Second Canadian Geotechnical Colloquium: Appropriate...

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Second Canadian Geotechnical Colloquium: Appropriate concepts and technology for unsaturated soils D. G. FREDLUND Department of Civil Eilgir~eering,Utliversity oJ'Saskatcl~ewatl,Saskaroorl, Sask., Carzada S7N 0 WO Received October 17, 1978 Accepted October 30, 1978 A practical science has not been fully developed for unsaturated soils for two main reasons. First, there has been the lack of an appropriate science with a theoretical base. Second, there has been the lack of an appropriate technology to render engineering practice financially viable. This paper presents concepts that can be used to develop an appropriate engineering practice for unsaturated soils. The nature of an unsaturated soil is first described along with the accom- panying stress conditions. The basic equations related to mechanical properties are then pro- posed. These are applied to practical problems such as earth pressure, limiting equilibrium, and volume change. An attempt is made to demonstrate the manner in which saturated soil mechanics must be extended when a soil is unsaturated. Two variables are required to describe the stress state of an unsaturated soil (e.g., (o - rr,) and (ti, - u,)). There is a smooth transition from the unsatu- rated case to the saturated case since the pore-air pressure becomes equal to the pore-water pressure as the degree of saturation approaches 100'/c. Therefore, the matrix suction (i.e., (u, - rr,)) goes to 0 and the pore-water pressure can be substituted for the pore-air pressure (i.e., (0 - rr,,)). The complete volumetric deformation of an unsaturated soil requires two three-dimensional constitutive surfaces. These converge to one two-dimensional relationship for a saturated soil. The shear strength for an unsaturated soil is a three-dimensional surface that reduces to the conventional Mohr-Coulomb envelope for a saturated soil. The manner of applying the volumetric deformation equations and the shear strength equa- tion to practical problems is demonstrated. For earth pressure and limiting equilibrium prob- lems, the unsaturated soil can be viewed as a saturated soil with an increased cohesion. The increase in cohesion is proportional to the matrix suction of the soil. For volume change problems it is necessary to have an indication of the relationship between the various soil moduli. There is a need for further experimental studies and case histories to substantiate the pro- posed concepts and theories. - Une science appliquee aux sols non saturCs ne s'est pas complktement dCveloppCe pour deux raisons essentielles. D'abord, il a manquC une base thCorique sufisante. Ensuite, une technolo- gie appropriCe pour dCvelopper des solutions techniques B un coGt raisonnable a fait dCfaut. Cet article prCsente des concepts qui peuvent servir de base B une bonne ingCniCrie des sols non saturCs. On dCcrit d'abord la nature d'un sol non saturC et les conditions de contraintes correspondantes. On propose ensuite les Cquations de base dCcrivant les propriCtCs mCcaniques. On applique ces Cquations ?i la solution de problbmes pratiques tels que la poussee des terres, I'Cquilibre lirnite, et les changements de volume. On tente de montrer comment la micanique des sols saturCs peut Stre Ctendue au cas d'un sol non saturC. Deux variables sont nCcessaires pour dCcrire I'Ctat de contrainte d'un sol non saturC, soit (o - 11,) et (11, - u,,). I1 y a transition continue du cas non saturC au cas satur6 puisque la pression de I'air dans les pores devient Cgale B la pression de l'eau interstitielle lorsque le degrC de saturation approche loo(,& Par consCquent la succion (rr, - I+,) devient nulle et on peut remplacer alors ar, par la pression interstitielle 11,. La dCformation volumique d'un sol non saturC nCcessite deux surfaces constituantes B trois dimensions pour @tre dCcrite complktement. Ces surfaces convergent en une seule relation B deux dimensions pour un sol saturC. La resistance au cisaillernent d'un sol non saturC est rep+ sentCe par une surface B trois dimensions qui se rCduit au critkre de Mohr-Coulomb habitue1 dans le cas du sol saturC. On montre de quelle manikre appliquer les Cquations de dCformation volumique et 1'Cquation de rCsistance au cisaillement B des problbmes pratiques. Pour les problkmes de poussCe des terres et d'Cquilibre limite, le sol non saturC peut &tre considCrC comnle un sol saturC ayant une cohesion accrue, I'accroissernent de cohCsion est proportionnel B la succion du sol. Pour les problkmes de changement de volume il est nCcessaire de connaitre la relation entre les differents modules du sol. Des Ctudes expCrimentales additionnelles et des Ctudes de cas sont nkcessaires pour confirmer les concepts et theories proposes. [Traduit par la revue] Can. Geotech. J., 16, 121-139 (1979) 0008-3674/79/010121-19$01.00/0 01979 National Research Council of Canada/Conseil national de recherches du Canada

Transcript of Second Canadian Geotechnical Colloquium: Appropriate...

Second Canadian Geotechnical Colloquium: Appropriate concepts and technology for unsaturated soils

D. G. FREDLUND Department of Civil Eilgir~eering, Utliversity oJ'Saskatcl~ewatl, Saskaroorl, Sask., Carzada S7N 0 WO

Received October 17, 1978 Accepted October 30, 1978

A practical science has not been fully developed for unsaturated soils for two main reasons. First, there has been the lack of an appropriate science with a theoretical base. Second, there has been the lack of an appropriate technology to render engineering practice financially viable.

This paper presents concepts that can be used to develop an appropriate engineering practice for unsaturated soils. The nature of an unsaturated soil is first described along with the accom- panying stress conditions. The basic equations related to mechanical properties are then pro- posed. These are applied to practical problems such as earth pressure, limiting equilibrium, and volume change.

An attempt is made to demonstrate the manner in which saturated soil mechanics must be extended when a soil is unsaturated. Two variables are required to describe the stress state of an unsaturated soil (e.g., (o - rr,) and ( t i , - u,)). There is a smooth transition from the unsatu- rated case to the saturated case since the pore-air pressure becomes equal to the pore-water pressure as the degree of saturation approaches 100'/c. Therefore, the matrix suction (i.e., (u, - rr,)) goes to 0 and the pore-water pressure can be substituted for the pore-air pressure (i.e., (0 - rr,,)).

