Three-Dimensional Anisotropic Shear Strength of Jointed Rock Masses

This study combines novel and previously published data to create a new method of estimating anisotropic rock-mass strength in three dimensions that considers both the orientation and the non-persistence of multiple joint sets. Equations are developed to evaluate joint persistence as a function of joint set spacing-to-length ratio, strength reduction as a function of joint persistence, and strength reduction as a function of dip and dip direction difference between a joint set and the shear plane. Vector rotations are used to calculate the dip and dip direction difference between the orientations of joint sets and the assumed shear plane orientation, allowing anisotropic rock-mass shear strength to be estimated for any three-dimensional direction of shear. The result is reduced strength for shear in the directions parallel to jointing and increased strength for shear in directions non-parallel to jointing, with a continuous function of strength in between the two extremes. Input parameters for the system are rock discontinuity set statistics, intact rock strength, discontinuity strength, and RQD. A slope instability case history example is back analyzed using the strength estimation procedure. A calculation sheet that implements the methodology is presented and shared for digital download. exponential distribution. Simulated shear paths subparallel to jointing were created and the percentage rock bridge required to create the shear path was measured. The percentage intact rock bridge along each shear path, Br, was measured using Eq. (1), where Lr is the rock bridge length and Lj is the joint length. An example fracture network simulation and shear path is shown in Figure 1. Eq. (2) was developed to fit the data and relate joint set spacing-to-length ratio to the percentage rock bridge along the shear path for spacing-to-length ratios less than 1, as shown in Figure 2. Br = Lr Lr+Lj (1) Fig. 1. Example fracture network with optimized shear path (red represents rock bridge), spacing-to-length ratio=0.13. Br = 1 − 0.5 Spacing Length (2) Fig. 2. Discontinuity spacing-to-length ratio versus measured percent intact rock bridge. 2.2. Strength Reduction as a Function of Joint Persistence The data from six studies were used to develop a relation for strength reduction as a function of joint persistence. Modeled or simulated jointing was non-continuous for all the studies. Strength reduction (RJR) was modeled as the percentage reduction between the joint/discontinuity strength (τj) and the intact rock strength (τi) as shown in Eq. (3), similar to the Jennings (1972) criteria. Data were transformed to shear-normal space to aid in direct comparison. Shaunik and Singh (2019) and Cheng et al. (2016) constructed plaster specimens with non-persistent joints and tested them under uniaxial stress. Bahaaddini et al. (2013, 2016) used calibrated PFC3D models to predict the effect of non-persistent joints on uniaxial strength. Fereshtenejad (2020) used 3D printing and casting technology to build plaster specimens with coplanar intermittent joints and tested them in direct shear. Hu et al. (2020) constructed cement mortar samples with discontinuous fractures and conducted direct shear tests. Eq. (4) was developed to fit the data and relate percentage rock bridge along the shear path to a linear strength weighting reduction factor as shown in Figure 3. τ = RJR ∗ τi + (1 − RJR) ∗ τj (3) RJR = Br 1.5 (4) Fig. 3. Percent rock bridge versus strength reduction factor. The probability of joint set occurrence (PO) may be combined with Eqs. (2) and (4) resulting in Eq. (5) that estimates the strength reduction factor (RJ) for sliding parallel to jointing based spacing, length, and PO. The PO of a joint set is determined by terrestrialor photogrammetry-based cell mapping (Nicholas and Sims, 2000), and is defined as the percentage of cells in which a joint set is observed. An example of calculated reduction factors using Eq. 5 is shown in Table 1. RJ = 1 − PO ∗ (1 − (1 − 0.5 Spacing Length )) (5) Table 1. Example Calculated Strength Reduction Factors, RJ Prob.of Occurrence, PO Mean Length (ft/m) Mean Spacing (ft/m) Strength Reduction Due to Jointing (RJ) Joint Set 1 1.00 18.6 5.8 0.09 Joint Set 2 0.45 20.0 4.2 0.57 Joint Set 3 0.33 11.1 1.2 0.68 Joint Set 4 0.90 13.3 8.3 0.28 2.3. Strength Reduction as a Function of Orientation Difference The data from two studies were used to develop a relation for strength reduction as a function of orientation 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 Pe rc en t R o ck B ri d ge , B r Spacing/Length Ratio Fracture Simulations Fit y = 1 0.5X 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Jo in t St re n gt h R ed u ct io n F ac to r, R JR Percent Rock Bridge, Br Shaunik and Singh (2019), Type A, θ=60° Bahaaddini et al. (2013), β=60° Bahaaddini et al. (2016), β=45-60°, e≥0.2mm Fereshtenejad (2020), plaster Cheng et al. (2016), plaster β=45° Hu et al. (2020) Fit