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Critical state framework for liquefaction of fine grained soils
Engineering Geology 117 (2011) 2–11
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Engineering Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n g g e o
Critical state framework for liquefaction of fine grained soils Balasingam Muhunthan a,⁎, Diana L. Worthen b a b
Department of Civil and Environmental Engineering, Washington State University, Pullman, WA 99164, USA PBS Engineering and Environmental, Vancouver, WA, USA
a r t i c l e
i n f o
Article history: Received 25 March 2010 Received in revised form 4 August 2010 Accepted 17 September 2010 Available online 30 October 2010 Keywords: Fine grained soils Liquefaction Critical state Liquidity index Stress ratio Earthquake
a b s t r a c t A consistent methodology based on the critical state framework to characterize the different regimes of finegrained soil behavior under earthquake loads is put forward. Shear strength and deformation behavior of soils depend in a major way on the combination of volume and confining stress. Depending on their combination, a soil aggregate may fracture into clastic debris, fail with fault planes, or yield plastically. This characterization of the class of limiting soil behavior is used to analyze the potential for large deformation and liquefaction in fine grained soils. The central piece of the proposed characterization is the (η, LI5) stability diagram where η = q/p′ and LI5 = LI + 0.5 log (p′/5). This diagram captures the effects of soil plasticity through liquidity index LI, confinement through mean normal effective stress p′, and shear stress q through the stress ratio η. The three regions of behavior; fracture, fault, and fold/yield are identified. Soils become susceptible to liquefaction when they shift into the fracture zone (LI5 ≤ 0.4), or if they plot outside of the stable yielding region. Under earthquake loading, the initial soil states will migrate into different regions in the stability diagram depending on their initial location, shear stress increment, and, pore pressure response. The final position of the soil state would dictate the type of limiting behavior expected in the field; fracture, rupture or yield. The final states which fall into the fracture region have the potential for catastrophic failures including “liquefaction”; the ones which fall onto the rupture region would experience the attainment of a peak stress ratio followed by softening along failure planes; the ones in the yield region would continue to yield in a stable manner. The latter two types of deformations while resulting in large deformation may not be of a catastrophic nature. The proposed characterization is used to examine the liquefaction susceptibility of fine grained soils from China, Taiwan, and Turkey. Use of simplified empirical criteria based on parameters such as plasticity index and fines contents may not capture the true nature of the type of undrained limiting behavior of fine grains soils in the field including liquefaction. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Recent field evidence of ground failure in low-plasticity silts and clays during strong earthquakes have highlighted the need to better characterize the strength loss and development of large deformations in a broad range of saturated soils, from sands to fine grained soils. In the absence of a fundamental understanding of the seismic behavior of fine-grained soils, their liquefaction potential is currently evaluated by the ‘Chinese criteria,’ which is based on their plasticity characteristics. First introduced by Wang (1979) based on the study of major earthquakes in China, these criteria were expanded by Seed and Idriss (1982) and have become the commonly accepted method for predicting fine-grained liquefaction susceptibility. More recent research has shown that the Chinese criteria are ineffective and should not be relied upon to safely predict liquefaction potential (Boulanger and Idriss, 2006; Bray and Sancio, 2006). Modifications to
⁎ Corresponding author. Tel.: + 1 509 335 3921; fax: + 1 509 335 7632. E-mail address: [email protected] (B. Muhunthan). 0013-7952/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2010.09.013
the Chinese criteria have been proposed for predicting the liquefaction potential of fine grained soils. The shear strength and deformation behavior of soil is quite sensitive to the combination of changes in volume and confining stress. Depending on their combination, a soil aggregate may fracture and crack into clastic debris, or fail with fault planes on which gouge material dilates and softens, or it can continue to yield and deform plastically. The current prediction models do not distinguish these distinct classes of soil behavior in the description of large deformation failures. This has led to many of the difficulties faced in the interpretation of liquefaction failures (Muhunthan and Schofield, 2000). Most often, evidence of large deformation observed from post reconnaissance efforts following an earthquake are lumped into the “liquefied” phenomenon which may not capture their true characteristics. This study makes use of critical state soil mechanics and puts forward a framework for fundamental understanding of the liquefaction behavior of sands, and fine-grained soils over a broad range of plasticity characteristics. It explicitly recognizes that soil is an aggregate of interlocking frictional particles and that the regimes of soil behavior depend in a major way on its density and effective
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pressure. Liquefaction and large deformation in fine grained soils are shown to belong to a class of failures that occur in the fracture region brought about by a combination of plasticity, confining stress and imposed shear. The observed field data on fine grained soil performance under past earthquakes are examined using this new framework. 2. Current fine grained soil liquefaction criteria Most current criteria to assess liquefaction susceptibility of fine grained soils stem from the Chinese criteria published by Wang (1979). Wang found that the common characteristics of the liquefied fine-grained soils were having less than 20% clay content (5 μm or less), liquid limit between 21 and 35, plasticity index between 4 and 14, and natural water content of 90% or above the liquid limit. The plasticity characteristics of the original Wang (1979) data are shown in Fig. 1. Seed and Idriss (1982) modified Wang's conclusions, and defined the liquefaction criteria of fine-grained soil to consist of clay fraction (5 μm or less) less than 15% by weight, liquid limit less than 35, and water content 90% or greater of the liquid limit. Liquefaction will occur only if all three criteria are met. Recent field observations have necessitated the need for reevaluation of the Chinese criteria. Three recent earthquakes, the 1994 Northridge, 1999 Kocaeli, and 1999 Chi-Chi earthquakes, all showed evidence of liquefaction in soils with greater than 15% clay content. This means the soils did not meet all three of the Chinese criteria, and yet still liquefied. Liquefaction of soils in Adapazari, Turkey during the 1999 Kocaeli earthquake resulted in much of the damage throughout the city (Erken, 2001), even though they typically had clay contents greater than 15% (Bray and Sancio, 2006). After the 1994 Northridge earthquake, Holzer et al. (1999) noted that soils which would have been determined safe from liquefaction by the Chinese criteria did in fact liquefy and lead to permanent ground deformations. Many recent researchers, such as Bray and Sancio (2006) and Boulanger and Idriss (2006), have recommended not relying on the Chinese criteria. The Chinese criteria are based solely on plasticity characteristics, and do not account for the confining stress of in-situ soil. One plasticity criterion is chosen to be applicable. Wang's research shows that soils with water content-liquid limit ratios less than 0.9 did not liquefy, except in one case with a ratio only slightly less than 0.9. Soils with water content-liquid limit ratios greater than 0.9 all liquefied, again except for one case, in which the ratio was only slightly greater. This criterion appears to be important in evaluating liquefaction susceptibility; however, the other two Chinese criteria may not be valid. Wang's data from 1979 does not show the behavior of any soil with a liquid limit greater than 35, and recent earthquakes have revealed liquefied soils with greater than 15% clay fractions. Andrews and Martin (2000) concluded that the criteria for liquefaction susceptibility should not include a water content-liquid limit ratio as a condition for silty soils. Additionally, they reduced the liquid limit criterion to LL b 32, and the clay fraction condition to b10%. They also used 2 μm as the definition of clay-sized particles, which is smaller than the previously used 5 μm. If a soil meets only one of the two criteria, further study is required to confidently evaluate
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liquefaction susceptibility. Polito (2001) determined three categories of liquefaction susceptibility, based on plasticity characteristics. A soil is liquefiable if LL b 25 and PI b 7; soils with 25 b LL b 35 and 7 b PI b 10 are potentially liquefiable; and 35 b LL b 50 and 10 b PI b 15 indicates a soil susceptible to cyclic mobility. Perlea (2000) recommended lab testing as the best evaluation method for liquefaction potential. Boulanger and Idriss (2006) have suggested that ‘sand-like’ finegrained soils and ‘clay-like’ fine-grained soils respond differently to cyclic loads. A sand-like soil is susceptible to liquefaction, whereas a clay-like soil will respond with cyclic failure. The plasticity index PI is used to determine the difference between the two categories. A PI b 7 indicates sand-like liquefaction susceptibility, while soils with a PI ≥ 7 are considered clay-like, prone to cyclic failure. 3. Critical state framework for soil behavior: fracture, rupture, and ductile regimes Critical state soil mechanics (Schofield and Wroth, 1968) is a framework that explicitly recognizes that soil is an aggregate of interlocking frictional particles and that the different regimes of soil behavior under shear depends in a major way on its density and effective pressure (Schofield, 1998, 2005). Detailed accounts of the basic principles, the features, and finite element applications of the CSSM framework have been presented in a number of publications. We present here only the salient features of the framework relevant to our discussion. The two invariant stress parameters used in CSSM are mean normal stress: ′
p =
1 ′ ′ ′ σ + σ2 + σ3 3 1
ð1Þ
and the deviator stress: 1 ′ ′ 2 ′ ′ 2 ′ ′ 2 1 q = pffiffiffi ðσ2 −σ3 Þ + ðσ3 −σ1 Þ + ðσ1 −σ2 Þ 2 2
ð2Þ
where σ1′, σ2′, σ3′ are the principal effective compressive stresses. For triaxial test conditions where, σ2′ = σ3′ Eqs. (1) and (2) reduce to p′ = 1/3(σ1′ + 2σ3′), and q = (σ1′ - σ3′), respectively. The two parameters p′ and q, and a third variable, the specific volume V = (1 + e), where e is the void ratio, define the state of a soil specimen. Roscoe et al. (1958) quote experimental evidence that the ultimate state of any soil specimen during a continuous remolding and shear flow will lie on a critical state line (CSL) as shown in Fig. 2. The figure also shows the isotropic consolidated line (ICL or the normal compression line NCL), and a fracture line whose characteristics are discussed later. The critical state line distinguishes the two distinct types of soil behavior. There are states for which the combinations of specific volume V and mean normal effective stress p′ lie further away from the origin than the line of critical states. These states are called contractive states; shearing there causes aggregates to compress to denser packing and emit water with ductile stable yielding of a test specimen. There are also states for which the combinations of specific volume V and mean normal effective stress p′ lie closer to the origin
Fig. 1. Chinese criteria of Wang (1979) (After Bray and Sancio, 2006).
