 21. Stress-strain response of low plasticity Fraser River silt (Wijewickreme and Sanin, 2005).

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Shear Strain, (%)

Sh

ea

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tre

ss,

t (k

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OCR = 1.0 CSR = 0.20

ec = 0.884

W = 36.2%

0

0.05

0.1

0.15

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CS

R

Figure 18. CSR vs. No. of cycles for Fraser river sand, DR =40% and ’c = 100 kPa (tests data from Sriskandakumar, 2004).

dense sand in that it has a gradual increase in strain with each cycle of stress rather than an abrupt increase as for a loose sand.

A great deal of high quality cyclic loading data for sandy soils is available and indicate the characteristic behaviour shown above. The CRR and post-liquefaction stiffness and strength increase with density, and the post-liquefaction strength conforms with critical state concepts.

Back analysis of field data gives post-liquefaction strengths that are significantly lower than the critical state strength determined from laboratory tests on undisturbed samples tested at the in situ state. The reason for this apparent difference is related to expansion of the soil skeleton associated with redistribution of water during and after shaking ,and will be discussed in detail in a later section of this paper.

3. STANDARD PRACTICE FOR LIQUEFACTION ASSESSMENT

In dealing with liquefaction, three aspects arise:

a) Will the design earthquake trigger liquefaction in significant zones of the foundation or earth structure, and if so,

b) Will the post-liquefaction strength be adequate to preserve stability and prevent a flow slide in the absence of inertia forces, and if so,

c) Will the displacements be tolerable?

The standard practice approach uses three separate analyses to respond to the three aspects; a triggering analysis, a flow slide analysis, and a displacement analysis.

3.1 Triggering Analysis

A triggering analysis involves comparing the cyclic stress ratio (CSR) caused by the design earthquake with the cyclic resistance ratio (CRR) that the soil has because of its density. The result is expressed in terms of a factor of safety against triggering liquefaction, Ftrig = CRR/CSR. Ftrig in the range 1 to 1.4 are generally considered acceptable and assure that seismic displacements will be small and tolerable (Byrne and Anderson, 1991, Youd et al., 2001).

The CSR is normally computed from an equivalent elastic dynamic analysis such as SHAKE using strain compatible moduli and damping , where CSR = 0.65 x max/'vo , and max is the maximum computed dynamic stress at a specific location during the design earthquake. CSR may also be computed from a simplified formula based on SHAKE analyses, but this approach is not recommended as the values are very sensitive to the specific site characteristics and the frequency content of the design earthquake.

CRR is generally computed indirectly from penetration resistance values and field experience during past

earthquakes. Cone, Standard penetration, or Becker penetration tests are commonly used with corrections for the magnitude of the design earthquake, stress level, and, static shear stress or bias. The CRR at any location can be expressed as:

CRR = CRR1 * Km * Kσ * Kα [1]

where CRR1 is obtained from the NCEER, 1997 workshop chart (Youd et al., 2001) shown in Figure 22, and Km, Kσ, Kα, are corrections for earthquake magnitude, effective overburden pressure, and static bias respectively. These are addressed in Youd et al., 2001.

3.2 Flow Slide Assessment

The factor of safety against a flow slide, Fflow is computed from standard limit equilibrium analysis procedures using a post-liquefaction strength in those zones predicted to liquefy from the triggering analysis. Post-liquefaction strengths are based on field experience during past earthquakes and are significantly lower than values obtained from direct testing of undisturbed samples at in situ void ratios. Post liquefaction strength may be expressed directly in terms of penetration resistance, Su

as suggested by Seed and Harder (1990) and shown in Figure 23, or as a strength ratio Su/ /'vo as shown in Figure 24 (Olson and Stark, 2002). It may be seen that the values of strength ratio are low, in the range 0.03 to 0.13. Computed factors of safety, Fflow > 1.2 to 1.25 are generally considered satisfactory (Byrne et. al., 1994).

Figure 22. MCEER Recommended chart for CRRevaluation based on (N1)60 (Youd et al., 2001).

3.3 Seismic Displacements

Seismic displacements that arise during shaking are commonly based on Newmark (1965) who modelled a potential sliding block of soil as a rigid mass resting on an inclined plane as shown in Figure 25. The acceleration that would just cause yielding and movement down the plane is called the yield acceleration (ay) in gravity units , and corresponds to the seismic coefficient, k, that would reduce the factor of safety to unity. This is readily obtained from limit equilibrium analyses described in Section 3.2 and can take the reduced strength associated with liquefaction into account.

