Evaluation of the coefficient of earth pressure at rest (K0) of a saturated-unsaturated colluvium soil

To predict the stress-strain behaviour of soils under loading it is relevant the knowledge of its natural stress state, expressed by the coefficient of earth pressure at rest (K0). There are correlations in the literature for K0 determination that comes from researches developed considering sedimentary soils, typically from temperate or cold regions. In dealing with residual and colluvium soils, typical of tropical regions, it is not appropriate to use these correlations, since K0 is affected by factors such as degree of weathering, laterization processes and suction, among others that also affect saturated sedimentary soils. This work analysed samples of a colluvium soil from a natural slope at the Pontifical Catholic University of Rio de Janeiro. For the determination of the coefficient of earth pressure at rest of the unsaturated colluvium soil, a flexible wall equipment, with a system of suction control was used. The influence of the net stress level on the value of the coefficient of earth pressure at rest was investigated under a constant suction of 10 kPa, following a loading and unloading cycle of applied vertical stress. The coefficient of earth pressure at rest of the saturated colluvium (null suction) was also determined using a stress-path, Bishop and Wesley type, servo-controlled triaxial equipment. The obtained results are presented and discussed.


Introduction
The necessity to predict the stress-strain behaviour of soils under loading makes it important to know the in situ stress state that occurs initially in the soil (Vaughan and Kwan [1]). The coefficient of earth pressure at rest is defined as the ratio between the horizontal and vertical effective stresses. Thus, it is a parameter that expresses the natural state acting on the soil. Determining such value correctly should, therefore, be a common practice in engineering.
Although the earth pressure theory is commonly used, few types of studies have been carried out to evaluate the influence of suction and other factors on the coefficient of earth pressure at rest of unsaturated residual and colluvium soils. Recently, Alvim and de Campos [2] evaluated the influence of suction and weathering degree on the K 0 value of a residual soil. Nhut and Adel [3] carried out tests for the determination of K 0 of a collapsible soil formed by kaolin and sand.
To evaluate the influence of the suction and of the net stress level on the coefficient of earth pressure at rest of the colluvium soil, laboratory tests were performed. An equipment with control of suction, similar to a triaxial equipment, developed at the Geotechnical and Environmental Laboratory of the Pontifical Catholic University of Rio de Janeiro (PUC-Rio) was employed for that. To obtain the coefficient of earth pressure at rest on the same soil in the saturated condition, CK 0 tests were Corresponding author: lorenagomesabrantes@hotmail.com performed on a servo controlled triaxial equipment developed at Imperial College, London.
The soil samples used in this work were retrieved from a natural slope located at the Experimental Campus II of the Pontifical Catholic University of Rio de Janeiro, in Rio de Janeiro, Brazil.

The coefficient of earth pressure at rest
Following the well-known Bishop's effective stress definition for unsaturated soils, given by: where ' and are, respectively, effective and total normal stresses, u a is air pressure, u w is water pressure and is the Bishop's parameter for effective stress, the following formulation is derived for the coefficient of earth pressure at rest, K 0 (Bishop [4]): where the sub-indexes h and v correspond, respectively to horizontal and vertical stresses.
To include the behaviour of over-consolidated soils in the K 0 formulation, Andrawes and El-Sohby [5] defined such parameter as (Equation 3): the ratio between the increase of the effective minor principal stress (horizontal effective stress) and the increase of the effective major principal stress (vertical effective stress), with the restriction of the deformation in the direction of the minor principal stresses.
Taking into account the Bishop's effective stress equation, and assuming the soil as being homogeneous and linear elastic, Lu and Likos [6] expanded the Hoke's law for unsaturated soils as indicated in Equations 4 to 6.
where: , , = strains in directions x, y and z; = vertical total stress; e = horizontal total stresses; ν = Poisson's ratio; = elasticity modulus; To satisfy the condition of null lateral deformation, the following equation applies: With that, it is possible to arrive at the following relationship: was that proposed by Lu and Likos [6] to define the coefficient of earth pressure at rest in unsaturated soils. This equation indicates that, theoretically, K0 would tend to reduce with an increase in the matric suction.
Following a different approach, which does not include any definition of effective stresses, Fredlund and Rahardjo [7] proposed the following equation to define K 0 in unsaturated materials: Fredlund and Rahardjo [7] do not say anything on the effect of the air pressure on the K0 value. However, if the values of the horizontal and vertical stresses are kept constant, an increase in the air value would imply in a decrease of the K0 value as defined by Equation 9.
In the present work, K0 values were computed using both Equations 2 and 9. To use Bishop's approach (Eq. 2) the value of was computed following the proposal by Khalili e Khabbaz [8] as indicated in Equation 10: where: S = current matric suction (equal to u a -u w ); S e = suction at the air entry point; r = material dependent parameter, adopted in this work as -0,55.

