The application of the numerical Discontinuous Deformation Analysis method (DDA) in rock engineering is discussed here. Following a review of recent 2D and 3D DDA validations, application of the method in analysis of natural rock slopes and underground openings are presented. Two interesting problems are explored in the applied section of this paper:

  1. the deformation of discontinuous overhanging rock slopes, and

  2. the stability of karstic caverns underneath active open pit mines.

The numerical DDA method is shown here to be a powerful tool for modeling dynamic rock mass deformation when the interaction between multiple discrete elements dictates the expected global deformation.

1. introduction

Recent developments in the validation and application of Discontinuous Deformation Analysis (DDA), originally developed by Shi and Goodman [1] are presented. We begin with a brief review of recently published 2D - DDA validations for cases of dynamic loading [2] along with new 3D - DDA validations for single and double face sliding [3]. Following these validations we present dynamic DDA applications in natural rock slopes and underground openings.

2. Dynamic DDA Validations
2.1 Single Face Sliding in 2D

A displacement based sliding block model was first proposed by Newmark [4] and Goodman and Seed [5], is now largely referred to as "Newmark" type analysis. Determination of the amount of displacement during an earthquake involves two steps [5]:

  1. Determination of horizontal acceleration required to initiate down slope motion, also known as "yield acceleration" (ay), which can be found by pseudo-static analysis, and

  2. Evaluation of the displacement developed during time intervals when yield acceleration is exceeded, by double-integration of the acceleration time-history, with the yield acceleration used as reference datum.

2.2 Single Face Sliding in 3D

The two dimensional formulation shown in Equation 4 has been expanded by Bakun Mazor [3]to three dimensions for a single block on an inclined plane, in order to validate 3D-DDA under three components of input motion. The analytical solution is based on the original static limit equilibrium formulation presented by Goodman and Shi [6]. A typical three dimensional model of a block on an incline is illustrated in Fig. 2(a). The dip and dip direction angles are, α = 20 ° and β = 90°, respectively. A Cartesian coordinate system (x,y,z) is defined where X is horizontal and points to east, Y is horizontal and points to north, and Z is vertical and points upward. In an unpublished report, Shi [7] refers only to the case of a block subjected to gravitational load, where the block velocity and the driving force have always the same sign. The same is true for the original equations published in the block theory text by Goodman and Shi [6]. For the complete set of equations of the three dimensional solution see [3]. The relative error of the new analytical solution and 3D-DDA method with respect to the existing Newmark's solution is shown in the lower panel of Figure 3(A) and are both found to be less the 3% in the final position.

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