Newmark modelled the mass on the inclined plane as a single-degree-of-freedom rigid plastic system, applied the design time history of acceleration at the base and solved the equation of motion to obtain the displacement of the mass caused by the shaking. Basically yielding at the

base and displacement down the plane is instigated each time the base acceleration exceeds the yield acceleration and causes the block to move in discontinuous steps during the period of strong shaking as shown in Figure25. Note that even the peak acceleration may only induce a small displacement as it lasts for only a fraction of a second. Peak accelerations greater than the yield acceleration imply an instantaneous factor of safety less than unity. This does not imply failure, but some limited displacement that can be calculated from the equation of motion and the prescribed earthquake motion. The standard practice approach of the past evaluated dynamic stability based on factor of safety and a seismic coefficient. This past procedure is generally not appropriate and should be replaced by a displacement approach such as the simple one proposed by Newmark.

Seismic displacements associated with liquefaction can also be estimated from empirical equations such as Youd et al. (2002). These equations are based on field

Figure 24. Normalized residual strength for liquefied sand vs. (N1)60 (Olson and Stark, 2002).

Figure 23. Residual strength for liquefied sand vs. (N1)60

(Seed and Harder, 1990).

Time (s)

Time (s)

Time (s)

a

(a)

(b)

(c)

(d)

Figure 25. Newmark-rigid block model of seismic slope motion (Modified from Day, 2002).

observations during many past earthquakes and can be a useful guide for existing conditions, but are of limited use for design of remedial measures.

Seismic displacements also occur due to dissipation of excess pore pressure. Since the earthquake loading is rapid, occurring in a matter of 10 to 100 seconds, it is generally assumed that these occur after earthquake shaking has ceased and are essentially vertical. A chart for estimating vertical strains associated with post liquefaction consolidation is shown in Figure 26 and indicates that strains could vary between about 0.5% and 5% depending on pre-earthquake relative density.

Figure 26. Relationships for volumetric reconsolidation strain as a function of equivalent uniform cyclic stress ratio, (N1)60-CS for Mw = 7.5 (Wu 2002).

The state-of-practice liquefaction procedures have a number of shortcomings: The three aspects of liquefaction; triggering, flow slide, and displacements are addressed in three separate analyses. In fact they are part of a single liquefaction response in which pore pressure rise and liquefaction occur at different rates and times in various zones of the earth structure causing the response of the structure to change as it softens. Redistribution of excess pore pressure may create more severe conditions, and finally dissipation and reconsolidation occurs as the soil regains its strength. No direct account of these aspects is considered in state-of practice procedures, and in particular no direct account of pore pressure rise and redistribution effects on response is considered. These shortcomings can lead to predicted response that is not realistic, and remedial designs that can be overly conservative, or unsafe depending on site conditions.

These aspects will be examined in state-of-art liquefaction analyses.

4. STATE-OF-ART TOTAL STRESS DYNAMIC ANALYSIS

This procedure addresses the liquefaction response taking pre-triggering, triggering, and post-triggering aspects into account in a single analysis that more

closely simulates conditions in the field. The earth structure of concern is modelled as a collection of discrete zones or elements. In the early stages of the analyses stiff moduli representing pre-triggering conditions are specified. The stress pulses in each zone are computed with time as they occur and are weighted depending on the size of the pulse as compared to the reference pulse that would cause liquefaction in 15 cycles. So a small pulse may account for a half cycle or less, whereas a large pulse may count for 2 or 3 cycles or more. When or if the cycle count reaches 15 in any element, it is deemed to have liquefied and given much reduced strength and stiffness properties consistent with post liquefaction response. In this way, the most severely loaded or looser zones will liquefy first and the extent of liquefaction will expand with further shaking as observed in centrifuge model tests. If sufficient elements liquefy and their residual strength is not adequate for stability, then a flow slide is predicted.