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Fig. 2. State of soils in V − ln p′ space.
than the line of critical states. These states are termed dilative states; shearing there causes aggregates to increase volume and suck in water, and ground slips at peak strength with unstable failures. In the critical state framework, the limits to stable states of yielding are defined by the state boundary surface in the 3-D, p′–q–V space. The 3-D state boundary surface can be represented in a 2-D space using appropriate normalization of the variables. Various normalizing parameters are used for this purpose (Atkinson, 1993). One such normalization suitable for the present work is that using pressure at ′ . The 2-D representation of the normalized state the critical state pcrit ′ − p′/pcrit ′ space is shown in Fig. 3. The boundary surface in the q/pcrit values shown on the horizontal axis correspond to data obtained from Weald clay (Schofield and Wroth, 1968; Schofield, 1980) but the discussion below applies to all soils albeit with slightly different numbers as appropriate. It is observed from Fig. 3 that the limiting soil behavior is not a straight line fit using the Mohr–Coulomb failure envelope as is often done in geotechnical practice. Soil behavior at the limiting state consists of three distinct classes of failure; the limiting lines OA and OG (Figure 3) indicate states of soils exhibiting fractures or cracks; AB and GE indicate soil states with fault ruptures on Hvorslev or Mohr–Coulomb planes; BD and ED indicate ductile yielding and folding of soils. Soil states falling on the fracture surface would result in the development of unstable fissures and crack openings. Heavily overconsolidated clays and overcompacted sands at low confining stresses
may reach this limiting state. The “no-tension” or “limiting tensile strain” criteria are the most widely used among the alternative theories to quantify tensile fracture (Schofield, 1980). For triaxial specimens, the no tension criterion with σ3′ = 0 results in p′ = σ1′ /3 or q/p′ = 3 and leads to vertical splitting cracks, which is the case of line OA. For horizontally spalling cracks, setting σ 1′ = 0 results in p′ = 2/3 σ3′ , q = −σ3′ , or q/p′ =1.5 which is the case of line OG. For clays or silty clays, Schofield (1980) suggested that the change from rupturing to tensile ′ , where pcrit ′ is the effective cracking occurs at a pressure p′ = 0.1 pcrit confining stress at the critical state (Figure 3). The corresponding line of the fracture states in V − ln p′ space is shown in Fig. 2. When the effective stress path crosses the fracture surface OA, the soil element begins to disintegrate into a clastic body and unstressed grains become free to slide apart. In that case, the average specific volume of the clastic mass could increase (due to large voids/cracks) and consequently its permeability could increase significantly and instantly. A significant shear stress at low confining stresses can cause the crossover of the crack-surface OA and a large increase in specific volume. When such a condition occurs, the opening within the soil body may consist of an extensive crack or a local pipe or channel. If such an opening (crack or channel) extends into the water body in the case of earth dams, it could lead to free flow of water toward the downstream slope resulting in catastrophic failures. During shearing, soil aggregates on the contractive side of the critical state line would continue to yield in a ductile stable manner as a continuum. Such plastic yielding of soils represented by the curve BD allows stress to extend a certain distance beyond the critical state (Figure 3). The plastic deformation associated with such yielding is quantified well by the Cam-clay model (Schofield and Wroth, 1968) among many others. For sands and clayey silts of low plasticity, stable yield (rupture and ductile) behavior occur only within a narrow band on both the denser and looser side of the critical state line (Figure 2). The different classes of soil behavior described in Fig. 3 using normalized stress-based variables can also be depicted in a normalized 2-D diagram involving a combination of volume and stress ratio variables. The shear stress ratio η = q/p′ and vλ = v + λ ln p′ where λ is the slope of the isotropic compression line (assumed parallel to the critical state line) in the V − ln p′ diagram (Figure 2) are chosen for this purpose. Drawing a fan of lines radiating from the origin at chosen angles in Fig. 3 and choosing the values of the intersection of these lines on the sectors OA & OG, AB &GE and BC & ′ and p′/pcrit. ′ . These values in turn ED, will yield the values of q/pcrit ′ can be used to determine the stress ratio η = q/p′ values. The p′/pcrit values along with the slope λ on the V − ln p′ diagram can be used to calculate the value of vλ = v + λ ln p′. The resulting diagram is as shown in Fig. 4. The three classes of failure for soils shown by the lines OA, AB, BD in the triaxial compression region, and by DE, EG, and GO in the triaxial extension region in Fig. 3 are now shown in Fig. 4 by the corresponding lines. Stable soil behavior is limited to the interior region bounded by these lines.
4. Critical state, Casagrande's, and Seed's view of liquefaction
Fig. 3. Schematic diagram of limiting states of soil behavior in normalized q/p′crit and ′ stress space. p′/pcrit
There are currently two widely quoted views of liquefaction; the first one describes liquefaction of an aggregate in states looser than the critical void ratio as if it were a phase transformation process, such as the melting of a solid and the change to a fluid (Casagrande, 1975). This view is amplified further in the steady-state methodology where the critical void ratio concept is replaced with the steady state of sands (Poulos, 1981). On the other hand, based on undrained cyclic triaxial tests, Seed and Lee (1966) defined liquefaction as a phase transition but now under the condition where pore pressure approaches the confining stress and effective stress drops to zero. Thus in the Casagrande or the steady-state based methodology, liquefaction occurs on the
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Fig. 4. Different classes of soil behavior in (vλ, η) space (modified after Schofield, 2005).
contractive side of the critical state line, while for the Seed based methodology it occurs on the dilative side. As discussed before, during shearing soil aggregates on the contractive side of the critical state line continue to yield in a ductile stable manner as a continuum. Such plastic yielding of soils is not of the flow type as envisioned in the steady state or Casagrande view, but is quantified well by the Cam-clay model (Schofield and Wroth, 1968) among many others. Soils in states beyond point D on the q = 0 axis (Figure 3 or Figure 4) or outside the isotropic normal compression line (Figure 2) exist in a meta-stable state because of external mechanisms contributed by cementation, structure, or capillary forces. This is the case with cemented quick clays, wind deposited loose sands (structure), and moist-tamped sands (capillary). Quick clay and other such lightly cemented or bonded soil aggregates (Sangrey, 1972) can stand with vertical faces to small cliffs in an undisturbed state with high water contents. When a quick clay avalanche occurs, it disintegrates into fissured debris, and as lumps of quick clay are remolded, their high water content becomes evident. When the debris is fully remolded, it forms a body of soil with more water than that of a soil at the same effective pressure on the isotropic normal compression line. This is not a change of grain positions, but rather the result of a loss of bonds (Muhunthan and Schofield, 2000). The notion of liquefaction as an event propagating with retrogressive slips in a meta-stable body of lightly cemented or bonded collapsing silt, causing quick clay flowslides, is consistent with critical state theory. This phenomenon is similar in the case of ultra loose moist tamped laboratory sand specimens collapsing upon triaxial shear, resulting in what has been loosely termed “liquefaction.” However, it is important to note that these special classes of soil fall outside the stable band of contractive soils. When an unloading effective stress path reaches the fracture regions OA or OG (Figure 3 or Figure 4) the continuum begins to disintegrate into a clastic body and unstressed grains become free to slide apart. In that case the average specific volume of the clastic mass and its permeability can increase greatly in a very short time. Whenever a soil is excavated or broken into clumps, for ease of handling or for mixing with water, an unloading stress path reduces a principal effective stress component to zero in a controlled manner. If, however, there is a hydraulic gradient across the soil
body at the time it cracks or crumbles, the event is less controlled and has the characteristics of a sudden hydraulic fracture or fluidization. Accordingly, following Schofield (1981) and Muhunthan and Schofield (2000), liquefaction is defined as a class of instability (channeling, piping, boiling, or fluidizing) seen in soil far on the dry side of critical states, near zero effective stress and in the presence of a high hydraulic gradient. This applies to the case of cyclic pore pressure in earthquakes as well as to static hydraulic fracture. The opening within the soil body may be an extensive crack or a local pipe or channel. In the case of a local pipe, water slowly following tortuous paths may be able to dislodge grains in a direction perpendicular to the axis in which the pipe is developing. If debris forms soft mud that blocks the channel, the crack or pipe will heal itself. If hydraulic pressure is transmitted along a pipe or crack to regions where the pressure gradients cause cracking faster than cracks heal, there is a sudden transmission of pressures and a body of crumbling soil can disintegrate into a sort of soil avalanche. Or several pipes can break through a sand layer and vigorous sand boils can occur. This is the class of liquefaction with which the Seed methodology was concerned. It is important to note, however, that the increase of excess pore pressure against the effective confining pressure is necessary but not alone sufficient. The formation of openings and the presence of a large hydraulic gradient, which could lead to disintegration of the continuum into clastic blocks of soil, is another important requirement. Sand-like materials reach near-zero shear strength as pore water pressure rises and effective stress approaches zero; whereas clay-like materials retain strength albeit smaller as effective stress approaches zero. As a result, the term liquefaction has been chosen by some researchers to apply only to sand-like materials and another term, cyclic softening to clay-like materials (clays and plastic silts). The critical state definition of liquefaction highlighted above encompasses the sand-like and clay-like behavior within the same definition. Sands have higher permeability and are prone to transfer the hydraulic gradients leading to “liquefaction” whereas clay like materials have low permeability with minimal chance of transmitting high gradients resulting only in softening. Thus, all forms of large failures including the various types of liquefaction can be described from a fundamental mechanistic point of view by the critical state framework.