The complete volumetric deformation of an unsaturated soil requires two three-dimensional constitutive surfaces. These converge to one two-dimensional relationship for a saturated soil. The shear strength for an unsaturated soil is a three-dimensional surface that reduces to the conventional Mohr-Coulomb envelope for a saturated soil.

The manner of applying the volumetric deformation equations and the shear strength equa- tion to practical problems is demonstrated. For earth pressure and limiting equilibrium prob- lems, the unsaturated soil can be viewed as a saturated soil with an increased cohesion. The increase in cohesion is proportional to the matrix suction of the soil. For volume change problems it is necessary to have an indication of the relationship between the various soil moduli.

There is a need for further experimental studies and case histories to substantiate the pro- posed concepts and theories.

-

Une science appliquee aux sols non saturCs ne s'est pas complktement dCveloppCe pour deux raisons essentielles. D'abord, il a manquC une base thCorique sufisante. Ensuite, une technolo- gie appropriCe pour dCvelopper des solutions techniques B un coGt raisonnable a fait dCfaut.

Cet article prCsente des concepts qui peuvent servir de base B une bonne ingCniCrie des sols non saturCs. On dCcrit d'abord la nature d'un sol non saturC et les conditions de contraintes correspondantes. On propose ensuite les Cquations de base dCcrivant les propriCtCs mCcaniques. On applique ces Cquations ?i la solution de problbmes pratiques tels que la poussee des terres, I'Cquilibre lirnite, et les changements de volume.

On tente de montrer comment la micanique des sols saturCs peut Stre Ctendue au cas d'un sol non saturC. Deux variables sont nCcessaires pour dCcrire I'Ctat de contrainte d'un sol non saturC, soit (o - 11,) et (11, - u,,). I1 y a transition continue du cas non saturC au cas satur6 puisque la pression de I'air dans les pores devient Cgale B la pression de l'eau interstitielle lorsque le degrC de saturation approche loo(,& Par consCquent la succion ( r r , - I+,) devient nulle et on peut remplacer alors ar, par la pression interstitielle 11,.

La dCformation volumique d'un sol non saturC nCcessite deux surfaces constituantes B trois dimensions pour @tre dCcrite complktement. Ces surfaces convergent en une seule relation B deux dimensions pour un sol saturC. La resistance au cisaillernent d'un sol non saturC est rep+ sentCe par une surface B trois dimensions qui se rCduit au critkre de Mohr-Coulomb habitue1 dans le cas du sol saturC.

On montre de quelle manikre appliquer les Cquations de dCformation volumique et 1'Cquation de rCsistance au cisaillement B des problbmes pratiques. Pour les problkmes de poussCe des terres et d'Cquilibre limite, le sol non saturC peut &tre considCrC comnle un sol saturC ayant une cohesion accrue, I'accroissernent de cohCsion est proportionnel B la succion du sol. Pour les problkmes de changement de volume il est nCcessaire de connaitre la relation entre les differents modules du sol.

Des Ctudes expCrimentales additionnelles et des Ctudes de cas sont nkcessaires pour confirmer les concepts et theories proposes.

[Traduit par la revue] Can. Geotech. J., 16, 121-139 (1979)

0008-3674/79/010121-19$01.00/0 01979 National Research Council of Canada/Conseil national de recherches du Canada

122 CAN. GEOTECH. J. VOL. 16, 1979

Introduction The success of the ~ract ice of soil mechanics can

be traced largely to the ability of engineers to relate observed soil behavior to stress conditions. This ability had led to the transmissibility of the science and a relatively consistent engineering practice. Al- though this has been true for saturated soils, such has not been the case for unsaturated soils.

The general field of soil mechanics can be sub- dividedinto that portion dealing with saturated soils and that portion dealing with unsaturated soils (Fig. 1). This differentiation is necessary because of their basic difference in nature and behavior. Whereas a saturated soil is a two-phase system, an unsaturated soil with a continuous air phase is a four-phase system (Fredlund and Morgenstern 1977) (Fig. 2). The air phase generally becomes continuous when the degree of saturation is less than approximately 85-90%. The four phases of an unsaturated soil are (i) solids, (ii) water, (iii) air, and (iv) contractile skin (or air-water interface). Soils commonly falling into this category are the natural, desiccated soils and compacted soils. The pore-water pressures of these soils are generally negative and the most prevalent problems are related to their expansion and shrink- age.

There has been difficulty in extending soil mechan- ics to embrace unsaturated soils. This has been borne out by the empirical nature of most research asso- ciated with unsaturated soils.

The question can be asked: "Why hasn't a practical science developed and flourished for unsaturated soils?" A cursory examination may suggest there is not a need for such a science. However, this is not the case when the problems associated with expansive soils are considered. Jones and Holtz (1973) reported that in the United States alone: "Each year, shrink-

ing and swelling soils inflict at least $2.3 billion in damages to houses, buildings, roads and pipelines- more than twice the damage from floods, hurricanes, tornadoes and earthquakes!" They also reported that 60% of the new houses built in the United States will experience minor damage during their useful lives, that 10% will experience significant damage- some beyond repair. In addition there is the need for reliable engineering design associated with the use of com~acted soils.

There appears to be two main reasons why a practical science has not developed for unsaturated soils (Holtz et al. 1974). First, there has been the lack of an appropriate science with a theoretical base. This commences with a lack of appreciation of the engineering problems and an inability to place the solution within a theoretical context. The stress con- ditions and mechanisms involved as well as the soil properties that must be measured have not been fully understood. The boundary conditions for an analysis are generally related to the environment and are difficult to predict. Research work has largely re- mained empirical in nature with little coherence. As a result there is poor communication amongst en- gineers and design procedures are not widely accepted and adhered to.

