1.- Introduction

     The bitter experiences left by earthquakes during the past, clearly have shown that seismic disturbances are among the most severe natural disaster that can hit an inhabited area. In spite of the important achievements developed by the earthquake engineering during the last two decades, unfortunately, the damages to properties and loss of human lives caused by strong earthquakes are still significant all around the world. In Table 1 are resumed the casualties and the estimated cost of the damages provoked by some of the strong motions that have occurred in the last 15 years. As can be observed, in all of these earthquakes heavy losses have taken place which apparently do not reflect the present knowledge of the Earthquake Engineering. Most probably this fact can be explained by two main reasons:

• Collapse of structures that either have not been designed by engineers or their construction have not been performed with an appropriate control. Also, here it is possible to include miscalculation and wrong design performed by engineers, but these are clear mistakes under the view of the Earthquake Engineering, and therefore they do not indicate any lack in the present engineering knowledge.

• Ground problems. Since any structure can be seen as the combination of elements that carry load to the ground, it is the ground that finally support all the loads including the structure itself, therefore any problem in the ground is instantaneously reflected in some extra disturbance in the structure.

2.- Seismic failures observed on saturated loose cohesionless soils

     During shaking it has been observed various types of ground deformation which in some cases become large enough to cause important damage in structures and man-made facilities. Such deformation of the ground can be identified as ground failure. The phenomenon called "liquefaction" is among the most catastrophic ground failure and in the past it has been observed in almost all the large earthquakes that have occurred, being for instance, extremely dramatic during the Kobe earthquake in Japan. In Fig. 1 is shown an example of ground failure due to the occurrence of liquefaction.

     It seems that the first time when the term liquefaction was used it corresponds to a paper by Hazen (1920) where the failure of the hydraulic fill sands in Calaberas Dam was described. On March 24th, 1918, the up-stream toe of the under construction Calaveras dam, located near San Francisco in California, suddenly flowed and approximately 800,000 cubic yards (731,520 m³) of material moved around 300 ft (91.44 m). Apparently at the time of the failure none special disturbance was noticed. In his paper, Hazen indicates the basic concept behind the phenomenon of liquefaction that still is up date: "When a granular material has its pores completely filled with water and is under pressure, two conditions may be recognized. In the first or normal case, the whole of the pressure is communicated through the material from particle to particle by bearings of the edges and points of the particles on each other. The water in the pores is under no pressure that interferes with this bearing. Under such conditions the frictional resistance of the material against sliding on itself may be assumed to be the same, or nearly the same, as it would be if the pores were not filled with water. In the second case, the water in the pores of the material is under pressure. The pressure of the water on the particles tends to hold them apart; and part of the pressure is transmitted through the water. To whatever extent this happens, the pressure transmitted by the edges and points of the particles is reduced. As water pressure is increased, the pressure on the edges is reduced and the friction resistance of the material becomes less. If the pressure of the water in the pores is great enough to carry all the load, it will have the effect of holding the particles apart and of producing a condition that is practically equivalent to that of quicksand...... An illustration of this can be seen in the sand on the seashore. Such sand, comparable to dune sand in size, is usually found to be saturated with water for a certain distance above the water level. This condition is maintained by capillarity. If a weight is slowly placed on this saturated sand, there is a slight settlement, the grains of the sand coming to firmer bearings, and the weight is carried. A sharp blow, as with the foot, however, liquefies a certain volume and makes quicksand. The condition of quicksand last for only few second until the surplus water can find its way out."

     At the present, the word liquefaction is often used in a broad sense for several phenomena where either loss in strength or stiffness reduction takes place in a saturated cohesionless soil mass inducing large deformation of the ground. However, in order to understand the actual soil behavior is of great importance to distinguish between these two different phenomena: true liquefaction or flow failure and cyclic mobility or strain softening.

     Cyclic mobility or strain softening is a phenomenon where there is a significant reduction in the stiffness of the soil mass associated with a rising in the pore water pressure caused by a dynamic loading. It is important to point out that during the occurrence of this phenomenon, the soil mass does not undergo any loss in strength, however, important deformations can be developed due to the degradation of stiffness. Perhaps, the most common outcome of the large build-up in pore water pressure is the appearance of sand blows in the ground surface usually called sand boils. The re-consolidation of the soil layers may be associated with large seepage-forced that induce upward water flow that can transport to the ground surface soil particles generating sand boils with their typical volcano shape. During seismic loading, the level of deformation underwent by the soil mass, due to cyclic mobility, can be unacceptable for some structure, consequently this phenomenon can generate an important amount of damage on structures. Most probably in all large earthquake this phenomenon has been the cause of a great number of losses.

     On the other hand, the true liquefaction or flow failure has been observed in loose saturated cohesionless soil mass and its main features are the large amount of soil involved in the failure, the short time of few minutes that it takes and the very flat slope that is finally reached by the soil mass, e.g. an angle of 3º to 4º respect to the horizontal. This kind of failure can be trigger not only by earthquakes, but also by any other disturbance that is fast enough to induce an undrained response of the soil mass. In Fig. 2 is shown the result of liquefaction in La Marquesa dam caused by the 1985 earthquake in the central part of Chile. Because the true liquefaction or flow failure involves a loss in strength, it is much more catastrophic and in general, the failure compromises a more significant amount of soil mass .

     The correct understanding of cyclic mobility and flow failure is of great importance because the applicable stability analysis is different in each case. For example, in structural analysis of a steel beam, first of all, the resistance is verified and calculation is done in terms of available and acting stresses. Then, the allowable deformation is checked. If the beam fails at the first step, there is obviously no reason to continue with the second verification. Similarly, for the global stability analysis of a saturated cohesionless soil mass subjected to an undrained loading condition, the first analysis has to be done in terms of the real available strength and the acting forces. If the soil mass is able to resist the acting forces, then the second step is the verification of the resulting deformations. If it is not, countermeasures in order to increase the strength of the material, or to reduce the driving forces have to be performed.