The procedure makes use of both state-of-practice triggering and residual strength charts and, thus is a logical extension of the current state-of-practice approach discussed in Section 3. The method is described in detail in Beaty and Byrne (1999), and Beaty (2001). An example of predicted displacement from such a liquefaction analysis is shown in Figure 27. Note that excess pore pressures are not computed in this approach. They are indirectly accounted for by prescribing much softer post-liquefaction stress-strain relations and residual strength values after liquefaction has been triggered.

5. STATE-OF-ART EFFECTIVE STRESS ANALYSES, FINN LOOSE COUPLED MODEL

The strength and stiffness of soil is governed by effective stress, and so it is desirable to evaluate seismic response of soil in terms of effective stresses. Pore pressure rise and liquefaction are caused by the tendency of soil to contract when subjected to cyclic shear loading. Martin et al. (1975) presented a 4 parameter shear-volume coupling model for predicting the increment of volumetric compaction per load cycle, dεp

v. Their data showed that dεp

v depended on both the level of shear strain in the current cycle as well as the accumulated volumetric strain from prior cycles, but was independent of effective stress level. The associated rise in pore pressure for saturated undrained conditions is du = dεp

v * Ke, where Ke is the elastic bulk modulus of the soil. Based on this concept Martin et al. developed the first dynamic effective stress model in which shear strains were evaluated in each element allowing the pore pressure rise at the end of each shear cycle to be computed. The total rise in pore pressure for undrained conditions is simply the sum of the pore pressure increments. The strength and moduli are reduced in each element in accordance with the drop in effective stress as pore pressure rise occurs. Dissipation of excess pore pressure can also be accounted for by allowing flow between elements. This approach is commonly referred to as the Finn model and was used in the TARA program (Finn et al., 1986).

Later versions of the TARA program (TARA 3F, Finn et al., 1995) and the FLAC Finn model are based on a simpler two parameter shear volume coupling model by Byrne (1991), dεp

v = γ * C1 * Exp(-C2 *εp

v/γ) where γ is the shear strain in the current cycle , εp

v is the accumulated volumetric strain from prior cycles, and C1 and C2 are soil parameters that depend on relative density. This procedure is referred to as a loose coupled approach because the pore pressure is updated at the end of each cycle or half cycle of shear strain rather than at every time step.

The Finn model can adequately capture the pore pressure rise up to the point of triggering of liquefaction for level ground conditions, but cannot simulate the post-liquefaction response where dilation and pore pressure drop occurs during each load cycle as depicted in Section 2. TARA 3F handles post-liquefaction conditions by switching to a total stress approach and specifying a residual strength as discussed in Section 3 and a nominal shear stiffness.

6. STATE-OF-ART EFFECTIVE STRESS ANALYSES, FULLY COUPLED

A number of fully coupled effective stress models exist for predicting seismic response and liquefaction including DYNAFLOW (Prevost, 1985) and UBCSAND (Byrne et al. 1995, Puebla et al., 1997; Beaty & Byrne, 1999, Byrne et al., 2004). Such models are based on plasticity concepts, and shear-volume coupling effects are computed at each time step rather than at the completion of each half cycle of strain. Computer programs such as FLAC (Itasca, 2005) have dynamic and flow capabilities and a range of

Figure 28. Elastic and plastic zones in Mohr-Coulomb model.

soil models. However, because the cyclic response of granular soil is complex, most soil models are either too complex and have too many parameters, or are too simple and do not allow predictions that are in reasonable agreement with laboratory tests.

UBCSAND is a modified form of the built-in Mohr model in FLAC. The Mohr model has basically 2 elastic parameters that specify shear and bulk moduli, and 3 plastic parameters, that specify friction, cohesion, and dilation angle. The model is elastic when the stress state is below the strength envelope, and plastic when the stress point is on the failure envelope as shown in Figure28. The plasticity model used assures that the stress point does not exceed the strength envelope.

’

Failure (Plastic)

Elastic zone

Figure 27. (a) finite difference mesh of the Lower San Fernando Dam, (b) Predicted distorted mesh and displacement vectors of 1971 liquefaction-induced upstream slope failure (Beaty and Byrne, 1999).

The UBCSAND modification to the Mohr model involves allowing plastic yielding below the strength envelope. In shear stress versus normal stress space, the yield loci are radial lines from the origin of stress space, and the directions of the plastic strains are as shown on Figure 29 when shear and normal strains are superposed upon the stress space.

