A New Stochastic Approach to Predict Peak and Residual Shear Strength of Natural Rock Discontinuities

Abstract

Natural discontinuities are known to play a key role in the stability of rock masses. However, it is a non-trivial task to estimate the shear strength of large discontinuities. Because of the inherent complexity to access to the full surface of the large in situ discontinuities, researchers or engineers tend to work on small-scale specimens. As a consequence, the results are often plagued by the well-known scale effect. A new approach is here proposed to predict shear strength of discontinuities. This approach has the potential to avoid the scale effect. The rationale of the approach is as follows: a major parameter that governs the shear strength of a discontinuity within a rock mass is roughness, which can be accounted for by surveying the discontinuity surface. However, this is typically not possible for discontinuities contained within the rock mass where only traces are visible. For natural surfaces, it can be assumed that traces are, to some extent, representative of the surface. It is here proposed to use the available 2D information (from a visible trace, referred to as a seed trace) and a random field model to create a large number of synthetic surfaces (3D data sets). The shear strength of each synthetic surface can then be estimated using a semi-analytical model. By using a large number of synthetic surfaces and a Monte Carlo strategy, a meaningful shear strength distribution can be obtained. This paper presents the validation of the semi-analytical mechanistic model required to support the new approach for prediction of discontinuity shear strength. The model can predict both peak and residual shear strength. The second part of the paper lays the foundation of a random field model to support the creation of synthetic surfaces having statistical properties in line with those of the data of the seed trace. The paper concludes that it is possible to obtain a reasonable estimate of peak and residual shear strength of the discontinuities tested from the information from a single trace, without having access to the whole surface.

Appendices

Appendix A: Example of Calculation of Peak Shear Strength

This example focuses on the last two decrements (#154 and #155) of calculation of shear strength for the simplified surface shown in Fig. 33. Coordinates of the points making the geometry are given in Table 4. Note that the dimensions of the surface (6 m in the X direction by 1 m in the Y direction) and the material properties (Table 5) have been chosen to simplify the example and are not necessarily representative of the surfaces tested in this research. Shearing occurs along the X axis with the top wall moving from left to right. As a consequence, the gradient of the facets, as represented in Fig. 33, coincides with the apparent dip β _{app_i} . Also, in order to simplify this example, the facets are not triangular but rectangular, which does not change the mechanics of the model. So, there are only six facets in the surface presented.

Fig. 33

Simplified surface geometry for an estimate of peak shear strength. The surface is made of six facets identified in the initial geometry (a). b Perspective view of the initial geometry showing the rectangular facets. Width of the surface (in Y direction): 1 m

Table 4 Coordinates of the points of the initial geometry

Table 5 Dimensions, material properties and load

On the initial geometry, facets 1 and 4 are the steepest at 41.99°. The model sets the starting value of β* at 41.9° (at the nearest 0.1° below the steepest facets), making facets 1 and 4 active from decrement #1.

At decrement 154, β* has reduced to 26.6° (153 decrements of 0.1°). Following the progressive facet modification strategy described in Sect. 3.5, facets 2 and 5 have steepened and have become active. The apparent dip of all facets is reported in Table 6.

Table 6 Model variables at decrement 154

So, at decrement #154, four facets are active (N _cf = 4) and the total force applied to the discontinuity (1200 kN, see Table 4) is sheared between the four facets. Consequently, f _{local_i} is equal to 300 kN and σ _{local_i} is 0.3 MPa. Using Eqs. (7) and (6) and the materials properties reported in Table 5, the forces required to shear each facet at its base (f _{shear_i} ) and to slide over it (f _{sliding_i} ) are found to be 410 and 422 kN, respectively. Since f _{shear_i}  < f _{sliding_i} , shearing takes place. This means that β* is further reduced to 26.5°, which also becomes the new apparent dip (β _{app_i} ) of all active facets.

As highlighted in Sect. 3.5, changing the apparent dip of facets 1, 2, 4 and 5 also affects the dip of facets 3 and 6 (see values in Table 7). In particular, we now have: β _{app_3}  = 26.69° ≥ β* = 26.5°, meaning that facet 3 is now active and that N _cf = 5. Note that the apparent dip of facet 3 can be checked from the coordinates of point D in Fig. 33 (that still prevail at decrement 155), and the apparent dip of facets 1 and 2 at decrement 155 (26.5°).