     The checkup for judging whether the cyclic mobility or liquefaction can or can not occur in a given deposit constitutes the first important task for ensuring safety of the ground against shaking. Therefore, the stability analysis of earth-structures where saturated cohesionless materials are involved requires two main issues: a) to establish whether the soil mass is in such state that can develop or not a loss in its shear strength and b) to evaluate the acting or driving forces and the residual strength of the soil mass. These tasks have been solved to some extent using the steady state concepts developed mainly by Casagrande (1938), Castro (1969) and Poulos (1971).

     It is important to mention that when a soil mass is in a such state that it can undergo a flow failure, the triggering of such failure may be caused by both static or dynamic loading, it only need to be fast enough to develop an undrained soil response. The literature provides a significant number of cases where flow failure has been triggered by static loading, for example, in the Duth province of Zeeland, between 1881 and 1946 more than 229 flow slides were registered. The total area of land that was lost by the occurrence of these slides is around 2.65 millions m² and the total volume of soil displaced is approximately 25 millions m³ (Koppejan et al.,1948). In these cases, the flow failures were not triggered by a seismic load, but likely by tidal currents.

     Similar type of failures occurred in norwegian fjords have been reported by Bjerrum (1971), where six very large submarine slope failure occurred in fine sand and coarse silt deposits occupying the head of the fjords. Based on the disastrous character and large dimensions of these failures, Bjerrum concluded that loose fine sands and coarse silts as those encountered in the norwegian fjords can loss their strength completely. As a result, the soil mass assume the character of a viscous liquid flowing downwards by a large distance in a relatively shot period of time.

     Also, this phenomenon has been observed in dams constructed using the hydraulic-fill method, as example, it is possible to mention at least four important cases. Firstly, the flow failure of the upstream shell of the Calaberas dams where 800 thousand cubic yards of material flowed by around 300 ft, Hazen (1920). Secondly, the failure of the right abutment of the upstream slope of the Fort Peck dam in Montana in September, 22, 1989, Middlebrooks (1942). In this slide around 10 millions cubic yards of material were displaced in about 4 minutes by a distance up to 1500 ft. The third case of flow failure is probably the best known and well documented, it occurred in the Lower San Fernando dam in Southern California. The failure involved both the upstream shell an the upper part of the downstream slope of the dam. The slide was triggered by the San Fernando earthquake in February, 9, 1971 with a magnitude of 6.6. Fig. 3 shows the cross-section through the dam before and after the earthquake (Seed et al.,1975). The slide seems to have occurred about one-half minute after the shaking and almost caused a large disaster in the populated downstream area where around 80 thousand people were living. In Chile also one dam has manifested the occurrence of flow failure, La Marquesa dam, which developed a significant amount of cracks and displacements in both slopes as in shown in Fig.4.

     Another cases of flow slide have been reported by Seed (1987), for example, a major flow slide occurred in a sand deposit along the bank of lake Merced in San Francisco during the earthquake of 1957. Because the excitation was so short, about 4 seconds, likely the slide took place after the earthquake had stopped. ther flow failure occurred during the Nigata earthquake of 1964 in the Uetsu railway embankment of 33 ft height. The liquefied sand flowed about 400 ft over ground which sloped at about 2º and came to rest at a slope angle of about 4º. During the Tokachi-Oki earthquake of 1968 in Japan, the Kona Numa Railway embankment of 10 ft high failed by flowing in both directions from the center line. The liquefied material flowed around 60 ft, coming to rest at a slope of about 4º.

     Other earth-structures that have shown to be very sensitive against earthquakes are the tailing dams, which are usually constructed by hydraulic-fill methods. Tailings are the waste materials resulting from mining operations after the rock has been crushed and the valuable minerals extracted from the ore. Generally, tailings materials can be classified as fine sands, silts or rock flour. In Table 2 are indicated Chilean tailing dams that have experimented flow failure due to seismic loading.
 
 

     As can be observed, tailing dams built using cohesionless soil as sandy materials may undergo important failures due to seismic disturbances. Probably one of the oldest flow failure in a tailing dam that has been reported in the literature occurred at El Teniente copper mine in Chile following the earthquake of October 1, 1928. The failure of Barahona tailing dam involved 4 millions tons of material that flowed along the valley, killing 54 persons (Aguero, 1929; Dobry and Alvarez, 1967). Later, after the earthquake on March, 28, 1965, El Cobre tailing dams located in Chile failed catastrophically and more than 2 millions tons of material flowed around 12 km in a few seconds, killing more than 200 people and destroying El Cobre town (Dobry and Alvarez 1967). At the time of the failure, the dams was about 33 m high and it had a downstream slope as steep as 35º to 40º respect to the horizontal (Finn 1980). Another well documented example of flow failure in tailing dam took place after the earthquake of January, 14, 1978, at the dike No 2 of Mochikoshi gold mine in Japan. The failure occurred 24 hours after the main earthquake, at the time when there was not any shaking, and a total volume of 3 thousand m³ of material flowed into the valley to a distance of about 240 m (Ishihara 1984).

     Recently, during the Chilean earthquake of March, 3, 1985, two tailing dams presented failed by liquefaction. Cerro Negro dam of 30 m in height failed and about 130 thousand tons of tailing material flowed into the valley for distances of about 8 Km, (Castro and Troncoso 1989). The other failure occurred in Veta de Agua No. 1 dam, which at the time of the earthquake had a maximum height of 15 m. According to a witness, the failure took place in the central part of the dam few seconds after the shaking had stopped. The tailing material stored in the pound traveled along the El Sauce creek for about 5 km.