It may be seen from Figure 29 that for a stress point below the constant volume friction angle (phase transformation), the plastic vector is directed to the right indicating contraction, while at phase transformation state the vector is vertical indicating no plastic volume change, above the phase transformation line the vector is directed to the left indicating expansion. This plastic shear-volume effect is taking place at all loading stages, and is given by the simple expression:

dεpv = dγ * (Sin(cv) – Sin(d)) [2]

where d is the friction angle developed at any stage of loading and varies between 0 and f, the peak friction angle.

For Fraser River sand the constant volume friction angle, cv is 33 degrees and all other parameters are related to relative density or Standard Penetration value, (N1)60.

Characteristic monotonic stress strain undrained response is shown in Figure 30 for very loose and dense sand. The UBCSAND model will capture this response. Note that the very loose sand strain softens and the model, predicts this behaviour with no need for any external load shedding device.

The results of cyclic simple shear constant volume tests are shown in Figure 31 together with predictions from UBCSAND. It may be seen that the predictions are in reasonable agreement with the measurements. Additional tests having a range of applied CSR values were also predicted, and a comparison of predicted and measured liquefaction resistance is shown in Figure 31c. Note thatthe model captures the stiff pre-liquefaction stage, the onset of liquefaction at the appropriate number of cycles, and the very much softer post-liquefaction response. Note also that the post-liquefaction resistance at large strain is high and there is no indication of a low strength ratio corresponding to the back calculated values from field case histories.

6.1 Analysis of Level and Sloping Ground Conditions

An effective stress analysis of a 10m high sand column resting on a rigid base with and without a non-liquefiable silt layer located at a depth of 4m was examined (Figure 32). The column represents an infinite slope with a one degree ground slope. An idealized harmonic acceleration motion was applied at the base (Figure 33) and time

Фd < Фcv

Фd = Фcv

Фd > ФcvPlastic strain vector

Yield locus

, dp

Figure 29. Moving yield loci and plastic strain increment vectors.

Dense

Sh

ear

Str

ess,

Loose

Shear Strain,

Shear Strain,

Exc

ess

Por

e P

ress

ure

, u

Loose

Dense

Loose

Dense

Normal Stress, ’

cv

S

hea

r S

tres

s,

Figure 30. Characteristic response of sand to undrained loading in terms of (a) stress-strain, (b) excess pore pressure, and (c) effective stress path.

-1

0

1

2

3

4

5

6

7

8

9

10

-1 0 1 2 3 4

-20

-10

0

10

20

-15 -10 -5 0 5 10 15Strain (%)

Str

ess

(k

Pa

)

PREDICTION

TEST DATA

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8No. of Cycles

Ru

RU = 0.95

TEST DATA

PREDICTION

0

0.05

0.1

0.15

0.2

0 5 10 15 20No. of Cycles

CS

R

TEST DATA

PREDICTION

Figure 31. Comparison of predicted and measured response for Fraser River Sand, a) stress-strain, b) & c) Ru & CSR vs. No. of cycles ( tests data from Sriskandakumar, 2004).

Figure 32. Ground conditions used in the study, (a) case I, uniform profile without low permeability sub-layer, (b) case II, profile with low permeability sub-layer (Seid-Karbasi and Byrne, 2006).

Firm impervious ground

Element:(1,13)(1,10)(1,5)(1,3)

(a) case I (b) case II

Groundwater table

Sand

Barrier

histories of excess pore pressure ratio (Ru) at various depths were predicted (Figure 34).

It may be seen that for conditions; (a) without, and (b) with the silt layer the peak excess pore pressure Ru is close to unity (zero effective stress state) during most of the shaking and then drops with time as excess pore pressures dissipate. However, at the upper location just below the silt layer (location (1,13) in Figure 32) the Ru

values are higher for (b), with the silt layer, and remain significantly higher for some time after shaking has stopped. This effect is caused by the presence of the silt layer. The excess pore pressure generated by the motion must drain upward. The silt layer acts as a barrier and curtails the upward flow causing expansion and accumulation of water beneath the silt.

In Figure 35 it may be seen that very high shear deformations are predicted directly beneath the silt layer where the Ru values were highest. The predicted maximum surface displacements are significantly higher with the presence of the silt layer (Figure 36). Note that displacements continue for some time after the shaking has ceased when the silt layer is present.