Second, there is the lack of a system for financial recovery for services rendered by the engineer. In the case of expansive soils problems, the possible liability to the engineer is often too great relative to the financial remuneration. Other areas of practice are far more profitable to consultants. The owner often reasons that the cost outweighs the risk. The hazard to life and injury is largely absent and for this reason little attention has been given to the problem by government agencies. Although the problem basically remains with the owner, it is the engineer

SOlL MECHANICS + SOlL MECHANICS

UNSATURATED / SOlL MECHANICS

FOUR - PHASE SYSTEM + U, GENERALLY

< 0

FIG. I . Categories of soil mechanics. *Dry sands are two-phase systems. * * Soils with occluded air bubbles are two- phase systems with a compressible pore fluid. ***Can be either swelling or collapsing soils.

FREDLUND 123

AIR

( A I R -WATER INTERFACE ) // SOIL PARTICLE

CONTRACTILE SKIN

FIG. 2. An element of unsaturated soil (continuous air phase).

who has the greatest potential for circumventing possible problems.

This paper will address the need for an appropriate technology for unsaturated soil behavior. Such a technology must: (i) be practical, (ii) not be too costly to employ, (iii) have a sound theoretical basis, and (iv) run parallel in concept to conventional saturated soil mechanics.

A macroscopic, phenomenological approach to unsaturated soil behavior holds the greatest potential for satisfying the above conditions. In other words, the science is developed around observable pheno- mena while adhering to multiphase continuum mechanics principles. This has proven to be the most successful approach for saturated soils and should be retained for unsaturated soils. Therefore, un- saturated soil behavior can be viewed as an extension of saturated soil mechanics and retain a smooth transition in rationale between the two cases.

This paper will propose the basis of an appropriate technology for unsaturated soil behavior. First, the nature of an unsaturated soil is described along with its stress conditions. Then the basic equations related to the mechanical properties are proposed. Finally, these are applied to earth pressure, limiting equilib- rium, and volume change problems. An attempt will be made to demonstrate the extension of the theory for saturated soil conditions to that for unsaturated soils. Due to the broad scope of the paper, each aspect will be dealt with in a cursory manner.

Nature of Unsaturated Soils

Most soil deposits are originally saturated. Lacus-

trine deposits, for example, are deposited at water contents above the liquid limit and then are con- solidated by the weight of the overlying sediments (Fig. 3). The drying up of the lake and subsequent evaporation of water commence desiccation of the sediments. The water table is drawn below the ground surface. The total stress on the sediments re- mains essentially constant while the pore-water pressure is reduced. The pore-water pressure becomes negative with respect to atmospheric pressure above the water table giving rise to consolidation and eventually desaturation of the sediments. Grasses, trees, and other plants start to grow on the surface, further drying out the soil by applying a tension to the water phase through evapo-transpiration. Most plants are capable of applying 1-2 MPa (10-20 atm) of tension to the water phase prior to reaching their wilting point. Evapo-transpiration results in further consolidation and desat~iration of the soil.

The tension in the water phase acts in all directions and can readily exceed the lateral confining pressure in the soil mass. At this point, a secondary mode of desaturation commences (i.e., cracking). When a soil is remolded in the compaction process, desatura- tion is also the result of artificially subjecting the soil structure to tensile stresses.

Year after year the deposit is subjected to varying and changing environmental conditions. These pro- duce changes in the pore-water pressure distribution, which in turn results in shrinking and swelling of the soil deposit. As a result of environmental changes, the pore-water pressure distribution can take on a wide variety of shapes (Fig. 3).

Generally, an unsaturated soil is considered to be a three-phase system. However, on the basis of the def- inition of a phase, the air-water interface should be considered as a fourth and independent phase. The air-water interface, commonly referred to as the 'contractile skin', qualifies as a phase since it has (i) differing properties from that of the contiguous materials and (ii) definite bounding surfaces.

The most distinctive property of the contractile skin is its ability to exert a tensile pull. It behaves like an elastic membrane under tension interwoven throughout the soil structure. It appears that most properties of the contractile skin are different from those of the contiguous water phase (Davies and Rideal 1963). For example, its density is reduced, its heat conductance is increased, and its birefringence data are similar to those of ice. The transition from the liquid water to the contractile skin has been shown to be distinct or jumpwise (Derjaguin 1965). It is interesting to note that insects such as the 'water strider' walk on top of the contractile skin and those

CAN. GEOTECH. J. VOL. 16, 1979

EVAPORATION EVAPO - TRANSPIRATION

SATURATION I \

EQUILIBRIUM WITH WATER TABLE

EXCESSIVE EVAPORATION

DEPOSITION

2 FLOODING OF DESICCATE- SOIL a

V) 0 I l- a

TOTAL STRESS PORE - AIR PORE -WATER ( u ) PRESSURE PRESSURE

( U o ) (U, )

FIG. 3. Stress distribution during d e s i c c a t i o n of a soil.

FIG. 4. Insects that l ive above and below the contractile skin (from M i l n e and Milne 1978): (a) water strider; (b) back- swimmer.

such as the 'backswimmer' walk on the bottom of the contractile skin (Milne and Milne 1978). The water strider would sink into the water were it not for the contractile skin whereas the backswimmer would pop out of the water (Fig. 4).

It is imperative to recognize an unsaturated soil as a four-phase system when performing a stress analysis on an element (Fredlund and Morgenstern

1977). From a behavioral standpoint, an unsaturated soil can be visualized as a mixture with two phases that come to equilibrium under applied stress gra- dients (i.e., soil particles and contractile skin) and two phases that flow under applied stress gradients (i.e., air and water).