     Others soil deposits that have shown to be highly susceptible to flow failure are the artificial sand islands and berms to support exploration drilling structures for hydrocarbons on the Canadian Beaufort Sea. These earth-structures have been performed by hydraulic-fill methods and mainly due to the large cost of densification, these deposits are uncompacted and therefore very loose. A well documented case of flow failure is that occurred in the Nerlerk berm. On July of 1983, bathymetric survey showed that a large slide had occurred. Then, between July 25 and August 2, 1984, three more slide took place and a fifth one occurred on August 4, 1984. These flow failures were triggered by the loading originated from the soil placement itself (Sladen et al.,1985).

     Another type of flow failure has been frequently observed in mountainous regions where accumulations of mountain debris have flown like a stream of lava during periods of saturation. This flow failures have been the cause of serious loss of life and property (Casagrande, 1938).

     Moreover, another ground failure typically observed during and after earthquakes are the lateral spreads which are the consequence of liquefaction in a layer beneath the ground surface. Lateral spreads result in lateral displacements induced by both gravitational driving forces and inertial forces generated by the earthquake. This kind of ground failure occurs in relatively gentle slopes and the movements are toward an incised free face, e.g., rivers, channels or roads. For example, more than 250 bridged were damaged by lateral spreading of floodplain deposits toward river channels, during the Alaska earthquake on 1964.

     As can be seen, large failures of cohesionless soil mass have occurred in the past and also recently which have involved significant losses, it seems that the words of professor Arthur Casagrande said 63 years ago are still up-date (Casagrande, 1936): "It is not an exaggeration to state that throughout history and into the present day faulty designs have caused more loss of life and property in the field of earth and foundation engineering than any other branch of engineering. In this field the largest share of these losses have been caused by failures of dams and dikes. Many times larger than the property losses due to failure has been the waste of money due to excessive over-designing".

     Therefore, it is clear that effort must be done in order to understand in more detail the soil behavior and specially the undrained response of cohensionless materials which can be unstable and deform largely during fast loading condition, e.g., against earthquakes.

     Because of the catastrophic consequences of a flow failure or liquefaction and cyclic mobility, their study is of great concern for engineers and in the last 30 years a significant amount of research has been conducted in this subject which have provided important insight of these phenomena. Nevertheless, still there are many questions to be answer regarding the soil behavior during undrained loading conditions, even though during the last past years, interest in the importance of developing a suitable methodology to analyze the stability of saturated poorly compacted sandy soil deposit has increased. This fact comes as a consequence of the occurrence of large landslides of natural slopes, dam failures and flow failures of hydraulic placement of artificial islands or reclaimed areas along the coasts. In many cases, failure has been caused by seismic load, but in others, it has been triggered by small and quick perturbations. Therefore, a better understanding and a more suitable characterization of the undrained response of saturated cohesionless materials are needed and consequently in the Geotechnical Section of IDIEM these issues are among the main areas of research..
 
 

3.- Geometrical behaviour of saturated cohesionless soils

3.1.- Volumetric Strains of the Soil Response

     In a general sense a soil mass can be seen as a granular material consisting of voids and unaggregated of mineral particles which have a mechanical and in some cases also, physico-chemical interaction to each other. The main feature of soils in relation to others engineering materials is that they are three-phase system; in general they are composed of solid, e.g., mineral particles, liquid, e.g., water and gas, e.g., air. Since air is much more compressible than water and both can flow from or into the mineral particles or soil skeleton, the soil behavior is quite dependent on the relative proportions of these three components. However, fully saturated soils are observed very frequently in nature, and beside, from engineering point of view, they are more sensitive to seismic loading than partial saturated soils. Therefore, the study of soils in a saturated state is often selected and in the present article it will be considered a soil mass always fully saturated.

     To get a general picture of the soil behavior, first of all it is necessary to have a clear understanding of the volumetric deformations that are associated with the response of a soil mass subjected to shear stresses under drained condition. During shearing, one of the most relevant difference between a conventional engineering material and a soil is related with the volume change. The current engineering materials, as for example, steel and concrete do not show any important volume change when they are subjected to shear stresses. Soils, however, can undergo a large amount of volume change depending on the initial state of stresses and density. This tendency in volume change has shown to have a tremendous effect on the strength of the soil mass.

     The volume change that takes places during loading is mainly due to the contraction or expansion of the voids into the soil mass and it was firstly pointed out by Reynols (1985) in the past century. Reynols showed that dense sands tend to expand increasing their total volume when they are subjected to shear stresses. This phenomenon was called by Reynols dilatancy. Nevertheless, only in the 1930s. From the observation of the volumetric strains on dense and loose sands, Casagrande realized the actual importance of the volumetric strain in the soil response developing the concept of "critical density or critical void ratio". Using direct shear tests, Casagrande observed that during shearing dense sand expands and therefore increases its void ratio, while very loose sand reduces its volume and accordingly its void ratio. In a dense sand, the grain are pretty well interlocked, thus any deformation causes a loosening up of the initial compact structure. On the other hand, very loose sand tends to contract in order to achieve a more stable structure. Based on this observation Casagrande developed the concept of the critical void ratio: when dense and loose sands are sheared in a drained condition, they change their void ratio until a common constant value is eventually reached. This ultimate common void ratio was termed the critical void ratio. At this state, the soil continues to deform under constant strength and constant volume, hence the soil behaves as a frictional fluid. In Fig. 5 is presented a typical result showing this behavior.

     Later in 1958, using the simple shear test, Roscoe and his co-workers presented a conclusive study proving the concept of critical void ratio and extended it to clayey soils (Roscoe et al.,1958). Typical test results using the simple shear box on 1 mm steel balls are shown in Fig. 6a in terms of void ratio and horizontal displacement for a constant normal stress of 1.41 kgf/cm² (20 lb/sq.in.). As can be observed, the volumetric strain can be either positive or negative depending upon the initial void ratio and the level of deformation, but when the ultimate state is achieved, the volume change stops and the soil deforms under constant-volume condition reaching the critical void ratio associated with the normal stress under which the test is performed. For the same tests, Fig. 6b shows the void ratio versus the shear stress developed throughout the tests (CVR stands for critical void ratio). From these results is readily apparent that for a constant normal stress an ultimate unique critical void ratio can be reached. Furthermore, at this state a condition where the granular material deforms under constant volume, constant normal stress and constant shear stress is achieved.
 