Table 7 Model variables at decrement 155

The local normal stress drops from 0.3 MPa at decrement 154 to 0.24 MPa at decrement 155. As a result, the forces required to shear each facet at its base (f _{shear_i} ) and to slide over it (f _{sliding_i} ) become 368 and 336 kN, respectively. At that stage, the force required to slide over the facets is less than that required to shear them, which marks the end of the iterations.

The peak shear force is computed as the sum of the sliding force over all active facets at the last decrement:

$$f_{\text{peak}} = \mathop \sum \limits_{i = 1}^{{N_{\text{cf}} }} f_{{{\text{sliding}}\_i}} = 5 \cdot 336 = 1840 \,{\text{kN}}$$

The peak shear strength is calculated as:

$$\tau_{{p{\text{ - predicted}}}} = \, f_{\text{peak}} /A_{\text{tot}} = 1.840\,{\text{MN/}}6\,{\text{m}}^{2} \sim0.31\,{\text{MPa}}$$

The residual shear strength is obtained by the following equation:

$$\tau_{{{\text{res - predicted}} }} = \frac{{f_{\text{peak}} - c \cdot N_{\text{cf}} \cdot A_{\text{ip}} }}{{A_{\text{tot}} }}$$

where N _cf = 4 (facet #3 became active but was not actually sheared). So we get:

$$\tau_{\text{res - predicted}} = \frac{1.84 - 0.2 \cdot 4 \cdot 1}{6 \cdot 1} = 0.17\;{\text{MPa}}$$

Figure 34 shows the final surface geometry after all steps of progressive shearing.

Fig. 34

Modified geometry at decrement 155 (final decrement). Dashed line shows the initial geometry, for comparison

Appendix B: Derivation of Correlation Length θ

Consider points (x, y, z) of a surface and let us define z(x, y) as the surface height at point (x, y).

The directional gradients are defined as:

$$i_{x} = \frac{{z\left( {x + \Delta x,y} \right) - z\left( {x,y} \right)}}{\Delta x}$$

(14)

$$i_{y} = \frac{{z\left( {x,y + \Delta y} \right) - z\left( {x,y} \right)}}{\Delta y}$$

(15)

For the sake of consistency with the equation presented in the core of the paper, let us consider one direction (either x or y) and drop the x or y subscript.

Using the definition of the gradients above, their variance ${\text{var}}\left[ i \right]$ can be expressed as:

$${\text{var}}\left[ i \right] = \sigma_{i}^{2} = \left( {\frac{1}{\Delta x}} \right)^{2} \cdot {\text{var}}\left[ {z\left( {x + \Delta x,y} \right) - z\left( {x,y} \right)} \right]$$

(16)

where σ _i is the standard deviation of gradients. Equation (16) then becomes:

$$ \sigma_{i}^{2} = \left( {\frac{1}{\Delta x}} \right)^{2} \cdot [2 \cdot \sigma_{z}^{2} - {\text{cov}}\left[ {z\left( {x + \Delta x,y} \right), z\left( {x,y} \right)} \right]$$

(17)

where cov is the covariance and σ _z is the standard deviation of heights, which are related by:

$${\text{cov}}\left[ {z\left( {x + \Delta x,y} \right),z\left( {x,y} \right)} \right] = \sigma_{z}^{2} \cdot \rho \left( {\Delta x} \right)$$

(18)

Combining Eqs. (17) and (18), we get:

$$\sigma_{i}^{2} = 2\left( {\frac{{\sigma_{z} }}{\Delta x}} \right)^{2} \cdot \left[ {1 - \rho \left( {\Delta x} \right)} \right]$$

(19)

Equation (19) can be reformulated as:

$$\rho \left( {\Delta x} \right) = 1 - \frac{1}{2} \cdot \left( {\frac{{\sigma_{i} \cdot \Delta x}}{{\sigma_{z} }}} \right)^{2}$$

(20)

A Gaussian correlation formulation was chosen for the correlation coefficient, which reads

$$\rho \left( d \right) = {\text{e}}^{{ - \pi \cdot \left( {\frac{d}{\theta }} \right)^{2} }}$$

(21)

Combining Eqs. (20) and (21), in which the condition d = ∆x (where ∆x is the spatial increment along the surface) is imposed, yields an estimate the correlation length θ which depends on the standard deviation of the height and gradients:

$$\theta = \Delta x \cdot \sqrt {\frac{ - \pi }{{\ln \left( {1 - \frac{1}{2}\left( {\frac{{\Delta x \cdot \sigma_{i} }}{{\sigma_{z} }}} \right)^{2} } \right)}}}$$

(22)