From the standpoint of the volume-weight rela- tions for an unsaturated soil, it is possible to consider

FREDLUND

0 10 2 0 30 4 0 50

WATER CONTENT ( % )

FIG. 5. Volume-weight relations for unsaturated soils.

the soil as a three-phase system since the volume of the contractile skin is small and its weight can be considered as part of the weight of water. All the commonly used volume-weight relations (for a selected specific gravity of solids) are shown in Fig. 5. The three most basic volume-weight variables are the void ratio, water content, and degree of satura- tion. Although relations between these basic variables are fixed for saturated soils, such is not the case for unsaturated soils. For an unsaturated soil, a change in volume-weight conditions can involve changes anywhere below the 100% saturation line. It now becomes necessary to know any two of the basic volume-weight variables to describe all volume- weight relations. This also applies for changes in one or two of the volume-weight variables. Therefore, two constitutive relations are required to describe the volume change associated with all phases of an un- saturated soil whereas only one is required for a saturated soil. This can also be expressed in terms of the volumetric continuity requirement for a referen- tial element of unsaturated soil.

111 AV/V = AV,/V + AV,/V

where V = total volume of the element of soil, V, = volume of water, and V, = volume of air.

Stress State Numerous so-called 'effective stress' equations

have been proposed for unsaturated soils (Bishop

1959; Croney et al. 1958; Lambe 1960; Aitchison 1961 ; Jennings 1961 ; Richards 1966). Common to all proposed equations is the incorporation of a soil parameter to couple together more than one stress variable. However, the description of the stress state should consist of independent stress variables, the number being dependent on the number of phases of the material. For example, the stress tensor for a one phase solid such as steel is the same as for "Jello". Likewise the effective stress variable for a saturated soil is independent of the physical properties of the soil.

Recently, two independent stress tensors have been proposed for unsaturated soils (Fredlund and Mor- genstern 1977). These can be supported by a stress analysis consistent with that used in multiphase continuum mechanics. The assumptions in the stress analysis are that the solids are incompressible and the soil is chemically inert. These assumptions are comparable to those associated with the stress state description for a saturated soil.

The stress analysis shows that any two of three possible stress variables can be used to describe the stress state of an unsaturated soil. Possible com- binations are: (1) (o - u,) and (u, - u,), (2) (o - u,) and (11, - u,), and ( 3 ) (a - u,) and (o - u,), where o = total stress, u, = pore-air pressure, and u, = pore-water pressure. After attempts to apply the stress state variables to earth pressure problems, shear strength problems, and volume change prob-

CAN. GEOTECH. J. VOL. 16, 1979

FIG. 6. The stress state variables for an unsaturated soil.

lems, the author has concluded that the (o - 11,) and (us - uw) combination is the most satisfactory. This combination is advantageous since the effects of total stress changes and pore-water pressure changes can be separated. This is beneficial both from a concep- tual and analytical standpoint. (The air pressure for most practical problems is atmospheric.)

The complete stress state at a point is shown in Fig. 6. Only the (o - 1 4 and (u, - 11,) combination will be used in the remainder of this paper. The (o - 11,) term is called the total stress1 and the (u, - 11,") term is called the matrix suction.

The two independent stress tensors are represented by

( 0 - 1 , 7x11

Tux (09 - lla) Zzx 7t.V (oz - u,)

for the first stress tensor and by

0 (11, - 11~) 0

for the second stress tensor. The first stress invariants of the first and second

stress tensors, respectively, are

[41 I11 = 01 + 0 2 + o3 - 3u:, I1 2 = 3(u, - u,)

The second stress invariants of the first and second stress tensors, respectively, are

'The term "total stress" is used to represent o relative to 11,.

The third stress invariants of the first and second stress tensors, respectively, are

where ol = major principal stress, oz = intermediate principal stress, and o3 = minor principal stress. Since the stress invariants of the second stress tensor are related, only one invariant is required. Therefore, the three-dimensional stress state can be described in terms of four stress invariants for an unsaturated soil as opposed to three stress invariants for a saturated soil.

The above stress state variables also provide a smooth transition in stress description when going from the unsaturated to the saturated soil case. As the degree of saturation approaches loo%, the pore- air pressure approaches the pore-water pressure. Therefore, the matrix suction term goes to 0 and the pore-air term in the first stress tensor becomes the pore-water pressure.

The state of stress for an unsaturated soil can be plotted on an extended Mohr type of diagram where the third orthogonal axis has been added to represent the matrix suction (Fig. 7).

From thermodynamic or total energy considera- tions of the pore water, its total suction, h, can be subdivided into a matrix suction component and an osmotic component, n (Aitchison and Richards 1965),

The relationship has been experimentally verified (Fig. 8) and changes in the osmotic component with water content are relatively small (Krahn and Fred- lund 1972). It is possible to use total suction as a substitute for the matrix suction variable. In this case either the changes in osmotic suction must be assumed to be small or else simulated between the field and the laboratory.

It is diflicult to measure matrix suction in the field due to cavitation of the measuring system at pore- water stresses approaching 100 kPa (1 atm) negative. In the laboratory these difficulties are circumvented using the axis-translation technique.

Recently psychrometers, which measure total suction, have received increased usage in evaluating the suction of a soil in the laboratory and the field (Fig. 9), The low cost of the psychrometers and the measuring equipment makes them attractive for engineering usage.2 However, their use is restricted

2Manufacturers of psychrometers are: Wescor, Inc., 459 South Main Street, Logan, UT 84321; and Emco, P.O. Box 34, Angola, IN 46703.

FREDLUND

FIG. 7. Extended Mohr diagram for unsaturated soils.

to soils with a suction greater than 100 kPa (1 atm).

Physical Properties

Various types of constitutive relations are required in soil mechanics. Each equation requires soil proper- ties that must be experimentally evaluated. Some of

0 TOTAL SUCTION ( PSYCHROMETER )

MATRIX SUCTION (PRESSURE PLATE )

A OSMOTIC SUCTION ( SQUEEZING TECHNIQUE 1

\ - OSMOTIC PLUS MATRIX SUCTION

0 1 I I I I I I 2 0 2 2 2 4 2 6 28 3 0 32

WATER CONTENT ( X )

FIG. 8. Total, matrix, and osmotic suction for Regina clay.

the constitutive relations required for an under- standing of unsaturated soil behavior are listed in Table 1. The details of the derivations associated with each type of constitutive relation are presented in the references shown in Table 1.