3.2.- Cyclic Undrained Soil Response

     The Valdivia earthquake occurred in 1960 in Chile induced strong soil deformation and the Nigata and Anchorage earthquakes occurred in 1964 in Japan and Alaska, respectively, left a severe damage on structures founded on saturated sandy soil deposits due to the excessive deformation that those materials developed during and immediately after the main shock. Professor Seed and his co-workers aware of this phenomenon started to study the characteristics of the cyclic response of sands under undrained condition (Seed and Lee (1966); Lee and Seed (1967); Lee and Seed (1967); Peacock and Seed (1968). An Undrained condition means that the natural tendency of volume change is not possible to occur because the load is too fast. When a load is fast enough there is no time for the occurrence of volumetric strains and therefore, pore water pressures are developed into the soil mass. A seismic disturbance is a typical load that is very fast generating essentially an undrained condition.

     During earthquakes, the main part of the soil deformation can be attributed to the upward propagation of shear waves. In Fig. 7 is considered an element of soil subjected to a normal stresses,  and  and to an initial horizontal shear stress, , which can be associated with either an external load produced by some structure or by the inclination of the ground surface as the case of a slope. Subsequently, the vertical propagation of shear waves induce an additional shear stress, , which is cyclic in nature. Therefore, the principal stress directions as well as their magnitude vary according on the seismic excitation. These conditions can best be reproduced in laboratory by a simple shear test on anisotropically consolidated specimens subjected to cyclic loading condition. However, a first approximation can be obtained by cyclic triaxial test, where the lateral total pressure remain constant while the axial stress is changed in .

     In spite of the quantitative difference between the soil response in cyclic triaxial test and cyclic simple shear, the general pattern is quite similar according experimental evidence reported by Peacock and Seed (1968). Fig. 8 shows typical cyclic triaxial tests followed by monotonic triaxial tests on loose and dense samples of Sacramento river sand (Seed and Lee 1966). In the case of loose sand, Dr = 38%, during the first eight cycles of loading, very small level of strain is observed, even though the pore water pressure built-up significantly; close to 50% of the initial effective confining pressure. During the application of the ninth stress cycle, the pore water pressure increases sharply reaching the initial effective confining pressure and the specimen deforms considerably. In the tenth stress cycle, the deformation of the sample exceeds 20% of the initial height and according to Seed and Lee (1966), over a wide range of strains the specimen could be observed to be in a fluid condition. When the cyclic loading was stopped, the effective stress on the specimen was zero because the pore pressure had reached the initial effective confining pressure. After the cyclic loading, the specimen was subjected to a monotonic loading under strain-controlled condition as shown in Fig. 8. It is readily apparent that the sample does not exhibit any resistance deforming continuously without change in pore pressure during a large level of strain, as large as 20%. However, thereafter the specimen develops a dilative behavior decreasing the pore pressure which lead to the development of shear resistance again.

     On the other hand, a typical result on dense specimen, Dr = 76.2%, is shown in Fig. 9. During the first 9 cycles, the deformation is very small although the pore water pressure has raised about 50% of the initial effective confining pressure. Closely to the 12 loading cycles, the pore water pressure starts to reach the initial effective confining pressure at the instant of zero deviator stress, in other word, at the time when there is no any shear stress acting on the sample. Associated to this condition it is observed that the strain amplitude increases markedly, but in contrast to loose samples, the level of strain gradually increases with the stress cycles. In fact, even though from the stress cycle number 13, the condition of zero effective stress is reached at each instant of zero shear stress, the axial strain of the sample does not exceed 10% after 20 stress cycles. Hence the response of dense sands does not show the sudden development of large strain observed in the case of loose samples.

     The subsequent application of monotonic loading indicates that the sample starts to dilate, and therefore, starts to regain its strength at a much smaller strain on the order of 5%. This is in contrast to loose sand which starts to offer some resistance after a large strain on the order of 20%.

     From a set of experimental results carried out on cyclic triaxial tests similar to those explained above, Seed and Lee introduced new criteria to define "liquefaction" (Seed and Lee 1966; Lee and Seed 1967). These criteria are as follows:

• Failure: Some level of strain which would be associated with a failure from a practical point of view.
• Complete liquefaction: When the sample deforms without shear resistance over a wide range of strain.
• Partial liquefaction: When a sample exhibits no resistance to deformation over a range of strain smaller than that defined as failure.
• Initial Liquefaction: When a soil first exhibits any degree of partial liquefaction during cyclic loading. This occurs when the pore water pressure reaches the initial effective confining pressure for the first time.

     Fig. 10 shows for three different densities, the applied cyclic loading versus the number of cycles required to cause failure according to the criteria defined above (Lee and Seed 1967). It is seen from this figure that the stress amplitude required to induce some level of strain increases with the density of the sand. The initial liquefaction can be induced regardless the density of the sample. In the case of loose samples, the condition of initial liquefaction and the condition of a level of axial strain of 20% are almost achieved simultaneously. However, in the case of dense samples, the number of cycles needed to reach these conditions are significantly different, between 200 to 500 times different.

