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Site investigations are usually carried out in geotechnical engineering to track the range of design parameters. Due to the inherent soil spatial variability and usually limited scope of site exploration programs, the design parameters are usually uncertain at locations where test samples are not taken. This uncertainty often propagates to the response of geotechnical structures such as soil slopes. This paper developed a conditional simulation framework to investigate the sampling efficiency (i.e., sampling location and sampling distance) in designing slopes in spatially varying soils. A performance-based sampling efficiency index is proposed to achieve this. It is found that the optimal location to take vertical samples are in the vicinity of slope crest to slope midpoint and the optimal distance within a limited exploration scope is such that the additional sample subdivides the influence domain to either side of the first vertical sample.

Spatial variability (i.e., heterogeneity) of soil properties can have significant effects on geotechnical structures [

Geotechnical investigations include field tests and laboratory experiments. For example, cone penetration test (CPT) is frequently performed in situ and triaxial tests carried out in the laboratory. In view of estimating the spatial correlation structure, CPT measurements are often preferable to conventional laboratory tests as the amount of data is usually much larger. For example, a database made of CPT measurements from Oslo, Norway, was used to estimate the vertical correlation statistics [

An obvious question arises regarding the design of the sampling strategy (i.e., sampling location and sampling distance between measurements) in the soil deposit for a particular geostructure. For example, if an embankment slope is to be constructed in a soil, a number of CPTs are proposed to be conducted along a straight line in the longitudinal direction (e.g.,

Li et al. [

For reducing the uncertainty in slope stability assessment, Yang et al. [

With application to two clay slope stability examples, it is demonstrated that the best locations for carrying out CPT tests can be identified by the proposed approach, thereby to increase confidence in a slope’s stability state (stable or not). The first example seeks to find the optimum locations for site investigations in the first phase, before moving on to find the optimal distance between tests for various site variabilities. The second example considers a slope with a foundation layer to see if the inclusion of a soil foundation layer changes the general findings. The idea is to further generalise the findings regarding the optimal locations and optimal distances of tests in the first example.

For simplicity, this paper focuses on applications involving only a single soil layer (i.e., a single layer characterised by a statistically homogeneous undrained shear strength), although the extension to multiple soil layers is straightforward. Moreover, the effect of random variation in the boundary locations between different soil layers can also be easily incorporated by conditioning to known boundary locations (e.g., corresponding to where the CPTs have been carried out). For example, Li et al. [

This study uses the method of Li et al., and the method description partly reproduces their wording [

Several methods can be used to simulate the unconditional random field [

Kriging [

Kriging interpolation can be used to obtain a best linear unbiased prediction of soil properties (

The weights

The Lagrange method can be used [

The error variance is then expressed as

For a mean following some trend, the modification to equation (

Equation (

Then, the estimation error variance (according to equation (

Note that the left-hand side matrix

In geotechnical engineering, it is common for a sampling strategy to follow some pattern [

Example illustration of CPT sampling strategy (

Let

Example CPT data grid (

Following the basic equation (equation (_{i} and CPT_{j} (where each CPT has

The right-hand side vector is organised as_{p}:

The unknown weight vector is formed as_{q}:

Note that equation (

The conditional simulation of geotechnical performance based on finite elements and the above kriging-based conditional random fields is presented here to show the workflow of the procedure. A flowchart for undertaking such a simulation is shown in Figure

The procedure for conditional RFEM simulation (#1: via equations (

Note that in order to use the direct CPT data, i.e., cone resistance and sleeve friction, in the analyses reported here, a conversion or transformation model is needed to relate the test measurement (e.g., cone resistance) to an appropriate design property (e.g., the undrained shear strength) [

To show the preservation of the measured values within each realisation, and to demonstrate the effect of conditioning the random fields, samples are to be taken in a 5 m high (

An example realisation

The four fields for realisation

The four fields for realisation

Figure

In order to quantify and assess the efficiency of a sampling design for slope stability, a performance-based sampling efficiency index, based on the standard deviations of conditional and unconditional simulation, is here defined as

Two simple slope stability examples are presented in this section, in order to demonstrate the capability of conditional simulation as a useful aid to geotechnical sampling design. In particular, the optimum locations and sampling distances for CPT profiles are investigated in assessing the stability of the first slope, in order to minimise the stability uncertainty of the slope founded on a firm base. The same sampling procedure is investigated in the second example, involving a slope with a foundation layer to investigate the influence of including a foundation layer.