Stress State Variables Versus Volume- Weight Relations

The most basic and important constitutive equa- tions are those relating the stress state variables and the volume-weight properties. Equations are re- quired for each of two phases of an unsaturated soil. The soil structure (or overall element) and the water phase are selected for presenting the constitutive relations. The soil is first assumed to behave as an isotropic, linear elastic material. The constitutive relations can be developed in a semi-empirical man- ner as an extension of the elasticity formulation used for saturated soils as follows.

Soil structure:

where E,, E ~ , E, = strain in the x, y, and z directions; El = Young's modulus with respect to (o - ~ a ) ;

PI = Poisson's ratio; and HI = elastic modulus with respect to (ua - u,).

128 CAN. GEOTECH. J. VOL. 16, 1979

FIG. 9. Soil psychrometer (compliments of Wescor, Inc.). Scale: 1 in. = + in.

Water phase:

where 0 , = change of volume of water in element

(i.e., AV,/V); HI1 = water phase modulus with respect to (o - u,); and R I = water phase modulus with respect to (u, - u,).

In order to retain consistency with saturated soil mechanics, the void ratio change will be used to represent the strain in the soil structure. Also, the matrix suction will be plotted in the same units as the total stress.

For one-dimensional or KO loading conditions, the arithmetic plot of void ratio versus stress state variables can be presented as a three-dimensional surface (Fig. 10). Generally, the surface will be non- linear; however, coefficients of compressibility can be defined for small stress increments. The slopes of particular interest are those associated with each of the stress state variables. Let the coefficient of com- pressibility with respect to the total stress be

[lo] at = - de/d(o - u,)

and the coefficient of compressibility with respect to matrix suction be

For corresponding stress ranges, at will approxi- mately equal a, when the degree of saturation is near 100%. There will be a slight difference in the co- efficients of compressibility since at is related to KO loading whereas a, is related to isotropic loading. As the degree of saturation decreases it has been found that at will become greater than a,. This shows that a change in (o - u,) is more effective in changing void ratio than a change in (u, - u,). In the plane of zero matrix suction (i.e., the saturation plane), the pore-air pressure becomes equal to the pore-water pressure and the coefficient of compres- sibility with respect to total stress is equal to the co- efficient of compressibility when the soil is saturated (i.e., a,). Coefficients of compressibility correspond- ing to rebound portions of the surface can be further subscripted with an s (i.e., at, and a,,). The proposed constitutive surface has been experimentally tested for uniqueness near a point (Fredlund and Morgen- stern 1976) and for uniqueness when larger stress increments are used (Matyas Radhakrishna 1968; Barden et al. 1969). The results indicate uniqueness as long as the deformation conditions are monotonic.

The constitutive surface can be linearized over a wider range of stress changes using the logarithm of the stress state variables (Fig. I I). The soil properties of interest are the compressive index with respect to the total stress, Ct, and the compressive index with respect to matrix suction, C,. The void ratio under

3For engineering purposes this procedure is superior to the p F representation of soil suction commonly used in soil science.

FREDLUND

L

MATRIX SUCTION

FIG. 10. Arithmetic representation of stress state variables versus void ratio.

TABLE 1. Types of constitutive relations

TY ~e Description Reference

Stress vs. stress

Stress vs. volume-weight (I) Relates the stress state variables to strains, deformations, Fredlund and Morgenstern 1976 and volume-weight properties such as void ratio, water content, and degree of saturation

(2) Density equations for air-water mixtures Fredlund 1976 (3) Compressibility equations for air-water mixtures Fredlund 1976

(1) Pore pressure parameters relating the norn~al stress Hasan and Fredlund 1977 components for undrained loading conditions

(2) Strength equations relating shear strength to stress state Fredlund et al. 1978 variables

Stress gradient vs. velocity (1) Flow laws for the pore air and pore water Hasan and Fredlund 1977

any set of stress conditions can be written in terms of an extended form of the equation commonly used for saturated soils:

[12] e = e o - C t l o g ((T - ~a)f la - 1lw)f - C,, log (0 - &)o (& - llw)~

where the f subscript represents final stress state and 0 represents the initial stress state. The range of appli- cability of the equation may be somewhat controlled by stress history. Once again, Ct is approximately equal to C,, when the degree of saturation approaches 100yo and Ct becomes equal to the conventional compressive index, C, (Fig. 12). At lower degrees of

saturation, Ct will be greater than C,. The quantita- tive relationship between the compressive indices must be evaluated experimentally. Unfortunately, the literature contains little if any complete data on compressive indices for soils with low degrees of saturation.

A similar constitutive surface exists for the water phase. Once again the nonlinear arithmetic plot of water content versus the stress state variables can be linearized on a logarithm of stress plot (Fig. 13). Let us define the coefficient of water content with respect to the total stress as

CAN. GEOTECH. J. VOL. 16, 1979

TOTAL STRESS

FIG. 11. Logarithm of stress state variables versus void ratio.

MATRIX SUCTION TEST RESULTS

8 0 - - ONE -DIMENSIONAL CONSOLIDATION RESULTS-

NOTE : NUMBERS IN BRACKETS INDICATE PRECONSOLIDATION - PRESSURE

60 - -

-

4 0 - -

I I 3 0 0.01 0 . I I .o 10

MATRIX SUCTION AND TOTAL SUCTION , k P a

FIG. 12. Comparison of recompression curves for matrix suction and total stress loading of initially saturated Regina clay (Fredlund 1964).

and the coefficient of water content with respect to sample. When bt is measured on the saturation plane, matrix suction as it becomes equal to at or a, provided bt is multiplied

~141 /I, = - d ~ / d ( ~ ~ , - u , ~ ) by the specific gravity of the soil solids, G,. On a logarithm of stress plot, the water content under any

At 100% saturation, bt approaches 6,. At lower set of stresses can be written as degrees of saturation, b, will generally be greater than bt since stress applied directly to the water phase

[I51 w = wo - Dtlog ( c - u J ~ - (lla - ~ w ) f

will be more effective in removing water from the (0 - &)o Og (Ua - UW)O

FREDLUND

FIG. 13. Arithmetic representation of stress state variables versus water content.

where w o = initial water content, Dt = water content index with respect to total stress, and Dm = water content index with respect to matrix suction.