     Others experimental results using different equipment, for example the hollow cylindrical apparatus, have been obtained. It seems that this type of cyclic torsional shear test permit to avoid the concentration of deformation in the top of the sample (Ishihara et al., 1975; Tatsuoka et al., 1982; Ishihara 1985 and Negase 1985, among others). Figs. 11 and 12 show typical results on relatively loose and dense sand using the cyclic torsional shear tests (Negase 1985; Ishihara 1985). As can be seen, quite similar behavior to that observed in the cyclic triaxial test is also noted in the cyclic torsional tests. Of special interest is the tremendous fluctuation in pore water pressure that a dense sand is able to develop after it starts to reach momentarily the condition of zero effective stress. This particular phenomena where the pore water pressure momentarily reaches the initial effective confining pressure and in connection a cyclically induces strains are developed in the specimen without cause a loss in strength, it is the so-called "cyclic mobility"

     The results of cyclic test shown above can be more clearly understood, if they are presented in terms of both stress-strain curves and effective stress-paths as they are in Figs. 13 and 14. In the case of loose specimen with Dr = 47%, there is a gradual migration of the effective stress-path toward the origin, which is more pronounced in the first cycle and after the phase transformation line is reached.

     When the cyclic effective stress path touches the phase transformation line, a significant change in the cyclic response takes place. During loading, the effective stress path is turned right upwards indicating a dilative response, while during unloading it is turned left toward the origin, what it means a strong contractive behavior associated with a large increase in pore water pressure. Once the phase transformation is crossed, this phenomenon is repeated onwards and at each cycle of loading and unloading, the effective stress path moves upward and downward closely along the failure line. Eventually it starts to pass though the origin, indicating a condition of zero effective stress at the instant of zero acting torsional shear stress. The same general pattern is observed in Fig. 14 for a sample with Dr = 75%, except that the phase transformation line is crossed during the first cycle and thereby the large change in the effective mean stress occurs much early than the specimen of loose sand.

     On the other hand, for the specimen with Dr = 47%, Fig. 13 shows the stress-strain relationship during the application of the cyclic torsional stress. It is observed during the first cycles relatively small loops, but after the phase transformation line is reached, the level of strain increases markedly and at each cycle the rate of deformation becomes larger and larger. For the specimen with Dr = 75%, Fig. 14 shows that there is a progressive rising in the level of strain but at decreasing rate indicating a stable behavior, even though the specimen has developed an important increment in the pore water pressure.

     The cyclic responses explained above can be classified as "cyclic mobility" because they only involve degradation of stiffness which can be significant or moderate depending upon the density of the sandy soil, among other factors. In the showed cases, the cyclic test results indicate that the soil response does not compromise any lost of strength. As long as the specimen is strained large enough, there is a tendency of the soil to dilate and regain stiffness and strength. In this context the cyclic mobility can be evaluated as the number of cyclic to reach certain amount of shear deformation under a constant amplitude of cyclic stress. The question that immediately arises it is concerned with the selection of the shear strain to be used for the evaluation of the cyclic mobility.

     From test results as those presented in Figs. 13 and 14, the maximum shear strain developed after a certain number of cycles can be associated with the amplitude of the applied cyclic stress. Fig. 15 shows for Fuji river sand at different relative densities, the relation between cyclic stress ratio, , and the maximum shear strain, , in percent after 10 cycles of loading (Ishihara 1985). Fig. 15 also indicates the range of deformation where pore pressures become equal to the initial effective confining pressure. From these results is readily apparent the narrow range of shear strain in single amplitude, between 2.5 to 3.5%, where the condition of 100% of pore pressure is reached. Thus, a 3% of cyclic shear strain in single amplitude can be a reasonable criterion to define cyclic mobility (Ishihara 1985). Nevertheless, it should be emphasized that the level of shear strain selected as representative of failure can be any. Obviously, the key factor is to select the amount of shear strain that really represent in someway the equivalent or actual level of shear strain that may cause failure of the ground from engineering point of view.

     Another feature of the cyclic mobility which can be observed from the results presented in Fig. 16, is that for a given numbers of cycles, medium to dense sands develop a large cyclic resistance. This characteristic is reconfirmed by the cyclic torsional test results on Toyoura sand presented in Fig. 17 (Tatsuoka et al., 1982). As can be seen, there is some threshold value of relative density above which the cyclic stress ratio that causes some amount of strain in a given number of cycles, increases drastically. For Toyoura sand under cyclic torsional simple shear test condition, the threshold relative density is around 80% and 85% for 10 and 20 stress cycles, respectively.

3.3.- Monotonic Undrained Soil Response

     As it was explained previously, there exist a failure that involve the flow of the soil mass, the true liquefaction. Under this state, it is postulated that the soil mass deforms continuously under constant normal stresses, constant shear stresses and constant volume, and also, any initial fabric or initial anisotropy is broken down, so that a new fabric is developed (Poulos, 1981).

     To show the steady state concepts a comprehensive series of triaxial tests were carried out on the Japanese standard Toyoura sand (Verdugo et al 1991; Ishihara 1993). It is important to mention that similar results have been obtained in cycloned tailings sands (Verdugo et al 1995). Toyoura sand is classified as a uniform clean fine sand consisting of subrounded to subangular particles, with a specific gravity of 2.65, mean grain size D₅₀ = 0.17 mm, uniform coefficient Cu = 2.0, and maximum and minimum void ratio, e_max = 0.977 and e_min = 0.597.

     For the same void ratio after consolidation, Fig. 18 shows the effect of the initial effective confining on the undrained soil response. It can be seen that depending upon the initial confining pressure, the soil response varies from a dilative to a contractive one. However, as long as the void ratio is the same, the ultimate shear or the undrained steady state strength is the same, regardless the initial level of pressure.

     The effect of the stress history on loose and dense specimens is shown in Figs. 19 and 20, respectively. In these tests two samples with identical initial states of density and pressure were tested under undrained loading. One sample was tested monotonically up to the ultimate state, while the second one was firstly subjected to a series of cycles of loading and unloading and then followed by a monotonic load until the ultimate state was reached. From these results it is possible to conclude that the ultimate condition or steady state strength achieved at large deformations is independent of the previous stress-history.