The predicted change in volumetric strains as a function of depth after 30 s are presented in Figure 37. It shows that the bottom 2/3 of the sand layer beneath the barrier contracts, while the top 1/3 expands, and that most of the expansion occurs directly beneath the barrier leading to a thin localized zone of high shear strain or a shear band.

The coupled stress-flow effective stress analysis indicates that layered sand deposits likely do not remain undrained during seismic loading and that expansion during and after shaking can occur leading to significantly lower stiffness and strength in thin zones directly beneath silt or clay layers of low permeability (Seid-Karbasi and Byrne, 2004; Naesgaard et al., 2005; and Seid-Karbasi, 2006). Without a silt layer, no significant expansion and no strain localization occurred.

Shaking tests conducted by Kokusho (1999) & (2003), show that the presence of a silt layer results in expansion and the formation of a water film at the base of the silt layer, and a flow failure some time after shaking has ceased, while uniform sand (without a silt layer) is stable during and after simulated seismic loading. This is demonstrated in cartoon form in Figure 38. Centrifuge tests conducted at The University of California, Davis with a thin silt layer within a sand model also show localization directly beneath the silt layer and the formation of a water-rich zone under simulated seismic loading, (Kulasingam, 2003; Malvick, 2005; and Malvick et al., 2005). Centrifuge tests at C-CORE in Newfoundland with simulated seismic loading also showed little slope movement for sand slope models without a silt barrier layer, however the presence of a silt barrier layer in a similar model resulted in a post-shaking flow failure (Phillips et al., 2004; Phillips and Coulter, 2005; and Naesgaard et al., 2005).

Figure 33. Acceleration time history for base input motion.

Further analyses predict that in layered deposits the pattern of contraction at the base of sand layers and expansion at the top occurs regardless of the thickness of the individual sand layers. This means that sand elements just below the barrier are being subjected to inflow causing them to expand regardless of the thickness of individual sand layers. Such expansion can greatly soften their response (Seid-Karbasi, 2006).

The results of monotonically loaded triaxial tests in which water is injected into the sample (inflow tests) are compared with results of conventional undrained tests in Figure 39a, b and c (Eliadorani, 2000). The principal stress difference versus axial strain is shown in (a) and indicates that a small amount of inflow causes a dramatic reduction in stress difference. The amount of inflow with axial strain is shown in (b) and indicates that a 1 or 2 % inflow or expansion strain can reduce the resistance to essentially zero. The stress paths are compared in (c) and indicate that 1 to 2% inflow can drive the sample to the zero effective stress state with zero shear resistance.

The predicted responses for both undrained and inflow conditions from the UBCSAND model are also shown on Figure 39 as the continuous lines, and are in remarkably good agreement with the measured data. The inflow tests were predicted using the same soil parameters as for the undrained case but with specified volumetric expansion conforming to the laboratory test as per Figure 39b. This indicates that the model can account for the effect of inflow. The numerical model simulations together with physical model tests in which a water film is observed to form at the base of barrier layers during or after shaking indicates that flow of water and expansion at the base of barrier sub layers within sand deposits is real and should be considered in liquefaction assessment.

The concept that the appropriate liquefaction response of sand can be determined from recovery and testing of undisturbed samples at the in situ void ratio is flawed for layered material in which a large contrast in permeability exists. The triggering resistance most likely can be obtained from such tests as there may be little time for drainage effects during the short period of strong shaking. However, the post-liquefaction response in field

Harmonic Motion

-3

-2

-1

0

1

2

3

0 1 2 3 4 5 6 7 8Time (s)

Acce

lera

tio

n (

m/s

^2)

Figure 34. Excess pore pressure ratio Ru vs. time at selected points with increasing depth, (a) case I, (b) case II (Seid-Karbasi and Byrne, 2006).

(1,13)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25Time (s)

Ru

(1,10)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25Time (s)

Ru

(1,3)

0

0.2

0.4

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1

0 5 10 15 20 25Time (s)

Ru

(1,13)

0

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0 5 10 15 20 25Time (s)

Ru

(1,10)

0

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1

0 5 10 15 20 25Time (s)

Ru

(1,3)

0

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0 5 10 15 20 25Time (s)

Ru

(a) (b)

-1

0

1

2

3

4

5

6

7

8

9

10

-1 0 1 2 3 4

(m)

(m)

Figure 35. Deformation pattern of soil profile (a) without barrier, case I (with max. lateral displacement of 0.95 m after 14 s), (b) with barrier, case II, with max. lateral displacement of 1.75 m after 30 s (Seid-Karbasi and Byrne, 2006).