Relatiotlship between Mocluli Four compressibilities or indices have been defined

to completely describe the volume-weight versus stress state variables for an unsaturated soil. The moduli are also different for increases and decreases of the stress state variables. All moduli tend towards one value for a saturated soil. For an unsaturated soil, four moduli are too much to evaluate on a routine basis for engineering purposes. Therefore a knowledge of the relationship of one modulus to another is highly advantageous.

First, the testing of a soil in the total stress plane (i.e., matrix suction equal to 0 ) can be accomplished in the conventional triaxial or one-dimensional con- solidation apparatus (Fig. 14). The two moduli on the total stress plane are approximately equal (i.e., at and bt). Note that the modulus with respect to water content must be multiplied by the specific gravity of the solids, Gs, in order for it to equal the modulus with respect to void ratio. This reduces the four unknown moduli to three.

Second, at all degrees of saturation, there is a relationship between the volume-weight properties as defined by the two constitutive surfaces.

Changes in any one of the variables can produce

changes in the other two variables. The change in water content can be written in terms of a change in degree of saturation and void ratio.

Likewise, a change in void ratio can be written as

If the change in degree of saturation is known or can be estimated, there is a fixed relationship between the modulus with respect to void ratio and water content. This in essence reduces the number of unknown moduli to two. For problems such as the prediction of heave, the final degree of saturation will be close to 100yo. Therefore, an analysis for the prediction of heave can be done in terms of a change in water content and the corresponding change in void ratio can be computed. In other words, there is a relationship between the moduli with respect to water content and void ratio.

For the case where the change in water content is predicted, let the ratio of the void ratio modulus with respect to the water content modulus, c,", be defined as follows:

where b = any coefficient of compressibility on the water content versus stress state variable plot and a = the corresponding coefficient of compressibility

132 CAN. GEOTECH. J. VOL. 16, 1979

e A little or no information is available on the relative FREE SWELL TEST L STRESS effects of the two stress state variables. This is an

area of much needed research.

Shear S tmzgth The shear strength of an unsaturated soil can be

written in an extended form of the Mohr-Coulomb diagram for a saturated soil (Fredlund et al. 1978).

[21] z = cf + (o - u,) tan $' + (u, - u,J tan $b

FIG. 14. Conventional consolidation test on an unsaturated soil.

on the void ratio versus stress state variable plot. Equation [19] is presented graphically in Fig. 15. For the case where the change in void ratio is

predicted, let the ratio of the water content modulus with respect to the void ratio modulus, c,, be defined as follows:

A shrinkage limit type of test also gives the re- lationship between the void ratio and water content constitu&e surfaces for the special case of changes in matrix suction when the external loads are 0 (Fig. 16). The specific bulk volume is the inverse of the dry density and is a measure of the void ratio. The slope of the shrinkage curve can be related to the ratio of the void ratio coefficient to the water content coefficient of compressibility.

Third, information is required on the relationship between the relative effects of a change in total stress and matrix suction. If this were known, the number of unknowns would be reduced to one. However,

where c' = cohesion intercept when the two stress variables are 0, 4' = the friction angle with respect to changes in the (o - u,) stress variable, and =

the friction angle with respect to changes in the (u, - 11,) stress variable.

Conceptually the equation can be visualized as a three-dimensional plot with matrix suction plotted on the third axis (Fig. 17). As the soil approaches 100% saturation, u, approaches 11, and the matrix suction component becomes 0, leaving the remainder of the equation equivalent to that for a saturated soil. The c' and 4' parameters can be evaluated in the conventional manner used for saturated soils.

The $ b can be computed by first plotting the dif- ference between the strength of the unsaturated and saturated soil (under similar net total confining pressures) versus matrix suction (Fredlund et al. 1978). This can be done using a stress point type of analysis where the change in the strength, Azd (corrected to a distance perpendicular to the failure plane), is plotted against matrix suction (Fig. 18). The equation of the resulting line is

where yf' = an angle similar to 4' but defined using the stress point procedure (i.e., sin $' = tan \y') and a = the angle between the change in shear strength and matrix suction (using the stress point method of analysis).

The angle cr. can be first converted t o a correspond- ing angle on the extended Mohr-Coulomb failure envelope and then to the friction angle with respect to matrix suction, $b.

The angle, $" can be determined for a soil when the shear strength (and matrix suction) is known at a matrix suction greater than zero. Few sets of data are available that allow a complete analysis for un- saturated soils. Typical results are presented in Table 2 from data obtained by Bishop et al. (1960).

The shear strength parameters indicate that a de- crease in pore-water pressure in an unsaturated soil

FREDLUND

C O L L A P S I N G

FIG. 15. Compressibility ratios versus volume-weight properties.

WATER CONTENT , PERCENT

(a, +a3 ) / 2 - u o

FIG. 17. Shear strength envelope for unsaturated soils.

It is possible to envisage the three-dimensional shear strength diagram as a series of two-dimensional Mohr-Coulomb type diagrams, one for each matrix suction (see Fig. 19). Therefore, the cohesion inter- cept is changed for each matrix suction while the friction angle remains equal to that on the saturation plane.

PI c = C' + (ua - u,") tan +b

FIG. 16. Specific bulk volume versus water content for where c = the cohesion intercept at a particular Regina clay. matrix suction.

Conceptually, an unsaturated soil has the same is not as effective in increasing shear strength as an internal angle of friction as a saturated soil. At in- increase in the total confining stress. creasing matrix suction values, the cohesion (or

CAN. GEOTECH. J. VOL. 16, 1979

FIG. 18. Change in shear strength due to matrix suction.