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The discussion presented above based on Fig. 4 using vλ values can be extended to fine grained soils using their plasticity characteristics as shown below. 4.1. Equivalent liquidity index and liquefaction of fine grained soils The effective critical pressure of most clays at their liquid limit is about 5 kPa and is about 500 kPa at the plastic limit (Schofield and Wroth, 1968, Fig. 6.14). Making use of this observation and using a judicious transformation Schofield (1980) converted the vλ values along the horizontal axis in Fig. 4 to equivalent liquidity index LI5 values. LI5 can be represented as LI5 = LI + 1/2 log (p′/5). Thus, LI5 equals the liquidity of soil as found in the ground plus a correction for mean effective stress p′. The resulting stability diagram showing the three (fracture, fault/slip, and fold/yield) regions is of the form as shown in Fig. 5. Using data from Weald clay (Parry, 1956; Schofield, 1980) these three failure regions on the compression side can be described as follows: Fracture η = 3 when
LI5 b 0:4
Fault=Slip η = 1:35 ðLI5 Þ−0:87 Fold=yield
ð3aÞ
when 0:4 b LI5 b 1:0
η = −2:53 ðLI5 Þ + 4:75 when 1:0 b LI5 b 1:9
ð3bÞ
effects including liquefaction. However, as detailed before, other requirements such as the formation of openings and the presence of high hydraulic gradients are needed to initiate disintegration of the continuum into clastic blocks of soil. Soils whether in their natural state or by choice of construction materials and methods would deform in a stable plastic manner when external work done by loads causes plastic flow to be dissipated in failure mechanisms (Schofield, 2005, 2006). The total work dissipated in the deforming soil when the critical state is reached will be approximately equal to the total displacement multiplied by the critical state ultimate load. In the case of Weald clay data, the critical state stress ratio mobilized during shear corresponds to η = M = 1 (Figure 5). Therefore, values of stress ratio η ≤ M = 1 would ensure stable plastic deformation regardless of the region on which the soil state lies. On the contrary, interlocking work is not automatically dissipated; it can be stored or transmitted from a region of interlocking to some other part of the soil. This would inevitably lead to disjointing of the continuum and would result in unstable behavior. Therefore, soils with η = M ≥ 1 and falling in regions given by Eqs. (3a) and (3b) are prone to unstable behavior but their nature is dependent on the characteristics of the region. Note that the value η = M = 1 was deduced from Weald clay data. It will be slightly different for other fine grained soils, but will not change the nature of the characteristic regions of soil behavior, or the methodology proposed here.
ð3cÞ 5. Analysis of field performance
Note that similar expressions for the extension side can also be derived but are not used here. The stability diagram in Fig. 5 is a means of characterizing fine grained soil behavior under various combinations of stress ratios, mean stress and plasticity characteristics. There is a general increase of limiting stress ratio η as equivalent liquidity LI5 falls, but this is not a continuous change because there is a change of limiting behavior from continuous yielding to discrete rupture and finally to fracture or stiff fissured soil at an equivalent density below 0.4. Soil states that fall into the fracture region are those that have the potential for catastrophic
We present an analysis of observed field data relating to studies of the liquefaction of fine-grained soils based on the proposed critical state framework. 5.1. Field data Data used in this study consists of those used in Wang's study (1979) and Standard Penetration Test (SPT) borings of post-earthquake liquefied sites in Turkey and Taiwan. Wang's data was for soils that
Fig. 5. Stability diagram: equivalent liquidity and different classes of soil behavior.
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liquefied in the Haicheng and Tangshan earthquakes, in 1975 and 1976, respectively. The SPT boring sites all included fine-grained soils. SPT data from six liquefied 1999 Kocaeli earthquake sites were downloaded from the website http://peer.berkeley.edu/publications/turkey/adapazari/ index.html. SPT boring logs from 35 sites that liquefied during the 1999 Chi-Chi earthquake were downloaded from the websites http://peer. berkeley.edu/lifelines/research_projects/3A02/ and http://www.ces. clemson.edu/chichi/TW-LIQ/In-situ-Test.htm (Papathanassiou, 2008). The moment magnitude Mw of the Kocaeli earthquake was 7.4 and of the Chi-Chi earthquake was 7.6. The SPT data for the Kocaeli earthquake sites were from the towns of Adapazari and Yalova, where the maximum ground accelerations (amax) were both recorded as 0.4 g (Papathanassiou, 2008). The borings for the Chi-Chi earthquake were performed in the towns of Yuanlin, Nantou, Wufeng, and Dachun, where the average PGA values were 0.18 g, 0.38 g, 0.67 g, and 0.19 g, respectively (Papathanassiou, 2008). 5.2. At-rest data on the stability diagram The liquidity index and the mean effective confining stress were determined from the field data. These values were in turn used to determine the equivalent LI5 values. The vertical effective stress was determined at the different depths and used to calculate the horizontal effective stress at rest. These values were in turn used to calculate the shear stress ratio η. Subsequently, the LI5 and η values were plotted on the stability diagram to determine the nature of soil stability. The field data used in this study for at rest conditions are shown in Fig. 6. The data in Fig. 6(a) are for states of soils from the database of Wang (1979). The solid diamond data points in this figure refer to states of soils that Wang (1979) identified as not liquefied under earthquake loading whereas the hollow diamond points indicate
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those that liquefied. It is evident that at-rest data identified as non liquefied falls within the stable region and below the η = M = 1 line. Most soils identified as liquefied fall outside the stable yield region atrest; thus they were in a meta-stable state to begin with and would have failed with catastrophic flow slides per the framework here. It is also of interest to note that some at-rest data points that lie inside the stability diagram were identified by Wang as liquefied after the earthquake. Their earthquake performance is discussed later in this study. Fig. 6(b) shows the field study data from the 1999 Kocaeli earthquake and the 1999 Chi-Chi earthquake on the (η, LI5) stability diagram. Most of these soils have LI5 N 0 and plot inside the stability diagram and below the η = M ( = 1) line. On the other hand, some of the data from these sites plot on the negative side of the LI5 axis. These soils are inherently prone to fracture and when disturbed by an earthquake or other causes would have resulted in unstable failures. There are also soil states at-rest that plot in the region outside the stable yielding region. As discussed before, these soils remain in a meta-stable state at-rest due to natural bonding and other causes but are prone to collapse and unstable failures under slight disturbance. 5.3. Earthquake effects on soil behavior The cyclic shear stress ratio CSR caused by an earthquake load can be determined by (Kramer, 1996): τ = σ ′v = 0:65ð amax = g Þðσv = σ ′v Þrd
ð4Þ
where τ = average shear stress, σ v′ = effective vertical stress, σv = total vertical stress, amax = maximum ground acceleration, g = acceleration due to gravity, and rd = depth reduction factor. For this study, the depth reduction factor recommended by Rauch (1997) is used. This factor can be written as: 4 3 2 rd = 1:0 + 1:6E 6 z −42z + 105z −4200z
ð5Þ
where z is the depth below the ground surface in meters. The above equation is valid to 30 m and all data used for this study is at a depth of 30 m or less. Since q = σ 1′ − σ 3′ and τ = (σ 1′ − σ 3′ )/2, the additional deviator stress on a sample caused by the applied shear stress of a cyclic load is twice the average shear stress; Δq = 2 τ. When Δq is added to the initial q, a state of stress will shift vertically upward in (η, p′) space. 5.4. Pore water pressure rise during earthquakes During an earthquake pore water pressure increases which in turn reduces the effective mean stress p′, and thus the state of stress will also shift to the left. For different values of the equivalent number of cycles of an earthquake N, Ansal and Erken (1989) developed an empirical model that relates cyclic stress ratio and pore water pressure ratio ru = Δu/σ ′, where Δu is the pore pressure rise, and σ ′ is effective normal stress. This relation is illustrated in Fig. 7. In order to use this figure, the equivalent number of cycles for a given earthquake needs to be determined. We chose to use the relationship shown in Fig. 8 as presented by Kramer (1996) for this purpose. Note that the pore pressure model used here is chosen for its simplicity over many others that are extant in the literature in order to illustrate the proposed methodology. 5.5. Changes to at-rest state under earthquakes on the stability diagram Fig. 6. Field study of in-situ soils at rest on the stability diagram. (a) Taiwan and Turkey data taken at varying depths. (b) Wang's data (♦ non liquefied, ◊ liquefied) at depth=1 meter.
