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# Analysis

## Successive Failure

The Successive Failure algorithm employed by RocSlope involves the following steps:

1. Determine all valid blocks formed by the intersection of joints with the geology.
2. Identify blocks which have a free face (i.e., daylights) on the geology.
3. For each daylighting block, determine its removability based on the constraints of its joint pyramid (Goodman and Shi, 1985)
4. For each removable block, compute the kinematics, taking into account self-weight, shear resistance along sliding joints, loading, water pressure, and supports. If the factor of safety of the block is less than the Design Factor of Safety defined in Project Settings, then the block is considered failed.
5. For each failed block, determine if any other blocks are face-face adjacent to it. The removal of failed blocks may present opportunities for blocks which are beside or behind (and in some cases, not immediately daylighting) to become removable due to the reduction of constraints forming its joint pyramid as well as reduction of shear resistance along the joint(s) (i.e., what was formerly a joint surface is now a free surface).
6. Repeat steps 3-5 until no further failed blocks are found.
The actual key block analysis and unraveling mechanism is very complex and unpredictable. The Successive Failure assumption provides a good estimate of the total scale, location, and depth of failure but does not simulate the coupling of blocks, rock bridging, or the actual order of failure. The iterations are meant to provide a grouping of blocks failed from steps 3-5 above.

If Successive Failure is turned off, then only immediately removable blocks which daylight are analyzed (i.e., steps 1-4). This is identical to the first iteration only of a Successive Failure analysis.

#### ASSUMPTIONS

Applications of various loads and supports with Successive Removal:

• If bolts intersect a failed daylighting block in a given failure iteration, then it is removed with the failed block and has no impact on block kinematics in subsequent failure iterations. The assumption is that the bolt has failed (pullout, stripping/failure of plate, tensile failure) and no longer provides any capacity. This is a conservative assumptions since in reality, some bolt capacity may still remain.
• Loads applied to the free surface do not transfer to the blocks below once the daylighting blocks are removed.
• Support pressure applied to the free surface do not transfer to the blocks below once the daylighting blocks are removed.
• Shotcrete applied to the free surface does not provide any capacity to blocks beyond the initial failure iteration. The assumption is that the shotcrete has failed, and the shotcrete along the shared edges between the failed blocks and the blocks in the next failure iteration is lost. This is a conservative assumption since in reality, some shotcrete capacity may still remain along intact edges.
• Ponded water level on the free faces are unaffected by removal of blocks. If a joint becomes a free surface due to elimination of failed blocks in preceding failure iterations, then the free surface is subjected to ponded water pressure forces (if any), and zero water pressure otherwise.
• Joint water pressure (with the exception of joints which become free surfaces) remain unchanged. The assumption is that the failure is immediately and water pressure does not have time to dissipate with the elimination of failed blocks in preceding failure iterations.

Successive Failure takes into account the possibility of key blocks whereby the stability of the entire slope may depend on them. When Successive Failure is turned on, RocSlope analyzes the progressive failure of the blocks, starting from blocks which are immediately removable and unstable on the surface of the slope and allowing the slope to 'unravel' subsequent to the elimination of unstable adjacent blocks.

## Design Factor of Safety

The Design Factor of Safety is applicable for Successive Failure and Probabilistic Analysis, as the threshold for what is considered a "safe" or "failed" block.

• A block with a Factor of Safety equal to or greater than the Design Factor of Safety is considered safe.
• A block with a Factor of Safety less than the Design Factor of Safety is considered failed.
• In Successive Failure, the Design Factor of Safety is used to determine the collection of failed blocks in each failure iteration.
• In Probabilistic Analysis, the Design Factor of Safety is used to in the determination of Probability of Failure (i.e., PF = number of failed blocks with FS < Design FS / number of sampled blocks) in each location.

## Joints Definition

Measured Joints and Synthetic Joints are modelled as planar disks with an orientation, radius, and center. These discs are discretized by a number of linear segments to form a polygon. The Rotation Angle and Number of Discretizations allows users to customize the shape of the planar structures. (i.e. Number of Discretizations = 3 for a triangle, 4 for a square, etc.)

Greater Number of Discretizations adds more computational complexity to Compute Blocks which may result in longer compute times and intersection issues.