In the two examples, results are both presented in terms of the uncertainty in the slope stability (with respect to the realised factor of safety). The strength reduction method [^{3}, Young’s modulus of

Finite element meshes and a series of numbered CPT locations for a slope cut problem (a) without and (b) with a foundation layer (dashed lines above the slope face indicate the excavated soil mass, numbers at the bottom of each mesh correspond to Gauss point locations within the corresponding column of finite elements, and an example CPT sample at location

Note that it is the

The first example considers constructing a 5 m high slope (Figure

A cross section through the proposed slope is shown in Figure

The RFEM simulations were carried out both with conditional and unconditional random fields, using

Simulation results in the first phase of example 1 (based on

Figure

Sampling efficiency indices at different sampling locations for various values of

Note that, for each _{h} in the above analysis, the same reference 2D random field is generated first for the rectangular soil block and then used to represent the “real” field variability when generating the conditional random fields in each RFEM analysis. The conditioning step is undertaken first for the rectangular soil block, and the conditional field is then mapped onto the finite element mesh (i.e., the sloped area). This is to ensure that they are consistent with practice, that is, sampling the ground first and then excavating the soil to form a slope. Hence, for CPT locations at

Figure

In some cases, a second phase of site investigation may be warranted. This study looks at the influence of a second CPT test (at position

Simulation results in the second phase of example 1 (based on

Figure

Sampling efficiency indices at different sampling locations for various values of

Uncertainty ratio for different values of

Conditional simulations considering different numbers of CPTs in the same domain (i.e., a shorter distance

The second example considers the excavation of a

The slope cross section and 40 possible locations to take the CPTs (

Similar to Figure

Sampling efficiency indices at different sampling locations for various values of

To further verify the above finding regarding the best location to sample the 2nd CPT in the 2nd phase of site investigation, 39 sets of conditional simulations for differing 2nd phase sampling locations

Sampling efficiency indices at different sampling locations for various values of

Conditional simulation can be used as a useful aid to geotechnical sampling design. Random fields conditioned upon CPT measurements are used in this paper to investigate the influence of CPT location and distance on slope stability uncertainty. The potential use of conditional simulation in geotechnical site exploration and cost-effective designs is illustrated through two numerical slope cut examples. The results showed that the confidence in the stability of a slope can be increased by conditional simulations honoring CPT measurement data. Indeed, the unconditional slope stability simulation based on the statistics of soil properties only (without explicit consideration of the constraints of CPT measurements at the sampling locations) results in a “prior” distribution of the factor of safety. In contrast, after the inclusion of the “real” spatial distribution of all CPTs, a “posterior” distribution of the structure performance (i.e., factor of safety in this paper) can be obtained. Therefore, the probability of failure (or reliability) from a conditional slope stability simulation can be viewed as a conditional probability of failure (or reliability). The updating and improvement of the probability density distributions of factor of safety in the two numerical examples clearly demonstrate this. For both slope examples, with or without a foundation layer, the conditional slope stability simulation led to a more cost-effective stability design (i.e., “effective” in the sense that the uncertainty in the response reduces) through eliminating unrealistic soil property realisations that are not consistent with measured CPT profiles.

The study also shows the updating of the response probability density function in a 2nd phase of site investigation and the improvement of the confidence in the probability of failure or survival in the 2nd phase. In fact, a site investigation may be carried out in multiple stages in many cases, with the initial analysis results (e.g., conditional simulation based on the 1st CPT) guiding further field tests (e.g., the 2nd CPT location or its distance from the 1st one). As demonstrated in the two examples, in the case of two-phase site investigation, it is possible to find out the optimum distance between the 2nd CPT and the 1st CPT, i.e., the optimal location for the additional testing in the 2nd phase. The potential use of the method in directing site exploration programs is highlighted and the efficient utilisation of field measurements is emphasized.

For the examples considered in this paper, the optimal sampling location lies near the slope crest to the midpoint of the slope, and the optimal sampling distance between CPTs for the 2nd phase was identified to equally subdivide the zones to the left or right of the 1st test. The general suggestion based on the current investigation is to sample at the best location in the first phase and then sample to either side of the first test at some locations that equally subdivide the mechanical influence domain to the left or right of the first test. This subdividing process goes on until the distance between CPTs is equal to or just less than half the horizontal scale of fluctuation [

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The first author appreciates the financial support of the National Natural Science Foundation of China (Grant no. 41807228) and the Fundamental Research Funds for the Central Universities (Grant no. 2652017071).