The rise in pore water pressure causes a reduction in the mean normal effective stress p′, an increase in the stress ratio η, and a decrease in equivalent liquidity LI5. These changes would shift the at-
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Fig. 7. Cyclic stress ratio-pore pressure relationship for different numbers of cycles (Ansal and Erken, 1989).
rest in situ state of stress into different regions. Fig. 9 illustrates schematically the potential changes in η and LI5 values for different initial at rest states, I–VI. These states are chosen to be representative in each of the characteristic regions described before. Soil state I is that of LI5 b 0; soils in this state are in a fractured state and would undergo large deformations with unstable failures upon shaking. Similarly, soils in state VI are outside the stable yield region as discussed before; they remain in this meta-stable state only due to cementation and other factors. Slight disturbance would make these soils crumble and often manifest in the form of catastrophic flow slides. However, prediction of their behavior is complex and dependent on the nature of such extraneous factors as the strength of bonds and fabric arrangements. Therefore, if the initial state of a soil falls in the regions of either state I or VI, it is prudent to assume that it is inherently unstable to begin with and prone to catastrophic failures upon even slight disturbance. Typical undrained effective stress paths under earthquake loading for soils with states II, III and IV (in region 0.4 b LI5 b 1), and V (in region
1 b LI5 b 1.9) are illustrated schematically in Fig. 9. It is possible that earthquake loading would move a soil in state II (closer to the lower boundary of LI5 = 0.4) of the rupture region, to the fracture region as shown. Such soil will fracture into clastic debris as described before. Soils in the region of states III and IV subjected to earthquake loading would exhibit rising peaks until they reach the rupture curve and subsequently soften towards the critical state. Finally, soil state V will plastically yield in a stable manner until the critical state is reached. We now apply the changes in state that would occur to the at-rest data presented in Fig. 6 for earthquakes with different amax values. The resulting soil state changes are shown in Fig. 10. Note that at-rest data for soils which fell in regions I and VI of the stability diagram were deemed to have failed in an unstable manner due to slight disturbances and thus are not shown any further. It is seen that all initial states move into different regions of stability with increase in amax with some data beginning to reach the fault/rupture region even with amax = 0.1 g. These states would attain peak stress ratios along Coulomb type rupture planes and begin to soften towards the critical state with prolonged shearing. Such softening behavior is dependent on the presence of gouge material inside fault planes. Therefore, it is rather difficult to predict their postpeak behavior. For practical purposes, states that reach the rupture line should be deemed to have undergone large deformations and treated as such. As emphasized before, however, failure of this type may not be characterized as liquefaction. Note that to avoid clustering of data points along the rupture line, once a state reached the rupture line, it was removed with the understanding that it had reached its limiting resistance and failed. It is of interest to note that the solid diamond data points originally identified by Wang as non liquefied continue to remain stable at an amax = 0.2 g. However, at amax = 0.5 g all but two of the data points remain inside the stable diagram. The others had reached the rupture line after attaining a peak stress ratio. The fact that they did not undergo large deformation (or “non liquefied” per Wang) leads us to hypothesize the presence of gouge material that was able to sustain prolonged shear deformation. This may not have been the case for soil states with hollow data points identified by Wang as liquefied. As discussed before, such failures along the rupture line may not, however, be classified as liquefaction in the framework proposed here. 5.6. Initial LI5 values and shift in stress ratio
Fig. 8. Number of equivalent uniform cycles Neq for earthquakes of different magnitude (Kramer, 1996).
The stress ratio η, which is plotted on the vertical axis of the behavior map, is one of the key factors in determining the safety against liquefaction of a soil with a given LI5. The changes in deviator shear stress Δq, as well as the rise in pore water pressure, determine the shift in the stress ratio. Fig. 11 shows the shifts in η for a range of LI5 and amax values. As depth increases, so does LI5 and as seen in Fig. 11, when LI5 is larger, the shift in η is smaller. A smaller shift in η makes a soil less likely to move into an unsafe zone thereby making it more likely to remain safe. The relationship between depth and the shift in η is shown in Fig. 12. Between 0 and 15 m below the ground surface, the shift in η under an applied stress is significant, especially at the shallower depths. Below about 15 m, the change is much smaller; negligible, even, for an amax = 0.5 g. Additionally, plasticity of soils affects its resistance to larger shifts in stress ratio. For a soil with a PI of about 4, reducing its water content by 2% would reduce its equivalent liquidity index by 50% and the corresponding confining stress by 90%, leading to a significant increase in the shear stress ratio η (Muhunthan and Pillai, 2009). The same 2% water content reduction in clay with a PI of 20, however, would result in an LI5 reduction by only about 10% and a corresponding minor decrease in p′ and increase in shear stress ratio η. This means that both the depth of the soil and its plasticity characteristics would affect how
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Fig. 9. Schematic illustration of effective stress paths under earthquakes for various initial states in the stability diagram.
easily η and LI5 could shift into the cracking zone, making the soil susceptible to catastrophic failures including “liquefaction.” 5.7. Conclusions This study has put forward a consistent methodology based on critical state framework to characterize the different regimes of
fine-grained soil behavior under earthquake loads. In the critical state view, liquefaction occurs when soil is on the dry side of critical states, near zero effective stress, and in the presence of high hydraulic gradients. In this view, liquefaction is one of a group of phenomena; including piping, boiling, and fluidization; with pipes and channels and hydraulic fractures, internal erosion and void migration.
Fig. 10. Field study data showing the effect of amax on stress ratio shift. **Depths vary for Taiwan and Turkey data (top) while depth is taken at 1 m for Wang's data (bottom).
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Fig. 11. Effect of LI5 on shift in stress ratio η.
The central piece of the proposed characterization of soil behavior is the stability diagram shown in Fig. 5. This diagram captures the effects of soil plasticity through the liquidity index LI, confinement through mean normal effective stress p′, and shear stress q through the stress ratio η. The equivalent liquidity LI5 = LI + 0.5 log (p′/5) is a useful parameter to capture the combined effects of LI and p′ on soil behavior. The three regions of limiting soil behavior; fracture, fault, and fold/yield, as a function of LI5 and η were identified using the data from Weald clay (Eqs. 3a–c). The proposed characterization is used to examine the liquefaction susceptibility of fine grained soils from China, Taiwan, and Turkey. Soils from these regions at-rest are shown to fall within the three
different regions. Therefore, their limiting behavior would be different. Some of the data from Taiwan and Turkey plotted on the negative side of the LI5 axis (Figure 6). These soils are prone to fracture and when disturbed by earthquake or other causes would have resulted in unstable failures. Natural cementation and interparticle bonding enable some soils to exist in conditions outside the LI5 boundary limit for stable yielding at-rest (Figure 6(a)). However, such soils are sensitive to disturbance and in danger of instability and liquefaction when external loads such as from an earthquake break the bonds and the structure collapses. Under earthquake loading, the initial states of soils would migrate into different regions in the stability diagram depending on their
Fig. 12. Effect of depth z on shift in stress ratio η.
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initial location, shear stress increment, and, pore pressure response. The migration paths under earthquake loads are illustrated in Fig. 9 for the various initial states. The final position of the state of soil would dictate the type of limiting behavior expected in the field; fracture, rupture or yield. The final states which fall into the fracture region have the potential for catastrophic failures including “liquefaction”; the ones which fall onto the rupture region would experience the attainment of the peak stress ratio followed by softening along failure planes; the ones in the yield region would continue to yield in a stable manner. The latter two types of deformations while resulting in large deformation may not be of a catastrophic nature. Acknowledgement The authors wish to thank Brenton Cook of the Department of Civil Engineering at Washington State University for his assistance with the preparation of the paper. References Andrews, D.C.A., Martin, G.R., 2000. Criteria for liquefaction of silty soils. Proc., 12th World Conference on Earthquake Engineering. NZ Society for Earthquake Engineering, Upper Hutt, New Zealand. Paper No. 0312. Ansal, A.M., Erken, A., 1989. Undrained behavior of clay under cyclic shear stresses. Journal of Geotechnical Engineering 115 (7), 968–983. Atkinson, J., 1993. An introduction to The Mechanics of Soils and Foundations through Critical State Soil Mechanics. McGraw-Hill. Boulanger, R.W., Idriss, I.M., 2006. Liquefaction susceptibility criteria for silts and clays. Journal of Geotechnical and Geoenvironmental Engineering 132 (11), 1413–1426. Bray, J.D., Sancio, R.B., 2006. Assessment of the liquefaction susceptibility of finegrained soils. Journal of Geotechnical and Geoenvironmental Engineering 132 (9), 1165–1177. Casagrande, A., 1975. Liquefaction and cyclic deformation of sands—a critical review. Proc., 5th Pan-American Conference. on Soil Mechanics and Foundation Engineering, Buenos Aires, Argentina. Erken, A., 2001. The role of geotechnical factors on observed damage in Adapazari during 1999 earthquake. 15th ICSMGE, TC4 Satellite conference on Lessons learned from Recent Strong Earthquakes, Istanbul, Turkey, pp. 29–32.