TABLE 2. Typical shear strength parameters for unsaturated soils

Soil description c' 4' 4° Initial water content (%)

Compacted shale 2.3 24.8 20.9 18.6 Boulder clay 1.4 27.3 24.0 11.6

(u,,-Ua) * Y h I: K p ( U , - U a ) 1 FIG. 19. Active and passive earth pressures for unsaturated soils.

internal isotropic confining pressure) is increased. constitutive relationships useful for describing the This treatment of the shear strength of unsaturated behavior of unsaturated soils. Therefore, an attempt soils is particularly useful when considering in situ has been made to describe those constitutive relations stress conditions and when performing limit equilib- of greatest interest. These relations will now be rium types of analysis. applied to earth pressure, limit equilibrium, and

volume change problems in unsaturated soils. Their Theoretical Unsaturated Soil Mechanics treatment will be mainly dealt with from a conceptual

In this paper it is not possible to discuss all the standpoint.

FREDLUND 135

Earth Presslnes Some of the earliest work in saturated soil mech-

anics dealt with earth pressures on retaining walls. However, there is little information on the earth pressures exerted on engineering structures by un- saturated soils. Swelling pressures exerted by expan- sive soils have been of concern but there has been lacking a general earth pressure theory for these soils.

The total vertical stress in a large, level mass of soil is computed in the same manner for both a saturated and unsaturated soil. The pore-air pressure is generally in equilibrium with atmospheric pressure. The pore-water pressure must be measured unless the water table is within the height of capillary rise.

The horizontal pressure can be written as a ratio of the vertical pressure. Let the coeficient of earth pressure at rest, KO, be defined as

The earth pressure at rest can be estimated assum- ing the soil is in a state of elastic equilibrium. Using the elasticity form of the constitutive relations for an unsaturated soil ([8]) and assuming the soil is laterally confined, the lateral pressure can be written as a ratio of the vertical stress:

[261 ( ~ 1 , - I!,) -

Km (ua - 1lw)

(0" - ua) 1 - p (0" - 11%)

where Km = E,/(l - p)H,. At saturation, the pore-air pressure approaches

the pore-water pressure and the equation reverts to that for a saturated soil. When the matrix suction becomes positive, the horizontal stress is reduced and is a function of depth. At shallow depths, the horizontal stresses will go to 0 and attempt to go negative. This will result in cracking of the soil commencing at ground surface.

The active and passive earth pressures of an un- saturated soil can be determined assuming the soil is in a state of plastic equilibrium. The active earth pressure of an ~~nsaturated soil is reduced from that of a saturated soil while the passive earth pressure is increased. The amount of the change in pressures can be visualized on a three-dimensional Mohr- Coulomb type of diagram (Fig. 19). The active earth pressure coefficient for any matrix suction can be written in a form similar to that for a saturated soil exhibiting both friction and cohesion:

where c = c' + (u, - I(,") tan $b.

The passive earth pressure coefficient can be written as

u,) tan2 (45 + $) + 2c tan (45 + $) In other words, the matrix suction affects the active

and passive earth pressure diagrams in the same manner as a soil having an increased cohesion (Fig. 20).

When the soil behind a retaining wall becomes moistened, the matrix suction is reduced and the pressure against the wall is increased. If the wall is allowed to move laterally, the final pressure upon saturation would correspond to that dictated by the saturation plane of the Mohr-Coulomb diagram. If no movement of the wall is allowed, the pressure against the wall would increase; however, it would not exceed the passive pressure state on the satura- tion plane. Other cases of deformation would have to be considered as a soil structure interaction problem.

Litniting Equilibriun~ Two common problems handled using limit

equilibrium analysis are slope stability and bearing capacity. In a slope stability analysis, the factor of safety is a measure of the reduction in shear strength required to produce a state of limiting equilibrium. For bearing capacity problems, the factor of safety is specified and the allowable bearing capacity is computed. In both of these cases, an un- saturated soil can be treated as a frictional soil with an increased cohesion intercept ([24]). Figure 21 demonstrates the increase in factor of safety for a soil with an increasing matrix suction.

From a theoretical standpoint, the conventional slope stability and bearing capacity charts can be used for unsaturated soils. At present, there is need for both experimental results and case histories to substantiate the theoretical conclusions.

Volurne Change The most common volume change problem asso-

ciated with unsaturated soils is the heave of light engineering structures. Often such structures are placed on a desiccated soil. Subsequent to construc- tion, moisture accumulates beneath the structure

(oh - lla) - resulting in a decrease in matrix suction and related [27] K, = - heave. The engineer desires to sample the soil at

(0" - la) various depths prior to construction and to be able

(o, - ua) tan2 45 - - - 2c tan 45 - -i- to predict the amount of heave likely to occur under ( ) ( various loading conditions. Figure 22 shows how the

CAN. GEOTECH. J. VOL. 16, 1979

SATURATED SOIL SATURATED SOIL

ACTIVE PRESSURES PASSIVE PRESSURES

FIG. 20. Active and passive earth pressures versus depth.

0.8 1 I I I I I 0.0 5 10 I5 2 0 25

I COHESION INTERCEPT ( k Po 1

I I I I I I I 0 10 2 0 3 0 4 0 SO 6 0 70

(Ua-U,) k P a

FIG. 21. Factor of safety of a simple slope versus matrix suction.

ya MATRIX SUCTION

LOG ( u - U , )

FIG. 22. Initial stress state when soil has undergone a drying and wetting history.

soil sampling can occur anywhere along a drying or wetting portion of the matrix suction plane. In the laboratory, the engineer would like to be able to re-establish the field state of stress and also obtain the moduli required for a heave analysis.