-1

0

1

2

3

4

5

6

7

8

9

10

-1 0 1 2 3 4

(m)

(m)

(a) (b)

0

0.5

1

1.5

2

0 5 10 15 20 25

Time (s)

X-d

is (

m) No Barrier

With Barrier

End of shaking

Figure 36. Surface lateral displacement vs. time for profiles with and without barrier (Seid-Karbasi and Byrne, 2006).

Figure 37. Profile of volumetric strain beneath the barrier layer, after 30s (Seid-Karbasi and Byrne, 2006).

4

5

6

7

8

9

10

-0.5 0 0.5 1

Dep

th (m

)

Barrier Base

Contraction Expansion

Volumetric strain (%)

Figure 38. Post shaking slope failure due to pore pressure redistribution (adapted from Kokusho, 1998).

Figure 39. Predictions of element undrained and partially drained triaxial test on Fraser River sand, (a) stress–strain, (b) volumetric strains, and (c) stress paths (from Atigh and Byrne 2004).

conditions will not be at constant volume, as flow of water into thin zones beneath low permeability silt or clay layers cause expansion and perhaps a water film to occur and result in a marked reduction in strength and stiffness. This explains why back calculated strengths from field case histories are significantly lower than undrained strengths from undisturbed samples. Strengths obtained from undisturbed samples of sand at the Lower San Fernando dam required a reduction factor of 20 (Castro et al., 1989) to bring them into agreement with back calculated values from the flow failure that occurred on the upstream face of dam. In addition, time for redistribution of generated excess pore water pressure and expansion to occur can explains why a number of field case histories, including the Lower San Fernando dam, failed some time after severe shaking (Seid-Karbasi and Byrne, 2004).

The lower than expected back calculated strengths from case histories do not imply that critical state principles do not apply to liquefaction induced flow slides. In fact it explains why greatly reduced strengths can occur. The sand element has expanded to a higher void ratio and hence a lower critical state strength. In the limit, the sand skeleton can only expand to its maximum void ratio at critical state. Further inflow cannot be absorbed by the element and will appear as a water film.

6.2 Drainage Effects

The effect of drains on liquefaction response is examined in Figure 40 for a sand stratum containing a silt barrier layer and an infinite slope having a gently sloping ground condition of 1 degree. The drain is represented by a single column of loose material with a coefficient of permeability 100 times that of the sand (see Figure 40a). The same ground motion was applied at the base as before. Contours of Ru values as well as flow vectors after 3.5 s of shaking are shown in Figure 40b. The results show that water flows horizontally into the drain and then up and out, and predicted Ru values are much lower than without the drain. The highest predicted Ru values occur beneath the left edge of the barrier as expected. Predicted Ru values with time are compared with and without the drain for the three cases in Figure 41. Predicted displacements with the drains were small.

A field experiment in which seismic shaking was simulated for conditions with and without drains is shown in Figure 42 and confirms that drains can be effective in reducing excess pore pressures. Centrifuge tests at C-CORE also confirmed the beneficial effects of drains in preventing flow failure (Phillips et al., 2004; Phillips, R., and Coulter, 2005; and Naesgaard et al., 2005). A post-shaking flow failure was generated in a model with a silt barrier; however flow failure did not occur in a subsequent similar model in which the barrier was perforated with drains (Figure 43).

Figure 40. Treated ground condition, (a) model with drain curtain (b) Ru (max) and flow vectors at 3.5s (Seid-Karbasi and Byrne, 2006).

Ru(max)0.650.700.750.800.850.900.951.00

(b)

k (m/s) 8.81 e-7 8.81 e-4 8.81 e-2

Vertical drain

(a)

Figure 41. Predicted time history of Ru at mid depth of loose sand for soil profile (a) without barrier layer, (b) with barrier layer and, (c) with barrier layer treated withdrain curtain (Seid-Karbasi, 2006).