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Holzer, T.L., Bennett, M.J., Ponti, D.J., Tinsley III, J.C., 1999. Liquefaction and soil failure during 1994 Northridge earthquake. Journal of Geotechnical and Geoenvironmental Engineering 125 (6), 438–452. Kramer, Steven L., 1996. Geotechnical Earthquake Engineering. Prentice Hall, New Jersey. Muhunthan, B., Pillai, V.S., 2009. Response of the authors to the discussion by Dr. J. S. Younger regarding ‘Teton Dam, USA: uncovering the crucial aspect of its failure. Civil Engineering 162 (CE2), 57-57 Proc. Institution of Civil Engineers. Muhunthan, B., Schofield, A.N., 2000. Liquefaction and dam failures. Proc. ASCE GeoDenver 2000, Geotechnical Special Publication #101, pp. 266–280. Papathanassiou, G., 2008. LPI-based approach for calibrating the severity of liquefaction-induced failures and for assessing the probability of liquefaction surface evidence. Engineering Geology 96, 94–104. Parry, R.H.G., 1956. Strength and Deformation of Clay. PhD Thesis, University of. London. Perlea, V.G., 2000. Liquefaction of cohesive soils. In: Pak, R.Y.S., Yamamuro, J. (Eds.), Proc., Soil Dynamics and Liquefaction 2000. Geotechnical Special Publication, 107. ASCE, Reston, Va., pp. 58–76. Polito, C., 2001. Plasticity based liquefaction criteria. Proc., 4th International Conference On Recent Advances In Geotechnical Earthquake Engineering and Soil Dynamics. University of Missouri-Rolla, Rolla, Mo. Paper No. 1.33. Poulos, S.J., 1981. The steady state of deformation. Journal of Geotechnical Engineering 107 (GT5), 553–562. Rauch, A.F., 1997. “EPOLLS: An Empirical Method for Predicting Surface Displacements Due to Liquefaction-Induced Lateral Spreading in Earthquakes,” PhD, thesis, Virginia Polytechnic Institute and State University at Blacksburg, Va. Roscoe, K.H., Schofield, A.N., Wroth, C.P., 1958. On the yielding of soils. Geotechnique 8 (1), 22–53. Sangrey, D.A., 1972. Naturally cemented sensitive soils. Geotechnique 22 (1), 139–152. Schofield, A.N., 1980. “Cambridge geotechnical centrifuge operations.” 20th Rankine Lecture. Geotechnique 30 (3), 227–268. Schofield, A.N., 1981. Dynamic and earthquake geotechnical centrifuge modeling. Proceedings Intl. Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, MO, pp. 1081–1100. Schofield, A.N., 1998. Mohr Coulomb error correction. Ground Engineering 31 (8), 30–32. Schofield, A.N., 2005. Disturbed soil properties and geotechnical design. Thomas Telford, London. Schofield, A.N., 2006. Interlocking, and peak and design strengths. Geotechnique 56 (5), 357–358. Schofield, A.N., Wroth, P., 1968. Critical State Soil Mechanics. McGraw-Hill, New York. Seed, H.B., Idriss, I.M., 1982. Ground motions and soil liquefaction during earthquakes. Earthquake Engineering Research Institute, Berkeley, Calif. Seed, H.B., Lee, K.L., 1966. Liquefaction of saturated sands during cyclic loading. ASCE Journal of Soil Mechanics and Foundations 92 (6), 105–134. Wang, W., 1979. Some findings in soil liquefaction. Water Conservancy and Hydroelectric Power Scientific Research Institute, Beijing.
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Engineering Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n g g e o
Critical state framework for liquefaction of fine grained soils Balasingam Muhunthan a,⁎, Diana L. Worthen b a b
Department of Civil and Environmental Engineering, Washington State University, Pullman, WA 99164, USA PBS Engineering and Environmental, Vancouver, WA, USA
a r t i c l e
i n f o
Article history: Received 25 March 2010 Received in revised form 4 August 2010 Accepted 17 September 2010 Available online 30 October 2010 Keywords: Fine grained soils Liquefaction Critical state Liquidity index Stress ratio Earthquake
a b s t r a c t A consistent methodology based on the critical state framework to characterize the different regimes of finegrained soil behavior under earthquake loads is put forward. Shear strength and deformation behavior of soils depend in a major way on the combination of volume and confining stress. Depending on their combination, a soil aggregate may fracture into clastic debris, fail with fault planes, or yield plastically. This characterization of the class of limiting soil behavior is used to analyze the potential for large deformation and liquefaction in fine grained soils. The central piece of the proposed characterization is the (η, LI5) stability diagram where η = q/p′ and LI5 = LI + 0.5 log (p′/5). This diagram captures the effects of soil plasticity through liquidity index LI, confinement through mean normal effective stress p′, and shear stress q through the stress ratio η. The three regions of behavior; fracture, fault, and fold/yield are identified. Soils become susceptible to liquefaction when they shift into the fracture zone (LI5 ≤ 0.4), or if they plot outside of the stable yielding region. Under earthquake loading, the initial soil states will migrate into different regions in the stability diagram depending on their initial location, shear stress increment, and, pore pressure response. The final position of the soil state would dictate the type of limiting behavior expected in the field; fracture, rupture or yield. The final states which fall into the fracture region have the potential for catastrophic failures including “liquefaction”; the ones which fall onto the rupture region would experience the attainment of a peak stress ratio followed by softening along failure planes; the ones in the yield region would continue to yield in a stable manner. The latter two types of deformations while resulting in large deformation may not be of a catastrophic nature. The proposed characterization is used to examine the liquefaction susceptibility of fine grained soils from China, Taiwan, and Turkey. Use of simplified empirical criteria based on parameters such as plasticity index and fines contents may not capture the true nature of the type of undrained limiting behavior of fine grains soils in the field including liquefaction. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Recent field evidence of ground failure in low-plasticity silts and clays during strong earthquakes have highlighted the need to better characterize the strength loss and development of large deformations in a broad range of saturated soils, from sands to fine grained soils. In the absence of a fundamental understanding of the seismic behavior of fine-grained soils, their liquefaction potential is currently evaluated by the ‘Chinese criteria,’ which is based on their plasticity characteristics. First introduced by Wang (1979) based on the study of major earthquakes in China, these criteria were expanded by Seed and Idriss (1982) and have become the commonly accepted method for predicting fine-grained liquefaction susceptibility. More recent research has shown that the Chinese criteria are ineffective and should not be relied upon to safely predict liquefaction potential (Boulanger and Idriss, 2006; Bray and Sancio, 2006). Modifications to
⁎ Corresponding author. Tel.: + 1 509 335 3921; fax: + 1 509 335 7632. E-mail address: [email protected] (B. Muhunthan). 0013-7952/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2010.09.013
the Chinese criteria have been proposed for predicting the liquefaction potential of fine grained soils. The shear strength and deformation behavior of soil is quite sensitive to the combination of changes in volume and confining stress. Depending on their combination, a soil aggregate may fracture and crack into clastic debris, or fail with fault planes on which gouge material dilates and softens, or it can continue to yield and deform plastically. The current prediction models do not distinguish these distinct classes of soil behavior in the description of large deformation failures. This has led to many of the difficulties faced in the interpretation of liquefaction failures (Muhunthan and Schofield, 2000). Most often, evidence of large deformation observed from post reconnaissance efforts following an earthquake are lumped into the “liquefied” phenomenon which may not capture their true characteristics. This study makes use of critical state soil mechanics and puts forward a framework for fundamental understanding of the liquefaction behavior of sands, and fine-grained soils over a broad range of plasticity characteristics. It explicitly recognizes that soil is an aggregate of interlocking frictional particles and that the regimes of soil behavior depend in a major way on its density and effective
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pressure. Liquefaction and large deformation in fine grained soils are shown to belong to a class of failures that occur in the fracture region brought about by a combination of plasticity, confining stress and imposed shear. The observed field data on fine grained soil performance under past earthquakes are examined using this new framework. 2. Current fine grained soil liquefaction criteria Most current criteria to assess liquefaction susceptibility of fine grained soils stem from the Chinese criteria published by Wang (1979). Wang found that the common characteristics of the liquefied fine-grained soils were having less than 20% clay content (5 μm or less), liquid limit between 21 and 35, plasticity index between 4 and 14, and natural water content of 90% or above the liquid limit. The plasticity characteristics of the original Wang (1979) data are shown in Fig. 1. Seed and Idriss (1982) modified Wang's conclusions, and defined the liquefaction criteria of fine-grained soil to consist of clay fraction (5 μm or less) less than 15% by weight, liquid limit less than 35, and water content 90% or greater of the liquid limit. Liquefaction will occur only if all three criteria are met. Recent field observations have necessitated the need for reevaluation of the Chinese criteria. Three recent earthquakes, the 1994 Northridge, 1999 Kocaeli, and 1999 Chi-Chi earthquakes, all showed evidence of liquefaction in soils with greater than 15% clay content. This means the soils did not meet all three of the Chinese criteria, and yet still liquefied. Liquefaction of soils in Adapazari, Turkey during the 1999 Kocaeli earthquake resulted in much of the damage throughout the city (Erken, 2001), even though they typically had clay contents greater than 15% (Bray and Sancio, 2006). After the 1994 Northridge earthquake, Holzer et al. (1999) noted that soils which would have been determined safe from liquefaction by the Chinese criteria did in fact liquefy and lead to permanent ground deformations. Many recent researchers, such as Bray and Sancio (2006) and Boulanger and Idriss (2006), have recommended not relying on the Chinese criteria. The Chinese criteria are based solely on plasticity characteristics, and do not account for the confining stress of in-situ soil. One plasticity criterion is chosen to be applicable. Wang's research shows that soils with water content-liquid limit ratios less than 0.9 did not liquefy, except in one case with a ratio only slightly less than 0.9. Soils with water content-liquid limit ratios greater than 0.9 all liquefied, again except for one case, in which the ratio was only slightly greater. This criterion appears to be important in evaluating liquefaction susceptibility; however, the other two Chinese criteria may not be valid. Wang's data from 1979 does not show the behavior of any soil with a liquid limit greater than 35, and recent earthquakes have revealed liquefied soils with greater than 15% clay fractions. Andrews and Martin (2000) concluded that the criteria for liquefaction susceptibility should not include a water content-liquid limit ratio as a condition for silty soils. Additionally, they reduced the liquid limit criterion to LL b 32, and the clay fraction condition to b10%. They also used 2 μm as the definition of clay-sized particles, which is smaller than the previously used 5 μm. If a soil meets only one of the two criteria, further study is required to confidently evaluate
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liquefaction susceptibility. Polito (2001) determined three categories of liquefaction susceptibility, based on plasticity characteristics. A soil is liquefiable if LL b 25 and PI b 7; soils with 25 b LL b 35 and 7 b PI b 10 are potentially liquefiable; and 35 b LL b 50 and 10 b PI b 15 indicates a soil susceptible to cyclic mobility. Perlea (2000) recommended lab testing as the best evaluation method for liquefaction potential. Boulanger and Idriss (2006) have suggested that ‘sand-like’ finegrained soils and ‘clay-like’ fine-grained soils respond differently to cyclic loads. A sand-like soil is susceptible to liquefaction, whereas a clay-like soil will respond with cyclic failure. The plasticity index PI is used to determine the difference between the two categories. A PI b 7 indicates sand-like liquefaction susceptibility, while soils with a PI ≥ 7 are considered clay-like, prone to cyclic failure. 3. Critical state framework for soil behavior: fracture, rupture, and ductile regimes Critical state soil mechanics (Schofield and Wroth, 1968) is a framework that explicitly recognizes that soil is an aggregate of interlocking frictional particles and that the different regimes of soil behavior under shear depends in a major way on its density and effective pressure (Schofield, 1998, 2005). Detailed accounts of the basic principles, the features, and finite element applications of the CSSM framework have been presented in a number of publications. We present here only the salient features of the framework relevant to our discussion. The two invariant stress parameters used in CSSM are mean normal stress: ′
p =
1 ′ ′ ′ σ + σ2 + σ3 3 1
ð1Þ
and the deviator stress: 1 ′ ′ 2 ′ ′ 2 ′ ′ 2 1 q = pffiffiffi ðσ2 −σ3 Þ + ðσ3 −σ1 Þ + ðσ1 −σ2 Þ 2 2
ð2Þ
where σ1′, σ2′, σ3′ are the principal effective compressive stresses. For triaxial test conditions where, σ2′ = σ3′ Eqs. (1) and (2) reduce to p′ = 1/3(σ1′ + 2σ3′), and q = (σ1′ - σ3′), respectively. The two parameters p′ and q, and a third variable, the specific volume V = (1 + e), where e is the void ratio, define the state of a soil specimen. Roscoe et al. (1958) quote experimental evidence that the ultimate state of any soil specimen during a continuous remolding and shear flow will lie on a critical state line (CSL) as shown in Fig. 2. The figure also shows the isotropic consolidated line (ICL or the normal compression line NCL), and a fracture line whose characteristics are discussed later. The critical state line distinguishes the two distinct types of soil behavior. There are states for which the combinations of specific volume V and mean normal effective stress p′ lie further away from the origin than the line of critical states. These states are called contractive states; shearing there causes aggregates to compress to denser packing and emit water with ductile stable yielding of a test specimen. There are also states for which the combinations of specific volume V and mean normal effective stress p′ lie closer to the origin
Fig. 1. Chinese criteria of Wang (1979) (After Bray and Sancio, 2006).