Devices and testing techniques
According to Ting et al [9], the laboratory methods for determining the coefficient of earth pressure at rest include flexible wall and rigid wall types of test. Flexible wall tests are those in which a rubber membrane confines the sample, such as in triaxial tests in which compression of the sample occurs while lateral deformation is completely impeded (Bishop [10]). Flexible wall tests have the advantage of the absence of friction between the soil and its confining wall, and the possibility of controlling the applied lateral and axial stresses while lateral deformation is kept null through monitoring radial deformations. In addition, measurement of lateral stresses is possible if an incompressible confining cell fluid and a rigid triaxial cell are used (e.g Ting et al [9], Bishop and Henkel [11]. Disadvantages are related to keep a radial strain null throughout the sample height when testing control is based on radial strain measurements (Ting et al. [9]).
Rigid wall tests comprise those in which a rigid lateral border confines the sample, such as the case of oedometric tests. In these tests, the requirement of null lateral deformation is typically obeyed. As pointed out by Ting et al [9], the main disadvantage of such tests is on the measurement of the horizontal stress due to the occurrence of lateral friction between the sample and the rigid confining wall of the oedometer ring.
Regardless the type of device (with flexible or rigid wall) a further testing complexity is related to the need of either to measure the suction within the unsaturated sample being tested or to control such suction.
In the present work, two equipment, both comprising flexible wall devices, were used.
For the determination of the coefficient of earth pressure at rest in the saturated colluvium soil it was used a servo controlled triaxial equipment, developed at Imperial College (IC), London. It comprises a stress path controlled triaxial cell, type Bishop & Wesley (see Figure  1), with an internal load cell for measuring and control axial (deviator) stress and an external displacement transducer for measuring axial displacements. Pressure transducers were used to measure and control cell and pore-pressures. An IC volume change device was employed to measure water volume changes of the 1,5" diameter x 3.0" height samples. Stress path testing was controlled through highresolution air pressure valves geared by step motors coupled to fine resolution gear boxes. The software TRIAX, developed by Toll [12], was employed to monitor all instrumentation and for full K0 testing control.
The following procedures were adopted in the development of the K0 tests in the triaxial equipment: i) Saturation using the back pressure technique, with increments of chamber pressure in steps of 10 kPa per hour up to reaching a Skempton B parameter of 0.95; ii) Isotropic consolidation to 30 kPa, in order to avoid an eventual failure of the sample by touching accidently the failure envelope in the very early stages of anisotropic consolidation. In this and all subsequent steps, the back pressure was kept constant and equal to 150 kPa, with drainage occurring both at the top and base of the triaxial specimen; iii) Constant mean stress consolidation up to a deviator stress failing in a K0 path estimated by using the wellknown Jaky's Equation -K0 = 1-sin '. This step was introduced in order to speed up the subsequent anisotropic consolidation main testing step. iv) Anisotropic consolidation following the K0 stress path up to a vertical effective stress of 200 kPa. Through the continuous measurement of the water volume variation of the saturated sample, the volumetric deformation of the specimen was defined. The condition of zero radial deformation was set by making the volumetric deformation equal to the monitored axial deformation. With such control criteria, the K0 path was followed applying increments of vertical stress of 10 kPa/h. As the sample tended to expand radially, the axial stress increment would be halted and the cell pressure would be increased, also at a rate of 10 kPa/h, until returning to the condition of zero radial strain. In the case of radial contraction, the axial stress would be incremented while keeping the cell pressure constant.
For the determination of the coefficient of earth pressure at rest under unsaturated conditions it was used a special cell developed at PUC-Rio by Daylac [13]. It comprises a suction controlled rigid triaxial chamber with internal diameter of 4.5", with the triaxial sample having a diameter of 4" and height of 30 mm. A self-supporting loading frame was employed for axial loading of the sample (see Figure 2). Deaired water was used as confining fluid and an internal load cell, with a piston inside the chamber of the same diameter of the sample, was used to monitor vertical stresses. Owing to that, as the specimen is consolidated, its volume variation is equal to the volume occupied by the advancement of the piston. Thus, the lateral deformation of the specimen was kept to zero. Suction was controlled using the axis translation technique. The water pressure was applied at the bottom of the specimen, through a saturated high air entry value porous stone of 500 bars. Air pressure was applied at the top of the sample, through a dry, coarse porous stone. Figure 3 shows a general view of the equipment set up, without the inclusion of the self-supporting loading frame.
Horizontal stress variation under increments of vertical stress and net stress was measured by a pressure transducer with 0.5 kPa resolution, inserted at mid-high of the rigid triaxial chamber. Corrections for fluid, tubing and chamber variations of volume due to pressure variations were applied to the measured horizontal stress through adequate preliminary calibrations.  The following procedure was adopted in the development of the unsaturated K0 tests: i) Sample set up at its natural water content; ii) Application of initial suction and monitoring of equilibrium of sample volume change through measurements of water entering or leaving the sample under zero axial loading; iii) Application of axial loading and unloading, in steps, and monitoring of sample volume change and of axial displacement changes until achieving equilibrium in each step. After equilibrium, measurement of the lateral pressure. Figure 4 and Table 1 show results of characterization tests performed on the tested material. According to the USCS classification system, the colluvium soil is a clay of high plasticity (CH). Indeed, it comprises a gap-graded soil, with a clay fraction of 60.1% and a coarse fraction of 33.0%.  Results of chemical analysis indicated that the colluvium presents a high degree of laterization. Mineralogical analysis showed that the main mineral present in the fine fraction is kaolinite, with traces of goethite. In the coarse fraction prevails quartz, with traces of grenade and magnetite. Table 2 shows average values of physical indexes of undisturbed samples of the colluvium.