Uniform

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20Time (s)

Ru

w ith Barrier

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20Time (s)

Ru

With Drain

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20Time (s)

Ru

(a)

(b)

(c)

Figure 42. Measured Ru in field liquefaction test for case (a) without drain and, (b) with drain (data from Chang et al. 2004).

-20

0

20

40

60

80

100

120

Time (sec)

Ru

(%)

0 1 2 3 4

Test without drainTest with drain

Figure 43 (a) Initial and displaced profile of centrifuge test CT5 with three drainage slots (b) numerical analysis of same, and (c) comparison of vertical displacement near crest with (CT5) and without (COSTA-C) drainage slots. Post-shaking flow initiated in the COSTA-C test at approximately 70s.

6.3 Back-Analysis of Lower San Fernando Dam Failure

In 1971 the Lower San Fernando Dam was shaken by a large earthquake with a peak velocity pulse of around 0.6m/s and peak ground acceleration of approximately 0.5 g (Seed, 1973; Seed et al., 1989; Castro et al., 1989,Castro, 1995). Approximately 20 to 30 s after the end of earthquake shaking the upstream face of the dam catastrophically failed leaving only 1.5m of free-board and putting a large population at risk (Figures 11, 12, 13). Extensive investigation and analyses of the dam were conducted following the event. From the studies it was concluded that the hydraulic fill soil within lower and central portions of the dam had liquefied and overlying portions of the dam had flowed out riding on the liquefied soil (Figure 14).

Numerous back-analyses of the dam have been conducted (Seed et al., 1973, Seed and Harder, 1990, Beaty, 2001) using both total stress and effective stress models, but none have emulated the post-shaking failure mechanism that was actually observed. By including low permeability barrier layers and underlying layers of higher permeability the post-shaking failure mechanism has been emulated, both in approximate geometry of the

failing block and in achieving ongoing failure after end-of-earthquake shaking (Naesgaard et. al., 2006).

The liquefiable portions of hydraulic fill soils were assumed to have (N1)60 values varying from 12 to 17 while the clayey core of the dam was given undrained shear strength of 20% of the vertical overburden pressure. The UBCSAND constitutive model was used for the potentially liquefiable portions of the dam while the Mohr Coulomb model was used for the clayey core and portions of the dam above the water table. Figure 44 shows the model grid, locations of low permeability barriers, and the final displaced shape at 119s when the analysis ended due to excessive distortion of the elements (the failing mass still had a velocity of 0.2 m/s). The identical analysis was also repeated (i) with ‘flow-off’ so as to emulate undrained behaviour and (ii) with ‘flow-on’ but without the low permeability barriers. In both cases deformations stopped at end-of-shaking with deformations much less (6 to 8m) than that when the barriers were present (>36m) (Figure 45). This clearly demonstrates the importance of considering the effects of pore-water redistribution. The analyses indicate that impermeable barriers and vertical and lateral migration of pore water play a key role in the failure mechanism.

Figure 44. Lower San Fernando Dam analysis (a) FLAC grid, (b) assumed (N1)60 and cohesion in core, (c) location of low permeability barriers with vertical permeability in cm/s, and (d) 5m lateral displacement contours at 120s (Naesgaard et al., 2006).

Figure 45. Lower San Fernando Dam numerical analysis: A = acceleration time history input at model base; B, C, & D = horizontal displacement of upstream face (point 'x' in figure 44) from (B) undrained silty barriers as illustrated analysis with flow-off, (C) analysis with flow-on but no barriers to vertical flow, and (D) analysis with flow-on and low permeability barriers as illustrated in figure 44c.

6.4 Effective Stress Summary

The results of coupled stress-flow effective stress analyses show that redistribution of void ratio can be very important in sand and gravel deposits containing low permeability silt or clay layers. Such layers can act as barriers to upward flow and dissipation of excess pore pressures generated by seismic loading. The effect is to cause an expansion of the soil in a thin zone directly beneath the barrier layer. The expansion concept is illustrated in Figure 46 in terms of critical state concepts. The initial state (eo-P'o) is assumed to lie below the critical state line at point A as shown on the figure. If Ru rises to unity during shaking then the stress point moves to point B if the element remains undrained and has zero effective stress. However, if it is sheared at constant void ratio it will dilate and fail at point C on the critical state line and will have a shear strength greater than the drained strength. If however, inflow takes place and the element expands, it can remain at the zero effective stress state and continue to expand until it finally reaches the critical state at zero effective stress, point D, and has zero shear strength. Any further inflow to the element will result in the formation of free water, or a water film at the sand-silt interface.