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Fig. 2. State of soils in V − ln p′ space.
than the line of critical states. These states are termed dilative states; shearing there causes aggregates to increase volume and suck in water, and ground slips at peak strength with unstable failures. In the critical state framework, the limits to stable states of yielding are defined by the state boundary surface in the 3-D, p′–q–V space. The 3-D state boundary surface can be represented in a 2-D space using appropriate normalization of the variables. Various normalizing parameters are used for this purpose (Atkinson, 1993). One such normalization suitable for the present work is that using pressure at ′ . The 2-D representation of the normalized state the critical state pcrit ′ − p′/pcrit ′ space is shown in Fig. 3. The boundary surface in the q/pcrit values shown on the horizontal axis correspond to data obtained from Weald clay (Schofield and Wroth, 1968; Schofield, 1980) but the discussion below applies to all soils albeit with slightly different numbers as appropriate. It is observed from Fig. 3 that the limiting soil behavior is not a straight line fit using the Mohr–Coulomb failure envelope as is often done in geotechnical practice. Soil behavior at the limiting state consists of three distinct classes of failure; the limiting lines OA and OG (Figure 3) indicate states of soils exhibiting fractures or cracks; AB and GE indicate soil states with fault ruptures on Hvorslev or Mohr–Coulomb planes; BD and ED indicate ductile yielding and folding of soils. Soil states falling on the fracture surface would result in the development of unstable fissures and crack openings. Heavily overconsolidated clays and overcompacted sands at low confining stresses
may reach this limiting state. The “no-tension” or “limiting tensile strain” criteria are the most widely used among the alternative theories to quantify tensile fracture (Schofield, 1980). For triaxial specimens, the no tension criterion with σ3′ = 0 results in p′ = σ1′ /3 or q/p′ = 3 and leads to vertical splitting cracks, which is the case of line OA. For horizontally spalling cracks, setting σ 1′ = 0 results in p′ = 2/3 σ3′ , q = −σ3′ , or q/p′ =1.5 which is the case of line OG. For clays or silty clays, Schofield (1980) suggested that the change from rupturing to tensile ′ , where pcrit ′ is the effective cracking occurs at a pressure p′ = 0.1 pcrit confining stress at the critical state (Figure 3). The corresponding line of the fracture states in V − ln p′ space is shown in Fig. 2. When the effective stress path crosses the fracture surface OA, the soil element begins to disintegrate into a clastic body and unstressed grains become free to slide apart. In that case, the average specific volume of the clastic mass could increase (due to large voids/cracks) and consequently its permeability could increase significantly and instantly. A significant shear stress at low confining stresses can cause the crossover of the crack-surface OA and a large increase in specific volume. When such a condition occurs, the opening within the soil body may consist of an extensive crack or a local pipe or channel. If such an opening (crack or channel) extends into the water body in the case of earth dams, it could lead to free flow of water toward the downstream slope resulting in catastrophic failures. During shearing, soil aggregates on the contractive side of the critical state line would continue to yield in a ductile stable manner as a continuum. Such plastic yielding of soils represented by the curve BD allows stress to extend a certain distance beyond the critical state (Figure 3). The plastic deformation associated with such yielding is quantified well by the Cam-clay model (Schofield and Wroth, 1968) among many others. For sands and clayey silts of low plasticity, stable yield (rupture and ductile) behavior occur only within a narrow band on both the denser and looser side of the critical state line (Figure 2). The different classes of soil behavior described in Fig. 3 using normalized stress-based variables can also be depicted in a normalized 2-D diagram involving a combination of volume and stress ratio variables. The shear stress ratio η = q/p′ and vλ = v + λ ln p′ where λ is the slope of the isotropic compression line (assumed parallel to the critical state line) in the V − ln p′ diagram (Figure 2) are chosen for this purpose. Drawing a fan of lines radiating from the origin at chosen angles in Fig. 3 and choosing the values of the intersection of these lines on the sectors OA & OG, AB &GE and BC & ′ and p′/pcrit. ′ . These values in turn ED, will yield the values of q/pcrit ′ can be used to determine the stress ratio η = q/p′ values. The p′/pcrit values along with the slope λ on the V − ln p′ diagram can be used to calculate the value of vλ = v + λ ln p′. The resulting diagram is as shown in Fig. 4. The three classes of failure for soils shown by the lines OA, AB, BD in the triaxial compression region, and by DE, EG, and GO in the triaxial extension region in Fig. 3 are now shown in Fig. 4 by the corresponding lines. Stable soil behavior is limited to the interior region bounded by these lines.
4. Critical state, Casagrande's, and Seed's view of liquefaction
Fig. 3. Schematic diagram of limiting states of soil behavior in normalized q/p′crit and ′ stress space. p′/pcrit
There are currently two widely quoted views of liquefaction; the first one describes liquefaction of an aggregate in states looser than the critical void ratio as if it were a phase transformation process, such as the melting of a solid and the change to a fluid (Casagrande, 1975). This view is amplified further in the steady-state methodology where the critical void ratio concept is replaced with the steady state of sands (Poulos, 1981). On the other hand, based on undrained cyclic triaxial tests, Seed and Lee (1966) defined liquefaction as a phase transition but now under the condition where pore pressure approaches the confining stress and effective stress drops to zero. Thus in the Casagrande or the steady-state based methodology, liquefaction occurs on the
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Fig. 4. Different classes of soil behavior in (vλ, η) space (modified after Schofield, 2005).
contractive side of the critical state line, while for the Seed based methodology it occurs on the dilative side. As discussed before, during shearing soil aggregates on the contractive side of the critical state line continue to yield in a ductile stable manner as a continuum. Such plastic yielding of soils is not of the flow type as envisioned in the steady state or Casagrande view, but is quantified well by the Cam-clay model (Schofield and Wroth, 1968) among many others. Soils in states beyond point D on the q = 0 axis (Figure 3 or Figure 4) or outside the isotropic normal compression line (Figure 2) exist in a meta-stable state because of external mechanisms contributed by cementation, structure, or capillary forces. This is the case with cemented quick clays, wind deposited loose sands (structure), and moist-tamped sands (capillary). Quick clay and other such lightly cemented or bonded soil aggregates (Sangrey, 1972) can stand with vertical faces to small cliffs in an undisturbed state with high water contents. When a quick clay avalanche occurs, it disintegrates into fissured debris, and as lumps of quick clay are remolded, their high water content becomes evident. When the debris is fully remolded, it forms a body of soil with more water than that of a soil at the same effective pressure on the isotropic normal compression line. This is not a change of grain positions, but rather the result of a loss of bonds (Muhunthan and Schofield, 2000). The notion of liquefaction as an event propagating with retrogressive slips in a meta-stable body of lightly cemented or bonded collapsing silt, causing quick clay flowslides, is consistent with critical state theory. This phenomenon is similar in the case of ultra loose moist tamped laboratory sand specimens collapsing upon triaxial shear, resulting in what has been loosely termed “liquefaction.” However, it is important to note that these special classes of soil fall outside the stable band of contractive soils. When an unloading effective stress path reaches the fracture regions OA or OG (Figure 3 or Figure 4) the continuum begins to disintegrate into a clastic body and unstressed grains become free to slide apart. In that case the average specific volume of the clastic mass and its permeability can increase greatly in a very short time. Whenever a soil is excavated or broken into clumps, for ease of handling or for mixing with water, an unloading stress path reduces a principal effective stress component to zero in a controlled manner. If, however, there is a hydraulic gradient across the soil
body at the time it cracks or crumbles, the event is less controlled and has the characteristics of a sudden hydraulic fracture or fluidization. Accordingly, following Schofield (1981) and Muhunthan and Schofield (2000), liquefaction is defined as a class of instability (channeling, piping, boiling, or fluidizing) seen in soil far on the dry side of critical states, near zero effective stress and in the presence of a high hydraulic gradient. This applies to the case of cyclic pore pressure in earthquakes as well as to static hydraulic fracture. The opening within the soil body may be an extensive crack or a local pipe or channel. In the case of a local pipe, water slowly following tortuous paths may be able to dislodge grains in a direction perpendicular to the axis in which the pipe is developing. If debris forms soft mud that blocks the channel, the crack or pipe will heal itself. If hydraulic pressure is transmitted along a pipe or crack to regions where the pressure gradients cause cracking faster than cracks heal, there is a sudden transmission of pressures and a body of crumbling soil can disintegrate into a sort of soil avalanche. Or several pipes can break through a sand layer and vigorous sand boils can occur. This is the class of liquefaction with which the Seed methodology was concerned. It is important to note, however, that the increase of excess pore pressure against the effective confining pressure is necessary but not alone sufficient. The formation of openings and the presence of a large hydraulic gradient, which could lead to disintegration of the continuum into clastic blocks of soil, is another important requirement. Sand-like materials reach near-zero shear strength as pore water pressure rises and effective stress approaches zero; whereas clay-like materials retain strength albeit smaller as effective stress approaches zero. As a result, the term liquefaction has been chosen by some researchers to apply only to sand-like materials and another term, cyclic softening to clay-like materials (clays and plastic silts). The critical state definition of liquefaction highlighted above encompasses the sand-like and clay-like behavior within the same definition. Sands have higher permeability and are prone to transfer the hydraulic gradients leading to “liquefaction” whereas clay like materials have low permeability with minimal chance of transmitting high gradients resulting only in softening. Thus, all forms of large failures including the various types of liquefaction can be described from a fundamental mechanistic point of view by the critical state framework.