Soil water retention curve
The soil water retention curve (SWRC) was obtained using the filter paper technique (e.g. Marinho [14]), following both drying and wetting paths.
Drying path was followed after saturation of subsamples by capillarity. Wetting path was followed after air drying of sub-samples (refer to Abrantes [15] for methodology details). Whatman 42 filter papers were used in all cases.
To obtain the SWRCs shown in Figure 5, the filter paper calibration equations proposed by Chandler and Gutierrez [16] were employed. The data points above 1 kPa of suction shown in this figure represent average results from filter papers put in contact with both top and bottom of each sub-sample. The data shown at the matric suction of 1 kPa represent, indeed, a condition of full soil saturation. Thus, the volumetric water content at such suction equals the average porosity of the initially saturated and dried sub-samples.
As indicated in Figure 5, the colluvium soil presents a double structure type of SWRC. Hysteresis occurs under wetting and drying conditions. The curves fitting the data points in Figure 5 were obtained according with the following equation, proposed by Gitirana and Fredlund [17]: where b1 and b2 = values of suction at the first and second points of air entry; res1 and res2 = values of suction at the first and second points of residual saturation; 1 = 1; 2 = Sres1; 3 = Sb ; 4 = Sres2 and =2exp[1/ln( j+1 / j ) = weight factors, j=1,2,3. Table 3 shows the fitting parameters of the SWRCs following Gitirana and Fredlund [17].

Results
The results of the investigations performed are presented and discussed considering, first, those associated to the saturated colluvium and, then, the unsaturated material.

Saturated Colluvium
Considering exclusively the anisotropic consolidation step of the CK0 triaxial test performed, Figure 6 shows that the condition of zero radial strain has been achieved. Thus, a K0 path was followed. As indicated in Figure 7, the obtained K0 value for the saturated colluvium soil was equal to 0.36.  Such K0 value differs from that obtained by Delcourt [18], which found a K 0 of 0.43 when testing a saturated colluvium from another site in Rio de Janeiro. Table 4 and Figure 8 show the values of the coefficient of earth pressure at rest, K0, obtained following both loading (increase of total vertical stress) and unloading (decrease of total vertical stress) paths. The shown K0 values were computed using both Equation 9 (named Fredlund in the figure and table) and Equation 2 (named Bishop in the figure and table). In the horizontal axis of Figure 8 are plotted values of the mean stresses p and p' defined as:

Unsaturated Colluvium
As mentioned at the end of item 2, to use Bishop's approach (Eq. 2) the value of was computed following the proposal by Khalili e Khabbaz [8]. In the present case, for the applied suction of 10 kPa and considering the SWRC shown in Fig. 5, a value of of 0,68 was adopted considering both the loading and unloading paths followed in the tests. It can be seen in Figure 8 that the K0 value decreases with the increase of the mean effective stress, either when defined in terms of net stress (Fredlund) or effective stress (Bishop). In addition, it can be seen that lower values of K0 are obtained as the axial stress is decreased (unloading). Furthermore, for the low suction level employed (10 kPa) little difference was found for the values of K0 computed as proposed by Fredlund and Rahardjo [7] and in terms of effective stresses as defined by Bishop [2].
Considering data available in the literature, Alvim and de Campos [2] report a K 0 variation between 0.94 and 0.51, under soil loading at a constant suction of 10 kPa, in an unsaturated residual mature soil. Such type of soil (mature residual soil) is considered, at PUC-Rio, as similar to a colluvium soil. Thus, as the K0 in the present work varied from 0.17 to 0.44 under loading, it appears that the origin / structural characteristics of the colluvium / mature soil may interfere in the laboratory-determined value of K0.

Conclusion
The laboratory tests for the determination of the coefficient of earth pressure at rest (K 0 ) of the colluvium soil showed that, under unsaturated conditions, K0 varies with both net stresses and effective stresses. For the low constant suction herein considered (10 kPa) variations of K0 obtained using both net stresses and effective stresses are negligible.
The value of K0 under unsaturated conditions varied during loading and unloading, being higher under unloading.
Under saturated condition, it was found a value of K 0 of 0.36 for the tested colluvium soil. Such value is higher than the average value of K0, equal to 0,26, obtained under a suction of 10 kPa, following a loading path. This is in agreement with what has been theoretically advanced in the literature, which indicates that the K0 value decreases with an increase of suction or decrease of the degree of saturation.
Further studies however are required to support the above findings. In particular, effects of suction and of structure of tropical soils on the K0 value are to be investigated.