The strength actually mobilized in any zone will depend on the in situ void ratio prior to shaking and the amount of inflow. The inflow in turn depends on the site stratigraphy as it reflects the presence and degree of permeability contrast, as well as the magnitude and duration of shaking. The wide variation in residual strength back

calculated from field case histories for given density conditions indicates that factors other than in situ density play an important role in post-liquefaction shear strength and should be considered as noted by Malvick et al. 2005 and Seid-Karbasi and Byrne, 2006).

7. SUMMARY AND CONCLUSIONS

Seismic loading has caused severe damage to civil engineering structures in past earthquakes. The design and retrofit of these structures in Canada is largely governed or guided by the National Building Code of Canada. NBCC 2005 has increased the seismic loading from an event having 10 % chance of exceedance in 50 years, to a 2475 year event having a 2% chance of exceedance in 50 years. This has lead to a doubling of the peak ground acceleration in Greater Vancouver, and large increases in other areas of Canada.

Liquefaction is caused by the tendency of granular soils to compact under cyclic shear loading. If compaction is prevented or curtailed by the presence of water that cannot escape, it causes pore pressure rise and softening of soil response. If pore pressure rise causes the effective stress to drop to zero, the stiffness and strength also drop to zero and the soil liquefies. However, unless the soil is very loose, it will dilate and recoverstrength and stiffness. In general shear strains are small and soil stiffness and strength remains high prior to liquefaction. Once liquefaction is triggered in significant zones of a foundation, strains and displacements become large and a possible flow side is a concern. This is the

Figure 46. Critical state conditions for undrained and inflow conditions

(Pcs)exp (Pcs)

ecs

e0 e0

D

C

B A

P

e

P0

CS Line

Expansion from inflow

Undrained cyclic loading

characteristic liquefaction response, and analyses procedures should take this into account.

The current state-of-practice approach for analysis and design for liquefaction is based on a three stage total stress approach in which; 1) triggering of liquefaction , 2) possibility of a flow slide, and 3) seismic displacements are examined in three separate analyses. Because earthquake loading is applied rapidly it is assumed that the process occurs undrained. This is generally a conservative assumption for assessing the triggering of liquefaction. However, for post-liquefaction conditions it may be highly unconservative as observed from field experience, physical model tests, as well as effective stress numerical modelling procedures.

State-of-art effective stress coupled stress-flow dynamic analyses show that for a uniform sand deposit, excess pore pressures generated by seismic shaking drain upward to the surface, and soil elements at all depths contract as water escapes to the surface, and liquefaction flow failures do not occur even for loose sand deposits. This is in accord with dynamic centrifuge modelling tests. However, if low permeability silt or clay layers are present, they impede the upward flow of water causing inflow and expansion in the sand directly beneath the low permeability barrier layer. This in turn can involve a very large reduction in residual strength during and after shaking and could result in flow failures even in medium dense sands. The amount of expansion depends on the density of the soil and the level and duration of shaking. Analyses indicate that the actual thickness of the sand layer itself is not a significant factor.

The results indicate that it is very important to identify the stratified nature of the soils present at a specific site, and take permeability contrasts and mesh size into account in modelling and analysis. The results also indicate that drains can be very effective in preventing the barrier layer effect. They may not prevent the triggering of liquefaction and the development of significant displacements due to reduced moduli, but they can remove the permeability contrast and prevent expansion and loss of strength beneath the barrier (prevent flow failure).

This work is still in the development stage and in practice residual strength obtained from back analysis of field case histories should still be considered at the end of shaking. However, there is very considerable scatter in the field case history data, and the effective stress analyses may be helpful in choosing post-liquefaction strength values that consider site and earthquake dependent characteristics in a rational manner.

ACKNOWLEDGEMENTS

The authors acknowledge funding from the National Scientific and Engineering Research Council through Strategic Liquefaction Grant No. NSERC 246394. The authors also acknowledge the development work on the UBCSAND program and analyses methods by Michael Beaty and Sung Sik Park, the centrifuge tests conducted

by Ryan Phillips at C-CORE and the insight into the flow liquefaction problem provided by the work of T. Kokusho.

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