FREDLUND

(a) NEGATIVE o POSITIVE

LAYER 2 - - - DESICCATED , \ \UNSATURATE0 CLAY LAYER 3 - - - \ ' I \ LAYER 4 - - -

I N I T I A L TOTAL STRESS

I N I T I A L PORE - A I R \ WATER PRESSURE 4 PRESSURE

(b) 0 ASPHALT r\

GRAVEL I \ \

LAYER I

F I N A L TOTAL STRESS

LAYER 3 - - - - LAYER 4 - - - -

HYDROSTATIC WATER

WATER TENSION ZERO PRESSURE

FIG. 23. Initial and final boundary conditions for heave analysis. (a) Initial boundary conditions. (b) Final boundary conditions.

An analysis for the prediction of total heave has three aspects (Fredlund 19756). First, the initial stress boundary conditions must be evaluated (Fig. 23a). The main variable of interest is the variation of pore- water pressure at various depths. The initial pore- water hressure can be evaluated by means of psy- chrometric measurements, axis-translation matrix suction tests or oedometer tests on undisturbed samples. The psychrometer is generally used in the lab6ratory; however, it can alsobe used in sitri where suctions are greater than 100 kPa (1 atm) and temper- ature variations are small. The psychrometer mea- sures total suction, which must be compensated for the osmotic suction component. When laboratory measurements are used, the suction must be divided into that portion due to overburden removal and that due to the in situ suction. This involves con- sideration of the pore pressure parameters that are outside the scope of this paper. The matrix suction measurement using the axis-translation technique on a high air-entry disc must also be corrected for that poFtion due to overburden unloading.

Two types of oedometer tests are commonly per- formed on expansive soils. These are referred to as the free swell and constant volume consolidation tests (Fredlund 1969). In either of these tests, filter paper should not be placed on each side of the sample

VOID RATIO

e o

COMPENSATION 1 ,FOR SAMPLE I %< UR8ANCE - --A----

CORRECTION FOR COMPRESSIBILITY OF APPARATUS \\

FIG. 24. Interpretation of the c o n s t a n t volume oedometer test.

and the test results should be corrected for the com- pressibility of the apparatus. As well, the effects of sample disturbance should be taken into account' This correction is not as significant for the free swell test as for the constant volume test. However, the constant volume test with a sample disturbance cor- rection appears to better represent it1 situ stress conditions (Fredlund 1975a). A Casagrande type of construction (Terzaghi and Peck 1948) appears to be satisfactory for compacted soils (Fig. 24); however, a detailed study has not been done on the best pro- cedure to compensate for sample disturbance in undisturbed, natural samples.

When a correction for sample disturbance is applied to the consolidation test results, the hori- zontal distance out to the constructed point of volume decrease is a measure of the overburden pressure plus the matrix suction. However, the matrix suction measured in the laboratory will be smaller than the in situ matrix suction since it is measured on the total stress plane. The laboratory measurement of matrix suction will be satisfactory for analysis purposes if the deformation moduli are also measured on the total stress plane.

Second, the final stress boundary conditions must be estimated. The final or equilibrium pore-water pressure is mainly dependent on the geographical and climatic environment. Generally, one of three as-

138 CAN. GEOTECH. J. VOL. 16. 1979

sumptions is made regarding the final pore-water Therefore, there is a fixed relationship between the pressures (Fig. 23b). change in void ratio and the change in water content

(i) It could be assumed that moisture will accum- (Fig. 15). ulate beneath the structure and eventually the water table will rise to some elevation near the surface. Summarv Therefore, the final pore-water pressure distribution is hydrostatic. This assumption will result in the largest predicted heave.

(ii) It could be assumed that the pore-water pres- sure is 0 with depth. Although this is an unsteady state final boundary condition, it may be applicable when considerable moisture has been made available from the surface of the soil.

(iii) It could be assumed that the final pore-water boundary condition is a state of slight tension in the water phase. The value of the pore-water tension is related to the soil type and climatic conditions (Aitchison and Richards 1965; Carpenter et al. 1974). Third. suitable constitutive relations are necessarv

to relate the initial and final stress boundary con: ditions. Since it is the amount of heave that is to be predicted, the main constitutive relationship of in- terest is the void ratio eauation. The void ratio change can be summed fo; increments of depth to get the total heave, Ah.

where i = soil layer numbers ranging from layer 1 to layer / I , hi = thickness of the ith soil layer, and ei = void ratio for the ith layer.

The change in void ratio, Aei, can be written as

[30] Aei = Ctlog ((J A Un)f (~ln - &)f + c m log (0 lla)~ (lln - ~w)o

Using the interpretation of the oedometer test pre- viously suggested, the stresses become additive since the two moduli are the same.

The soil properties must be compatible with the loading increase or decrease being considered. A de- tailed recommended procedure for a heave analysis will not be presented; however, any procedure for heave should be consistent with the general theoreti- cal content presented.

If the change in water content is also of interest, the water content constitutive relationship can be used. It is also possible to predict the change in water content by some analysis and then relate this to a change in void ratio (Johnson 1977). For the heave analysis, the degree of saturation generally changes from some initial value up to approximately 100%.

The paper has presented some of the basic con- cepts and technology required to place engineering practice associated with unsaturated soils on a science base similar to that available for saturated soils. The unsaturated soil is recognized as a four- phase system with two phases that come to equilib- rium under the application of a stress gradient (i.e., soil particles and contractile skin) and two phases that flow under the application of a stress gradient (i.e., air and water). The necessary stress variables for describing the stress state are presented, followed by proposed volume change and shear strength con- stitutive relations. These relations are applied t o earth pressure, limit equilibrium, and volume change problems. In all cases, unsaturated soil mechanics is viewed as an extension of saturated soil mechanics while still retaining a smooth transition between the two cases. Further research is required to expand the proposed concepts to other problems associated with unsaturated soils.

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FREDLUND 139

DERJAGUIN, B. V. 1965. Recent research into the properties of water in thin films and in micro-capillaries. Society for Experimental Biology Symposia, XIX. The State and Move- ment of Water in Living Organisms. Cambridge University Press, London, England.

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