The blog of Alvaro Boiero
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The blog of Alvaro Boiero
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"Live as if you were going to die tomorrow. Learn as if you were going to live forever" Mahatma Gandhi
The lateral earth pressure: the “ko” condition
When designing a retaining structure, one of the main parameters to be considered is the lateral earth pressure coefficient, since this coefficient is related to the stresses due to the earth thrust that the structure is expected to support, and to the type of deformations expected for the structure. In this context, what is the resting condition "Ko" and how is the value of this coefficient estimated?
Content
Horizontal stresses in the soil
The stresses in a soil mass are produced by two components: 1) the self-weight of the soil; and 2) the external loads applied to the soil. Although the actual distribution of these stresses in the soil mass is quite complicated, when the ground surface is approximately horizontal and no significant changes in the nature of the soil are observed in the horizontal direction (as is approximately the case for sedimentary soils), the stress system is greatly simplified, and the stresses are termed geostatic (Lambe & Whitman, 1969).
According to the above, the vertical geostatic stresses can be estimated by knowing the variation of the unit weight (γ) of the soil with depth (z), according to the expression shown below:
However, the relationship between the vertical (σv) and horizontal (σh) stresses is not hydrostatic, and is expressed by the lateral stress or lateral pressure coefficient (K):
This relationship is valid even when the stress conditions are not close to the geostatic case. In general, the value of K can vary in a wide range, depending on the stress conditions in the ground (higher or lower horizontal compression, either by effects related to overconsolidation, or by anthropogenic action), as shown in Figure 1.
Figure 1 Relationship between vertical and horizontal geostatic stresses (Source: Lambe & Whitman, 1969).
As can be seen from the previous figure, for normally consolidated soils the value of σh is less than that of σv, which implies that K is less than 1. This occurs because, as sedimentation occurs over an extensive area, there is no appreciable horizontal compression. The opposite occurs for overconsolidated soils, which have had significant loading in their geological past or have been subjected to desiccation processes, for which K is greater than 1.
At rest “Ko” condition
The rest condition, for which the lateral pressure coefficient is represented as Ko, corresponds to the stress state of an undisturbed soil, i.e., without the presence of external stresses or structures that modify its natural stress state.
In what cases would this stress condition occur in a retaining structure? Well, this case is plausible only in a wall of high stiffness, which does not rotate or deform significantly, a condition for which no lateral deformations would occur. In this case, the stress condition would be very similar to that of the natural ground, and therefore we would have K ≈ Ko (Coduto, 2001).
From a practical point of view, designing a wall under these conditions is rare. However, the value of Ko is essential for carrying out stress path analysis (a topic that I will soon address in another post), since in this type of analysis it is necessary to know the original state of stress in the ground, in order to predict with a higher degree of accuracy the performance of the structure under study (Holtz et al, 2011).
Estimation of “Ko” coefficient for SANDS
The determination of σh is not easy, since it is virtually impossible to install a pressure cell in situ, without producing some kind of densification or alteration of the ground around it. Therefore, the value of Ko is estimated either from theoretical analysis or from laboratory tests.
In addition, knowing certain strength or stress state parameters of the soil, it is possible to employ correlations, which allow estimating Ko values sufficiently accurate for most practical purposes.
Thus, for the case of normally consolidated sands, the well-known theoretical relationship between Ko and the angle of internal friction (φ) proposed by Jaky (1948) can be used:
According to the above expression, Ko values between 0.40 and 0.45 can be considered adequate for preliminary estimates.
For overconsolidated sands, Ko is somewhat higher than for normally consolidated sands. Alpan (1967) suggests that the increase in Ko can be related to the overconsolidation ratio (OCR), according to the following expression:
The value of Ko-NC can be estimated from the equation of Jaky (1948) mentioned above, while the value of the empirical coefficient h can vary from 0.40 for relatively low stiffness sands to 0.60 for very dense sands (Schmertmann, 1975).
