Block Theory

Background

Currently, the projects involves the construction of rock (slope, foundation, underground caverns) have become increasingly frequent. Rock engineering analysis, i.e. through various means and ways to get a correct understanding of the laws of rock deformation and failure, to determine the stability conditions of rock mass, predict future changes, and devise effective measures to deal with the project, becomes more and more important. Block Theory is one of the most famous traditional analysis methods for these purposes in computational rock mechanics.

Brief History

• Beginning: Stereographic Method for Stability Analysis of Discontinuous Rocks in Scientia Sinica (Series A) 1977, by Gen-hua Shi
• Grow up: A Geometric Method for Stability Analysis of Discontinuous Rocks in Scientia Sinica (Series A) 1982, by Gen-hua Shi
• Establishment: Application of Block Theory to Simulated Joint Trace Maps in Proceedings of International Symposium on Fundamentals of Rock Joints, 1985, by Gen-hua Shi, Richard E. Goodman and J. P. Tinucci
• Mature: Richard E. Goodman and Gen-hua Shi, Block Theory and its Application to Rock Engineering, published in 1985

Main Concepts

Analysis Subject

The rock blocks created by intersections of discontinuities, including:

• joint surfaces (planes): Geological interfaces formed with a certain direction, size, shape and characteristics in the rock body
• free surfaces (planes): Contact faces between the rock and the outside air (e.g. the excavation surfaces)

Basic Assumptions

1. All the surfaces are assumed to be perfectly planar (describe block morphology by linear vector equations)
2. Joint surfaces are assumed to extend entirely through the volume of interest (infinite plane)
3. Blocks defined by the system of joint faces are assumed to be rigid
4. The discontinuities and the excavation surfaces (joint and free planes) are assumed to be determined as input parameters (the influence of variations in the joint set orientations will not be pursued in Block Theory)

Block and Pyramid

• Block: space region surrounded by the actual planes (the intersection of half spaces), represent the shape and size of the actual rock block
• Pyramid: the intersection of half-spaces defined by the planes moved to the origin

Block Classification

• Infinite Block: connected to the base rock mass, provides no hazard to an excavation as long as it is incapable of internal cracking
• Finite Block: cut by the joint and free planes, are divisible into removable and nonremovable types
• Nonremovable Block: has tapered shape and cannot be removed from the rock mass
• Removable Block: nontapered, finite block
• Stable Block: stable even without friction with respect to the resultant force
• Potential Key Block: stable with sufficient friction, potentially unstable
• Key Block: not only removable but oriented in an unsafe manner so that it is likely to move unless restraint is provided

Examples (2D):

Analysis Method

Mathematical Descriptions

• Plane:

${\vec {x}}\cdot {\vec {n}}=D\rightarrow Ax+By+Cz=D$

• Half-space:

${\vec {x}}\cdot {\vec {n}}\geq D\rightarrow Ax+By+Cz\geq D$

• Block:

$\left\{{\begin{array}{c}A_{1}x+B_{1}y+C_{1}z\geq D_{1}\\A_{2}x+B_{2}y+C_{2}z\geq D_{2}\\\cdots \\A_{n}x+B_{n}y+C_{n}z\geq D_{n}\end{array}}\right.$