     Fig. 21 shows triaxial test results in terms of void ratio and mean stress for the ultimate state for both drained and undrained loading conditions. It can be seen that independent of the drainage condition, the same ultimate state is reached defining a unique steady state line in the e-log p plane.

     It is also important to mention that all the triaxial tests performed for a wide range of void ratios, confining pressures, initial fabrics and drainage conditions systematically developed an angle of internal friction at the ultimate state very closed to 31.5^o. This experimental fact confirms other results indicating that the angle of internal friction developed at the ultimate state is a material property and accordingly a constant value for a given sandy soil.

     The experimental results presented above support the steady state concept which states that there always exists an ending line in the e-q-p space, so-called steady state line, where a soil element must be located if it reaches the ultimate state. Fig. 22 shows this line for Toyoura sand in the e-p plane. The data that are shown has been obtained from drained and undrained monotonic loading and undrained cyclic loading. Thus it is possible to conclude that if a Toyoura sand sample is stressed at large deformations it should achieved an ultimate state located in this line.

     Saturated deposits of cohesionless materials have been shown to undergo liquefaction or flow-type of failure during earthquakes. This type of failure has been observed in natural and artificial slopes of cohesionless soils, as for instance, tailing dams. In this regard, the failure of the El Cobre tailing dam following the 1965 La Ligua earthquake, which killed 200 people, emphasize the importance of careful studies concerning the sandy soil response associated to flow failure. The true liquefaction or flow failure generates a large level of deformation where the steady state condition is developed. Under this state it is postulated that any initial fabric or initial anisotropy is broken down and finally a new fabric is developed when the steady state of deformation is achieved. However, the experimental results reported by different researchers are not yet conclusive and in some cases even contradictory.
 

3.4.- Effect of Inherent Anisotropy on the Soil Response

     It is important to keep in mind that during the creation of any soil deposit, the sedimentation mechanism of the soil particles is affected by the gravity force. Thus, the soil particles are deposited with a preferential orientation making the soil structure anisotropic. This initial anisotropy caused by the geological process of deposition was named Inherent Anisotropy by Casagrande and Carrillo (1944). Depending upon the environmental conditions existing during the sedimentation process, the inherent anisotropy may be very important and the soil response for small deformations can be affected in a significant manner.

     Considering the importance of the undrained response in the evaluation of the seismic response, it is clear that efforts must be made in order to figure out the factors that control it. Hence, the effect of the inherent anisotropy or initial structure on the soil response is presented.

     The inherent anisotropy is basically caused by the orientation of the soil particles in some preferential directions. Experimental measurements performed by Oda et al., (1978) on the particle orientations frequency in samples deposited under water and air are shown in Fig. 23. As can be seen, there are an important number of particles orientated with their major axis close to the horizontal indicating that the initial soil structure is anisotropic. Similar results have been shown by Arthur et al. (1972) and Mitchell et al. (1976) , among many others.

     Fig. 24 shows the stress-strain curves and the volumetric soil response for samples prepared by different methods that generates different inherent anisotropies in the soil mass. These results reported by Mitchell (1976) up to an 8% of axial deformation indicate that different inherent anisotropies may originate an important influence in the general soil response. Therefore, different degree of inherent anisotropies are associated to different soil responses.

     For a given inherent anisotropy, the effect of the orientation of the principal stress respect to the plane of deposition (bedding plane) on the soil response has been studied by Oda et al., (1978). Typical results obtained on plane strain tests up to 8% of strain are shown in Fig. 25. It is clear seen that depending upon the inclination of s₁ respect to the bedding plane, the stress-strain curves, the volumetric strains and the peak strength can be very different. These results suggest that the rotation of the principal stresses produces an effect on the soil response which has been confirmed by Gutierrez (1989) and others. Hence, at least for a medium level of strain, the inherent anisotropy produces different soil responses.

     Also, the cyclic response is significantly affected by the initial soil particle arrangement. Test results obtained from samples prepared by different procedures are presented in Fig. 26. As can be observed, there is a tremendous difference in the cyclic strength depending upon the sample preparation technique, suggesting that the initial fabric play an important role in the soil response.

     A study on the effect of fabric on the ultimate strength was carried out by the author. First a series of direct shear tests on samples prepared by different means were performed in order to investigate whether the adopted methods of sample preparation induced or not different initial soil structures. In doing so, it was assumed that the peak angle of internal friction should be directly affected by the initial structure of the soil mass. Hence, in these tests only the peak stress ratio, , was analyzed. Two different sandy soils were tested. The grain size distribution curves are shown in Fig. 27. The soil named S-12 contains a 12% of low plastic fines, it has a specific gravity Gs = 2.73, and maximum and minimum void ratios of 1.133 and 0.596, respectively. The soil named S-20 contains a 20% of low plastic fines, it has a specific gravity Gs = 2.66, and maximum and minimum void ratios of 1.111 and 0.547, respectively.

     The test scheme to investigate the effect of the initial anisotropy on the steady state consisted of a series of triaxial tests carried out under drained and undrained conditions of loading using samples prepared by means of different sample preparation methods. Lubricated as well as enlarged end plates were used in order to minimize non-homogeneities in the strain distribution throughout the samples and specially to avoid the development of shear bands. The axial load, pore water pressure, axial deformation and volumetric strain were measured by electrical transducers and automatically stored in a personal computer. All the tests were conducted under strain-controlled condition of loading. The deformation rates were 1 and 0.5 mm/min for undrained and drained tests, respectively. The initial dimensions of the specimens were 5 cm in diameter and 10 cm in height. The saturation of the sample was considered sufficient when the B-value was greater than 0.95. The sample void ratios were evaluated through the measurement of the water content after the tests were ended, Verdugo et al. (1991).