As for the OCR value, its determination is complicated for sands. There are some correlations based on the value of the cone tip resistance qc reported by Kulhawy & Mayne (1990, available at https://www.epri.com/research/products/EL-6800), which can be used to estimate this value.
Estimation of “Ko” coefficient for CLAYS
For the case of normally consolidated clays, Brooker & Ireland (1965) report that the Ko coefficient varies with the plasticity index (PI), according to the following expression, also included in Figure 2.
On the other hand, the relationship between Ko and OCR can be estimated by employing the effective internal friction angle (φ'), usually obtained for clays from isotropically undrained consolidated triaxial tests (CIU), or from drained consolidated tests (CD). Mayne & Kulhawy (1982) suggest using the following expression for overconsolidated clays:
The OCR value can be determined from consolidation tests on undisturbed samples in the laboratory, or estimated using the cone tip strength qc presented by Kulhawy & Mayne (1990, available at https://www.epri.com/research/products/EL-6800).
Figure 2 Relationship between effective internal friction angle φ' and plastic index PI for normally consolidated clays. (Source: Massarsch, 1979, CB Holtz et al, 2011).
According to the above, it is evident that the value of Ko is highly dependent on the stress history of the deposit, especially for the case of overconsolidated clays.
What about the determination of “Ko” for mixed soils?
As I have already discussed in the post The Unified Soil Classification System (USCS) in Geotechnical Engineering, the presence of clay particles (smaller than 2 μm in size) exerts a notable influence on the behavior of the soil.
So, what methodologies should be used to estimate the Ko coefficient of, say, a clayey sand (SC), or a silt (ML)? And for a silty clay (CL-ML)?
Considering numerous investigations developed in the last 30 years (Kenney, 1977; Lupini et al, 1981; Skempton, 1985; Kumar & Wood, 1999; Vallejo, 2001; Mitchell & Soga, 2005; Biscontin et al, 2007, among others), the answer depends on the colloid content (i.e., the content of small clay particles). Basically, from a practical point of view, the USCS must be complemented with an accurate estimation (through a hydrometry test) of the colloid content of the soil sample.
Thus, if this content is higher than 20%-25%, the soil will present a behavior totally dominated by the clay fraction, so the expressions shown in section 4 of this post should be used to estimate Ko. Otherwise, the methods mentioned in section 3 of this post should be used.
References
- Biscontin, G., Simonetta, C. Pestana, J. & Simonini, M. (2007) “Unified compression model for Venice Lagoon natural silts”, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, pp. 932-942.
- Coduto, D. (2001) “Foundation Design: Principles and Practices”. Second Edition. Prentice Hall. New Jersey, USA.
- Holtz, R.; Kovacs, W. & Sheahan, T. (2011) “An Introduction to Geotechnical Engineering”. Second Edition. Prentice Hall. New Jersey, USA.
- Kenney, T. (1977) “Residual strength of mineral mixtures”, In: Proceedings 9th International Conference on Soil Mechanics and Foundation Engineering, Tokyo, Japan, 1, pp. 155-160.
- Kumar, G. & Wood, M. (1999) “Fall cone and compression tests on clay-gravel mixtures”, Geotechnique, 49, pp. 727-739.
- Lambe, W. & Whitman, R. (1969) “Soil Mechanics”. John Wiley & Sons, Inc. New York, USA.
- Lupini, J., Skinner, A. & Vaughan, P. (1981) “The drained residual strength of cohesive soils”, Geotechnique, 31, pp. 181-212.
- Mitchell, J. & Soga, K. (2005) «Fundamentals of soil behavior”, 3rd ed., John Wiley & Sons, New Jersey, USA.
- Skempton, A. (1985) “Residual strength of clays in landslides, folded strata and the laboratory”, Geotechnique, 35, pp. 3-18.
- Vallejo, L. (2001) “Interpretation of the limits in shear strength in binary granular mixtures”, Canadian Geotechnical Journal, 38, pp. 1097-1104.