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The discussion presented above based on Fig. 4 using vλ values can be extended to fine grained soils using their plasticity characteristics as shown below. 4.1. Equivalent liquidity index and liquefaction of fine grained soils The effective critical pressure of most clays at their liquid limit is about 5 kPa and is about 500 kPa at the plastic limit (Schofield and Wroth, 1968, Fig. 6.14). Making use of this observation and using a judicious transformation Schofield (1980) converted the vλ values along the horizontal axis in Fig. 4 to equivalent liquidity index LI5 values. LI5 can be represented as LI5 = LI + 1/2 log (p′/5). Thus, LI5 equals the liquidity of soil as found in the ground plus a correction for mean effective stress p′. The resulting stability diagram showing the three (fracture, fault/slip, and fold/yield) regions is of the form as shown in Fig. 5. Using data from Weald clay (Parry, 1956; Schofield, 1980) these three failure regions on the compression side can be described as follows: Fracture η = 3 when
LI5 b 0:4
Fault=Slip η = 1:35 ðLI5 Þ−0:87 Fold=yield
ð3aÞ
when 0:4 b LI5 b 1:0
η = −2:53 ðLI5 Þ + 4:75 when 1:0 b LI5 b 1:9
ð3bÞ
effects including liquefaction. However, as detailed before, other requirements such as the formation of openings and the presence of high hydraulic gradients are needed to initiate disintegration of the continuum into clastic blocks of soil. Soils whether in their natural state or by choice of construction materials and methods would deform in a stable plastic manner when external work done by loads causes plastic flow to be dissipated in failure mechanisms (Schofield, 2005, 2006). The total work dissipated in the deforming soil when the critical state is reached will be approximately equal to the total displacement multiplied by the critical state ultimate load. In the case of Weald clay data, the critical state stress ratio mobilized during shear corresponds to η = M = 1 (Figure 5). Therefore, values of stress ratio η ≤ M = 1 would ensure stable plastic deformation regardless of the region on which the soil state lies. On the contrary, interlocking work is not automatically dissipated; it can be stored or transmitted from a region of interlocking to some other part of the soil. This would inevitably lead to disjointing of the continuum and would result in unstable behavior. Therefore, soils with η = M ≥ 1 and falling in regions given by Eqs. (3a) and (3b) are prone to unstable behavior but their nature is dependent on the characteristics of the region. Note that the value η = M = 1 was deduced from Weald clay data. It will be slightly different for other fine grained soils, but will not change the nature of the characteristic regions of soil behavior, or the methodology proposed here.
ð3cÞ 5. Analysis of field performance
Note that similar expressions for the extension side can also be derived but are not used here. The stability diagram in Fig. 5 is a means of characterizing fine grained soil behavior under various combinations of stress ratios, mean stress and plasticity characteristics. There is a general increase of limiting stress ratio η as equivalent liquidity LI5 falls, but this is not a continuous change because there is a change of limiting behavior from continuous yielding to discrete rupture and finally to fracture or stiff fissured soil at an equivalent density below 0.4. Soil states that fall into the fracture region are those that have the potential for catastrophic
We present an analysis of observed field data relating to studies of the liquefaction of fine-grained soils based on the proposed critical state framework. 5.1. Field data Data used in this study consists of those used in Wang's study (1979) and Standard Penetration Test (SPT) borings of post-earthquake liquefied sites in Turkey and Taiwan. Wang's data was for soils that
Fig. 5. Stability diagram: equivalent liquidity and different classes of soil behavior.
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liquefied in the Haicheng and Tangshan earthquakes, in 1975 and 1976, respectively. The SPT boring sites all included fine-grained soils. SPT data from six liquefied 1999 Kocaeli earthquake sites were downloaded from the website http://peer.berkeley.edu/publications/turkey/adapazari/ index.html. SPT boring logs from 35 sites that liquefied during the 1999 Chi-Chi earthquake were downloaded from the websites http://peer. berkeley.edu/lifelines/research_projects/3A02/ and http://www.ces. clemson.edu/chichi/TW-LIQ/In-situ-Test.htm (Papathanassiou, 2008). The moment magnitude Mw of the Kocaeli earthquake was 7.4 and of the Chi-Chi earthquake was 7.6. The SPT data for the Kocaeli earthquake sites were from the towns of Adapazari and Yalova, where the maximum ground accelerations (amax) were both recorded as 0.4 g (Papathanassiou, 2008). The borings for the Chi-Chi earthquake were performed in the towns of Yuanlin, Nantou, Wufeng, and Dachun, where the average PGA values were 0.18 g, 0.38 g, 0.67 g, and 0.19 g, respectively (Papathanassiou, 2008). 5.2. At-rest data on the stability diagram The liquidity index and the mean effective confining stress were determined from the field data. These values were in turn used to determine the equivalent LI5 values. The vertical effective stress was determined at the different depths and used to calculate the horizontal effective stress at rest. These values were in turn used to calculate the shear stress ratio η. Subsequently, the LI5 and η values were plotted on the stability diagram to determine the nature of soil stability. The field data used in this study for at rest conditions are shown in Fig. 6. The data in Fig. 6(a) are for states of soils from the database of Wang (1979). The solid diamond data points in this figure refer to states of soils that Wang (1979) identified as not liquefied under earthquake loading whereas the hollow diamond points indicate
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those that liquefied. It is evident that at-rest data identified as non liquefied falls within the stable region and below the η = M = 1 line. Most soils identified as liquefied fall outside the stable yield region atrest; thus they were in a meta-stable state to begin with and would have failed with catastrophic flow slides per the framework here. It is also of interest to note that some at-rest data points that lie inside the stability diagram were identified by Wang as liquefied after the earthquake. Their earthquake performance is discussed later in this study. Fig. 6(b) shows the field study data from the 1999 Kocaeli earthquake and the 1999 Chi-Chi earthquake on the (η, LI5) stability diagram. Most of these soils have LI5 N 0 and plot inside the stability diagram and below the η = M ( = 1) line. On the other hand, some of the data from these sites plot on the negative side of the LI5 axis. These soils are inherently prone to fracture and when disturbed by an earthquake or other causes would have resulted in unstable failures. There are also soil states at-rest that plot in the region outside the stable yielding region. As discussed before, these soils remain in a meta-stable state at-rest due to natural bonding and other causes but are prone to collapse and unstable failures under slight disturbance. 5.3. Earthquake effects on soil behavior The cyclic shear stress ratio CSR caused by an earthquake load can be determined by (Kramer, 1996): τ = σ ′v = 0:65ð amax = g Þðσv = σ ′v Þrd
ð4Þ
where τ = average shear stress, σ v′ = effective vertical stress, σv = total vertical stress, amax = maximum ground acceleration, g = acceleration due to gravity, and rd = depth reduction factor. For this study, the depth reduction factor recommended by Rauch (1997) is used. This factor can be written as: 4 3 2 rd = 1:0 + 1:6E 6 z −42z + 105z −4200z
ð5Þ
where z is the depth below the ground surface in meters. The above equation is valid to 30 m and all data used for this study is at a depth of 30 m or less. Since q = σ 1′ − σ 3′ and τ = (σ 1′ − σ 3′ )/2, the additional deviator stress on a sample caused by the applied shear stress of a cyclic load is twice the average shear stress; Δq = 2 τ. When Δq is added to the initial q, a state of stress will shift vertically upward in (η, p′) space. 5.4. Pore water pressure rise during earthquakes During an earthquake pore water pressure increases which in turn reduces the effective mean stress p′, and thus the state of stress will also shift to the left. For different values of the equivalent number of cycles of an earthquake N, Ansal and Erken (1989) developed an empirical model that relates cyclic stress ratio and pore water pressure ratio ru = Δu/σ ′, where Δu is the pore pressure rise, and σ ′ is effective normal stress. This relation is illustrated in Fig. 7. In order to use this figure, the equivalent number of cycles for a given earthquake needs to be determined. We chose to use the relationship shown in Fig. 8 as presented by Kramer (1996) for this purpose. Note that the pore pressure model used here is chosen for its simplicity over many others that are extant in the literature in order to illustrate the proposed methodology. 5.5. Changes to at-rest state under earthquakes on the stability diagram Fig. 6. Field study of in-situ soils at rest on the stability diagram. (a) Taiwan and Turkey data taken at varying depths. (b) Wang's data (♦ non liquefied, ◊ liquefied) at depth=1 meter.