     The samples were prepared by the methods of wet tamping and water sedimentation with different angles of the bedding plane. The first one uses oven dry soil that is well mixed with distilled water in a proportion of 5% in weight, then the soil is compacted inside a split mold in six layers with the same height and amount of wet soil. The moist soil is then strewed by fingers inside the split mold and spread out uniformly with a rod of 2 mm in diameter. Each layer is gently compacted by means of a metal mass held by a rod until the preestablished height is reached. After the last layer is compacted, the split mold is open and the specimens installed in the triaxial chamber, saturated, consolidated and tested. In the case of the water sedimentation method, dry soil is deposited inside a box full of water by means of a funnel. The box is illustrated in Fig. 28 and consists of an inclined base that permit the setting of any angle .

     After the soil has been deposited, the box is frozen in a camera under -35^oC, then the frozen block of soil is removed from the box and samples are trimmed with a bedding plane inclined in an  angle respect to the horizontal. Thereafter, the samples are installed in the triaxial chamber and tested. It is important to note that by this method, the soil is deposited continuously under water without causing appreciable segregation of the material.

     Fig. 29 summarize the results of the direct shear tests performed on the sandy soil S-12 in terms of relative density and maximum mobilized angle of internal friction, . It can be seen that for the range of relative density used,  is significantly affected by the adopted method of sample preparation. These experimental results show that the methods of sample preparation used generate different inherent anisotropies.

     It is interesting to note that the wet tamping likely produces a distribution of soil particle that is nearly random, while the water sedimentation induces an arrangement of soil particle markedly affected by the selected angle of deposition, . This is confirmed by the higher peak strength showed by the samples prepared by wet tamping, and by the lowest strength developed by the samples with a parallel inclination of particles respect to the plane of shear .

     On the other hand, the results of the triaxial tests have shown that the frictional resistance developed during the condition of steady state for the two soil tested under different condition of loading and sample preparation is a constant value. For the sandy soils S-12 and S-20, the angles of internal friction mobilized during the condition of steady state were 39^o and 38^o, respectively. These results confirm that during the occurrence of the steady state, the mobilized angle of internal friction is constant and a material property.

     Regarding the steady state strength of the sandy soils S-12 and S-20, Figs. 30a and 30b summarize all the data obtained for different condition of loading and sample preparation. Although some scatter is observed, these results show that the steady state strength is not affected by the initial soil structure. Therefore, for the homogeneous samples tested, the large deformations associated to the steady state condition were able to erase the initial arrangement of soil particle.

     The comparison between the steady state lines computed from reconstituted specimens and steady state data points from "undisturbed" samples provide another source of information concerning the effect of the inherent anisotropy on the steady state. Figs. 31a and 31b show published data in terms of void ratio and undrained steady state strength for both "undisturbed" and reconstituted specimens. From these results it is readily apparent that there is a strong difference between the steady state strength of "undisturbed" and reconstituted samples. This was explained by Poulos et al (1985) by the differences in the grain size distribution curves between these samples. However, the grain size distribution of the reconstituted samples corresponds to a kind of average gradation. Therefore, if there were no effect of the initial soil structure, the data points obtained from "undisturbed" samples should be well distributed above and below the steady state line obtained from the batch of soil with the average grain size. Otherwise, it means that the steady state strength is affected by the initial soil structure of the "undisturbed" samples. This result can be explained by the fact that natural deposits usually present stratified structure, and then, "undisturbed" samples posses non-homogeneous structures that can not be fully broken down even at large deformation.

     To explain the differences between the experimental results obtained in this study and the analysis performed on "undisturbed" sample, it is useful to visualize different initial structures with different arrangements of soil particles like those illustrated in Fig. 32. It is important to keep in mind that all the arrangements of the soil particles that are sketched in this figure are associated with the same soil.

     The cases shown in Figs. 32a to 32c correspond to homogeneous arrangements of particles in the sense that the same distribution of particle orientations, particle sizes and number of contacts are repeated uniformly throughout the whole soil mass, while the cases illustrated in Figs. 32d to 32f correspond to non-uniform distribution of soil particles.

     The experimental results presented in this study indicate that arrangement of soil particles as those illustrated in Figs. 32a to 32c can be broken down by the level of deformation, and a new special rearrangement of soil particles can be achieved during the occurrence of the steady state. On the other hand, for those arrangement of particles indicated in Figs. 32d to 32f, any amount of deformation will not be able to create a new common arrangement of soil particles, so that different steady state conditions are developed in each case.

     Therefore, for a better understanding of the soil behavior it seems to be convenient to divide the initial arrangement of soil particle in two groups:
 

a) Arrangements that can be completely erased under large deformation, so that any soil property evaluated at sufficiently large strains can be expected to be independent of the initial particle arrangement, and

b) Arrangements that can not be fully erased even when the soil mass is subjected to large strains, for instance, cemented soil mass, locked sands, and in general a non-homogeneous soil mass, so that any soil property measured at large strains, would be affected by the initial particle arrangements.

     Hence, as a conclusion it is possible to say that for homogeneous soil mass, the steady state strength is independent of the initial inherent anisotropy or initial particle arrangements. This result confirms the concept that during the steady state condition a new arrangement of particles is achieved breaking down any previous one. However, the steady state strength of non homogeneous soil mass is strongly dependent on the initial configuration of particles. The reason is that even large deformations can not to erase initial heterogenities.
 

4.- State model for undrained behaviour of sand

4.1.-Constitutive Model

     For cohesionless materials, the steady state concept has been used mainly in the study of the instability of sandy soils under large deformation, but less work in its application to the development of constitutive models has been done. In order to reliably estimate the stability of a cohesionless soil mass, it is important to be able to predict the behaviour of the soil at all stages of loading. Accordingly, a state model that is able to accurately represent the undrained behaviour up to the ultimate condition is presented. The main element of the model is based on the observation that the peak strength of sand (in both drained and undrained conditions) is not a unique parameter, but instead is dependent on the initial density and confining stress level at which the sand is sheared. Bolton (1986) have shown that the peak angle of friction f_p of several sands is related to the stress level p and relative density D_r as:

where  is the friction angle at the steady state. Been and Jefferies (1985) have also shown that the peak friction angles of many sands is function of the state parameter  which is the vertical distance of a sample's void ratio from the steady state line in the e-p plane.

