Development of three-dimensional numerical manifold method for jointed rock slope stability analysis

Highlights

• An advanced hybrid continuum-discontinuum numerical approach called 3-D numerical manifold method (NMM) has been developed for rock engineering applications.

• A 3-D contact detection and modeling algorithm has been customized for 3-D NMM.

• The validity and accuracy of the 3-D NMM has been verified through a series of examples.

• Potential application of 3-D NMM to rock engineering applications is clearly seen through the last slope example

• The distinctions of the 3-D NMM compared with other slope analysis tools are highlighted

Abstract

This paper presents a three-dimensional numerical manifold method (NMM) for jointed rock slope stability analysis. The main barriers for realistic rock slope stability analysis are discussed, and the unique features of the NMM which make it stand out in slope analysis are emphasized. The framework of the 3-D NMM is established via a regularly-patterned tetrahedral mesh-based mathematical cover system. A contact detection algorithm is then customized to fit the 3-D NMM characteristics. The developed code is first calibrated by two simple cases: inclined plane sliding and tetrahedral wedge sliding, and then applied to a more complex slope scenario. The complete failure process involving large displacement and rotation of multiple interaction blocks is exhibited dynamically. Two typical stabilization/protection techniques are also investigated. The developed 3-D NMM could be potentially used to find the trigger/failure mechanism of jointed rock slopes, so as to optimize the slope stabilization or protection design.

Introduction

Stability analyses are performed routinely to assess the design of excavated slopes (e.g. open pit mining, road cuts, etc.) and the equilibrium conditions of natural slopes [1]. In the past, most of the rock slope stability calculations were undertaken either graphically or using a hand-held calculator. Nowadays, a variety of tools are available for engineers and geologists, ranging from the simple infinite slope and planar failure limit equilibrium techniques to sophisticated hybrid finite-discrete element codes.

Among those approaches, the limit equilibrium-based methods remain the most popular option in rock slope engineering. These conventional methods, however, are limited to simple slope geometries, basic loading condition and rigid body assumption, thus provide relatively little insight into slope failure mechanisms [2]. Practical rock slope stability problems are intricate in the aspects of topological geometry, material anisotropy, non-linear behavior, in situ stress and the presence of several coupled processes [1]. Numerical modeling techniques help to forward the approximate solutions, which would have never been possible by using the conventional techniques.

In general, the numerical methods for rock mechanics analysis are underdeveloped compared to its demand in practical rock slope engineering. Main barriers include:

• Difficulty in geometrical modeling of rock masses. Rock is a natural geological material. Different from any artificial material, it is largely discontinuous with the presence of various discontinuities such as bedding planes, joints, shear zones, and faults [3];

• Difficulty in description of mechanical interaction among discontinuities. In most cases, the properties and engineering behavior of rock masses are governed by discontinuities. The accuracy of the numerical analysis results largely depends on properly handling with interactions among discontinuities. In numerical code, it is referred to as a contact problem, the treatment of which is extremely difficult, especially in 3-D analysis. It is because discontinuities may be infinite or finite and may terminate inside a block, and blocks may be convex or non-convex;

• Complexity of rock slope failure process. The rock slope failure is a complex process which involves both opening/sliding of existing joints, fracturing of intact rock and large movement of discrete blocks. Neither continuum-based methods nor discontinuum-based method can realize the whole failure process.

Current numerical methods could be categorized into three groups: continuum, discontinuum and hybrid modeling. Continuum modeling is the most suitable for analyzing slopes that are comprised by massive intact rock, weak rocks, and soil-like or heavily fractured rock masses by considering them as equivalent continuum media with the equivalent properties established by a homogenization process [1]. This type of modeling avoids the explicit representation of complex discontinuity network as well as complex contact detection and modeling among multiple blocks, thus does not suffer from the first two difficulties mentioned above. The accuracy of the approximation solution highly depends on the delicacy of the constitutive model of equivalent medium. Continuum modeling is limited to pre-failure stage of rock mass. Most continuum codes also incorporate major discrete fractures such as faults and bedding planes but limited to the cases where only a few fractures are present and it is not possible for fracture opening and complete block detachment. The continuum approaches used in rock slope stability include the finite difference and finite element methods. In recent years, a vast majority of published continuum rock slope analyses have used finite difference codes FLAC [4] and FLAC3D [5].

When the failure mechanism of the rock slope is controlled by discontinuities, a discontinuum modeling is considered more appropriate. Discontinuum methods (e.g., the discrete element methods (DEM)) explicitly represent the discontinuities and treat the rock slope as an assemblage of distinct and interacting blocks, that are subjected to external loads and are expected to undergo significant motion with time, thus are extremely suitable for post-failure movement of discrete bocks. Cundall and Hart [6] define the DEM as those that: (1) allow finite displacements and rotations of discrete bodies, including detachments; and (2) automatically recognize new contacts between bodies during calculations. Typical examples include the explicit DEM – the distinct element codes UDEC [7], 3DEC [8] and the implicit DEM – discontinuous deformation analysis (DDA) [9], [10], [11], [12], [13], [14], [15], [16]. In addition, particle methods such as PFC 2D/3D [17], [18], distinct lattice spring model [19] are also subordinated to discontinuum modeling approaches.

Although separate continuum and discontinuum analyses provide useful sights to rock slope stability problems, practical rock slope failures often involve mechanisms related to both open/sliding along pre-existing discontinuities as well as brittle fracturing of intact rock. Hybrid codes coupling these two techniques (i.e. continuum and discontinuum) were developed to optimally benefit from the advantages of both methods. For example, the hybrid finite-discrete element code ELFEN [20], [21], [22], allows modeling both intact rock behavior and the development of fractures. It uses a finite element mesh to represent the joint bounded blocks coupled together with discrete elements. If the stresses within the rock slope meet the failure criteria within the finite element continuum, a discrete fracture is initiated. Adaptive remeshing allows the propagation of the cracks to be simulated through the finite element mesh.

The numerical manifold method (NMM) [23], which is considered as a combination of the FEM and the DDA [9], provides another perspective of hybrid modeling. Due to the two independent cover system and the truncated discontinuous shape functions, continuous rock, fractured rock and assemblage of discrete rock blocks can be modeled in a unified form. Since initially proposed in 1991 [23], 2-D NMM has already been extensively developed and utilized in abundant studies [24], [25], [26], [27], [28], [29], [30], [31], [32]. Continuous versions of 3-D NMM have been developed in [33], [34], [35]. Heuristic formulation of the 3-D NMM together with a preliminary contact algorithm was derived by Cheng and Zhang [36]. This paper further enhances the 3-D NMM as a promising tool for jointed rock slope stability analysis.

The reminder of the paper is organized as follows. Section 2 briefly interprets the basic concepts of the NMM, clarifies its relation to the FEM and the DDA, and emphasizes the unique features of the NMM which makes it suitable for slope problems. A regularly-patterned tetrahedron mesh-based mathematical cover system and the corresponding approximations are established for the 3-D NMM in Section 3. Section 4 customizes a contact algorithm to fit the 3-D NMM characteristics. The developed code is employed to simulate three typical rock slope problems in Section 5. Among them, two simple examples are used to calibrate the code, while a complex scenario is simulated to demonstrate the capability of the 3-D NMM in capturing the rock slope failure process and deign of stabilization/protection measures. Section 6 finally draws the conclusions.