The rise in pore water pressure causes a reduction in the mean normal effective stress p′, an increase in the stress ratio η, and a decrease in equivalent liquidity LI5. These changes would shift the at-
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Fig. 7. Cyclic stress ratio-pore pressure relationship for different numbers of cycles (Ansal and Erken, 1989).
rest in situ state of stress into different regions. Fig. 9 illustrates schematically the potential changes in η and LI5 values for different initial at rest states, I–VI. These states are chosen to be representative in each of the characteristic regions described before. Soil state I is that of LI5 b 0; soils in this state are in a fractured state and would undergo large deformations with unstable failures upon shaking. Similarly, soils in state VI are outside the stable yield region as discussed before; they remain in this meta-stable state only due to cementation and other factors. Slight disturbance would make these soils crumble and often manifest in the form of catastrophic flow slides. However, prediction of their behavior is complex and dependent on the nature of such extraneous factors as the strength of bonds and fabric arrangements. Therefore, if the initial state of a soil falls in the regions of either state I or VI, it is prudent to assume that it is inherently unstable to begin with and prone to catastrophic failures upon even slight disturbance. Typical undrained effective stress paths under earthquake loading for soils with states II, III and IV (in region 0.4 b LI5 b 1), and V (in region
1 b LI5 b 1.9) are illustrated schematically in Fig. 9. It is possible that earthquake loading would move a soil in state II (closer to the lower boundary of LI5 = 0.4) of the rupture region, to the fracture region as shown. Such soil will fracture into clastic debris as described before. Soils in the region of states III and IV subjected to earthquake loading would exhibit rising peaks until they reach the rupture curve and subsequently soften towards the critical state. Finally, soil state V will plastically yield in a stable manner until the critical state is reached. We now apply the changes in state that would occur to the at-rest data presented in Fig. 6 for earthquakes with different amax values. The resulting soil state changes are shown in Fig. 10. Note that at-rest data for soils which fell in regions I and VI of the stability diagram were deemed to have failed in an unstable manner due to slight disturbances and thus are not shown any further. It is seen that all initial states move into different regions of stability with increase in amax with some data beginning to reach the fault/rupture region even with amax = 0.1 g. These states would attain peak stress ratios along Coulomb type rupture planes and begin to soften towards the critical state with prolonged shearing. Such softening behavior is dependent on the presence of gouge material inside fault planes. Therefore, it is rather difficult to predict their postpeak behavior. For practical purposes, states that reach the rupture line should be deemed to have undergone large deformations and treated as such. As emphasized before, however, failure of this type may not be characterized as liquefaction. Note that to avoid clustering of data points along the rupture line, once a state reached the rupture line, it was removed with the understanding that it had reached its limiting resistance and failed. It is of interest to note that the solid diamond data points originally identified by Wang as non liquefied continue to remain stable at an amax = 0.2 g. However, at amax = 0.5 g all but two of the data points remain inside the stable diagram. The others had reached the rupture line after attaining a peak stress ratio. The fact that they did not undergo large deformation (or “non liquefied” per Wang) leads us to hypothesize the presence of gouge material that was able to sustain prolonged shear deformation. This may not have been the case for soil states with hollow data points identified by Wang as liquefied. As discussed before, such failures along the rupture line may not, however, be classified as liquefaction in the framework proposed here. 5.6. Initial LI5 values and shift in stress ratio
Fig. 8. Number of equivalent uniform cycles Neq for earthquakes of different magnitude (Kramer, 1996).
The stress ratio η, which is plotted on the vertical axis of the behavior map, is one of the key factors in determining the safety against liquefaction of a soil with a given LI5. The changes in deviator shear stress Δq, as well as the rise in pore water pressure, determine the shift in the stress ratio. Fig. 11 shows the shifts in η for a range of LI5 and amax values. As depth increases, so does LI5 and as seen in Fig. 11, when LI5 is larger, the shift in η is smaller. A smaller shift in η makes a soil less likely to move into an unsafe zone thereby making it more likely to remain safe. The relationship between depth and the shift in η is shown in Fig. 12. Between 0 and 15 m below the ground surface, the shift in η under an applied stress is significant, especially at the shallower depths. Below about 15 m, the change is much smaller; negligible, even, for an amax = 0.5 g. Additionally, plasticity of soils affects its resistance to larger shifts in stress ratio. For a soil with a PI of about 4, reducing its water content by 2% would reduce its equivalent liquidity index by 50% and the corresponding confining stress by 90%, leading to a significant increase in the shear stress ratio η (Muhunthan and Pillai, 2009). The same 2% water content reduction in clay with a PI of 20, however, would result in an LI5 reduction by only about 10% and a corresponding minor decrease in p′ and increase in shear stress ratio η. This means that both the depth of the soil and its plasticity characteristics would affect how
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Fig. 9. Schematic illustration of effective stress paths under earthquakes for various initial states in the stability diagram.
easily η and LI5 could shift into the cracking zone, making the soil susceptible to catastrophic failures including “liquefaction.” 5.7. Conclusions This study has put forward a consistent methodology based on critical state framework to characterize the different regimes of
fine-grained soil behavior under earthquake loads. In the critical state view, liquefaction occurs when soil is on the dry side of critical states, near zero effective stress, and in the presence of high hydraulic gradients. In this view, liquefaction is one of a group of phenomena; including piping, boiling, and fluidization; with pipes and channels and hydraulic fractures, internal erosion and void migration.
Fig. 10. Field study data showing the effect of amax on stress ratio shift. **Depths vary for Taiwan and Turkey data (top) while depth is taken at 1 m for Wang's data (bottom).
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Fig. 11. Effect of LI5 on shift in stress ratio η.
The central piece of the proposed characterization of soil behavior is the stability diagram shown in Fig. 5. This diagram captures the effects of soil plasticity through the liquidity index LI, confinement through mean normal effective stress p′, and shear stress q through the stress ratio η. The equivalent liquidity LI5 = LI + 0.5 log (p′/5) is a useful parameter to capture the combined effects of LI and p′ on soil behavior. The three regions of limiting soil behavior; fracture, fault, and fold/yield, as a function of LI5 and η were identified using the data from Weald clay (Eqs. 3a–c). The proposed characterization is used to examine the liquefaction susceptibility of fine grained soils from China, Taiwan, and Turkey. Soils from these regions at-rest are shown to fall within the three
different regions. Therefore, their limiting behavior would be different. Some of the data from Taiwan and Turkey plotted on the negative side of the LI5 axis (Figure 6). These soils are prone to fracture and when disturbed by earthquake or other causes would have resulted in unstable failures. Natural cementation and interparticle bonding enable some soils to exist in conditions outside the LI5 boundary limit for stable yielding at-rest (Figure 6(a)). However, such soils are sensitive to disturbance and in danger of instability and liquefaction when external loads such as from an earthquake break the bonds and the structure collapses. Under earthquake loading, the initial states of soils would migrate into different regions in the stability diagram depending on their
Fig. 12. Effect of depth z on shift in stress ratio η.
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initial location, shear stress increment, and, pore pressure response. The migration paths under earthquake loads are illustrated in Fig. 9 for the various initial states. The final position of the state of soil would dictate the type of limiting behavior expected in the field; fracture, rupture or yield. The final states which fall into the fracture region have the potential for catastrophic failures including “liquefaction”; the ones which fall onto the rupture region would experience the attainment of the peak stress ratio followed by softening along failure planes; the ones in the yield region would continue to yield in a stable manner. The latter two types of deformations while resulting in large deformation may not be of a catastrophic nature. Acknowledgement The authors wish to thank Brenton Cook of the Department of Civil Engineering at Washington State University for his assistance with the preparation of the paper. References Andrews, D.C.A., Martin, G.R., 2000. Criteria for liquefaction of silty soils. Proc., 12th World Conference on Earthquake Engineering. NZ Society for Earthquake Engineering, Upper Hutt, New Zealand. Paper No. 0312. Ansal, A.M., Erken, A., 1989. Undrained behavior of clay under cyclic shear stresses. Journal of Geotechnical Engineering 115 (7), 968–983. Atkinson, J., 1993. An introduction to The Mechanics of Soils and Foundations through Critical State Soil Mechanics. McGraw-Hill. Boulanger, R.W., Idriss, I.M., 2006. Liquefaction susceptibility criteria for silts and clays. Journal of Geotechnical and Geoenvironmental Engineering 132 (11), 1413–1426. Bray, J.D., Sancio, R.B., 2006. Assessment of the liquefaction susceptibility of finegrained soils. Journal of Geotechnical and Geoenvironmental Engineering 132 (9), 1165–1177. Casagrande, A., 1975. Liquefaction and cyclic deformation of sands—a critical review. Proc., 5th Pan-American Conference. on Soil Mechanics and Foundation Engineering, Buenos Aires, Argentina. Erken, A., 2001. The role of geotechnical factors on observed damage in Adapazari during 1999 earthquake. 15th ICSMGE, TC4 Satellite conference on Lessons learned from Recent Strong Earthquakes, Istanbul, Turkey, pp. 29–32.
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