     The state model will be presented in terms of the stress and strain increment variables in the conventional triaxial tests:

     The first assumption used in the formulation of the model is that elastic strains are negligible and that the total and plastic strains are the same. For monotonic loading condition, to which the model will be applied, the assumption of zero elastic deformation has no serious consequences but greatly simplifies the formulation of the model. The elements of the model are summarised as follows:
 

Peak Strength as Function of Confining Stress

     The variation of peak strength with confining stress follows that of Bolton (1986), and Been and Jefferies (1985). Since the model will be applied to undrained loading, the void ratio is constant and only the variation of peak stress ratio  with mean stress during loading will be modelled:
 

(2)

Shear Strain Hardening

     The variation of stress ratio with shear deformation is based on the widely used hyperbolic relation:

(3)

where G_o is the value of G at the reference stress p_o and  is a material parameter.

     Differentiating Eq. (3) gives:

Stress-Dilatancy Relation

     The usual cam-clay type stress-dilatancy relation:

where is the volumetric strain increment due to dilatancy, was found to be inaccurate for sands specially for low values of the stress ratio . A new micro-mechanically motivated stress-dilatancy relation was therefore developed:

or,

     This stress-dilatancy relation gives zero dilatancy at  and at 
 

Consolidation Strain

     The volumetric strain  due to changes in mean stress is calculated as:

where B is the bulk modulus. Experimental results also indicate that the bulk modulus is dependent on the mean stress. Such stress dependency can be approximated by the relation:

where  is a material parameter. For many sands, .
 

Pore Pressure Change

     During undrained loading, the total volumetric strain increment:

is equal to zero. Substituting, Eqs. (8) and (9) in Eq. (11) and solving for dp:

Calculation Procedure

     For a given increment of shear strain, , the changes in shear stress ratio  and pore pressure dp are calculated respectively from Eqs. (5) and (12). With these increments, the parameters  are updated. When the sand reaches the steady state (p=p_cr), no changes in stresses are allowed , and the sand continuously deforms at constant stresses. The model requires 7 material parameters: .
 

4.2.- Applications of the Model to a Tailing Dam

     To illustrate the value of the model, the above equations are used to simulate the results of undrained triaxial compression tests on dense Toyoura sand. The tests were conducted over a wide range of confining pressures. Comparisons of the model predictions and the experimental stress-strain curves and effective stress paths for the different tests are shown in Fig. 33.

     In modelling the experimental results, only one set of material properties was used for all the tests (Table 1). Even though only one set of parameters was used, the model predictions fit each of the individual test results rather accurately. The main feature of the numerical results is that it represents very well the softening (decrease in stress ratio ) as the shear strain is increased.
 
 

     The state model was converted to 2D plane strain condition and implemented in the computer code FLAC (Fast Langrangian Analysis of Continua; ITASCA, 1993). FLAC is a dynamic code which uses an explicit time integration scheme and a large strain formulation, and is therefore well suited for calculating dynamic stability problems.

     As an illustration of the use of the state model, a 1:2 (height to width ratio) 7 m slope of a tailings dam built by the dowstream method was used as a case study. The dam was subjected to a 0.1g horizontal acceleration under undrained condition. The material parameters are given in Table 1 except for p_cr which was set to 0.5 MPa to simulate a medium density tailings sand. The results of the analysis are shown at the top of Fig. 34 in terms of the displacement vectors and the contours of horizontal velocities. For comparison, the same slope was analysed using a Mohr-Coulomb model with a failure angle of  and cohesion c = 0 MPa. The results using the Mohr-Coulomb model is shown at the bottom of Fig.34. One can see significant differences in the displacement vectors and the contours of horizontal velocity from the two analyses. The analysis using the state model show large and deeply extending soil mass movement and a potentially liquefiable zone below and to the right of the toe of the slope. The deep extent of soil movement is a direct consequence of the fact that the deeper the sand is, the more contractive it will tend to behave during shearing (due to higher overburden pressure). On the other hand the analysis using the Mohr-Coulomb model shows a shallow zone of potential slip almost parallel to the slope surface. The comparisons of the results from the models illustrate that traditional slope stability analysis based on the so-called c-f approach may not be sufficient in estimating the stability of slopes on potentially liquefiable soils. It is therefore important to accurately model the undrained behaviour of sand at all stages of loading before and up to the steady state.

5.- Conclusions

     Experimental evidence has been presented indicating that the undrained ultimate strength is only function of the initial void ratio, regardless the level of confining stress. In addition it has been shown that the initial fabric only affect the ultimate strength in the case of heterogeneous soil mass.

     Analysis of flow failure only need to accomplish the ultimate strength and the static shear stresses. However, for a complete analysis of the seismic response including the prediction of the level of deformation, it is needed a constitutive model with the incorporation of the pore water pressure development.

     A simple model incorporating the steady state concept was developed for the undrained behaviour of sand. The main feature of the model is the use of a pressure-dependent peak stress ratio parameter which approaches the steady state. The model was shown to be capable of accurately modelling the undrained behaviour of sand over a wide stress region, and was used in the finite element analysis of seismic stability of the dowstream slope of a tailing dam. The slope stability analysis showed the importance of accurately modelling the behaviour of sand, and the inadequacy of the traditional c-f analysis in estimating slope stability.
 

Acknowledgments

     The authors gratefully acknowledge the financial support provided by FONDECYT for Grant No. 1931191 and 1970440 that made possible part of the presented results. Also the author acknowledges the cooperation provided by the Instituto de Investigacion y Ensayes de Materiales, IDIEM, University of Chile.
 

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