Key Blocks
Key Blocks finds the blocks that can come loose from the real surface of a slope or excavation, using block theory (Goodman & Shi 1985). It works from either of two sources:
- The project Sets. At every point it takes the face's local orientation and every three sets' mean planes. It keeps the least stable removable block they cut, and maps that block's factor of safety, volume and mode. This is quick, and works on meshes and point clouds.
- A DFN (Discrete Fracture Network). The DFN's fractures cut the rock behind a mesh into real blocks: finite fractures, blocks of any shape, closed off by fractures and the surface. Each block exposed on the surface is tested for falling and sliding. The run also gives the in-situ block size distribution. This needs a mesh.
Overhangs and the roofs and walls of excavations are included in both.
Running it
- Prepare the source. For the project Sets, tag the planes with Sets, by hand or with Auto-cluster; at least three are needed. For a DFN, generate one whose domain covers the part of the mesh you want to analyse. Either way, give each Set its friction angle in the Sets list. A DFN set uses the friction of the project Set with the same name, and anything without one uses 30°.
- On a mesh's or a point cloud's menu choose Attributes & Analysis → Key Blocks.... Open a report first to get the tables in it.
| Field | Description |
|---|---|
| Blocks from | Project Sets (mean planes, on the face at each point), or one of the project's DFNs, listed as DFN: name (meshes only). |
| Face fitted over (m) | Sets only. The radius the local face is fitted over, as in Slope Stability. Default twenty point spacings, at least 1 m. |
| Largest block (m) | Sets only. The longest side of a block's trace on the face. Every block is scaled to this. Set it from the sets' spacing and persistence, since no block can be larger than they allow. Default 3 m. |
| Surface cell (m) | DFN only. The mesh is simplified to cubes of this size before the cut. Finer follows the surface more closely but takes longer. Default five point spacings, at least 0.5 m. The rock itself is cut on a grid at least three times this, coarsened further on a large box (the report gives the size): a fracture parts a cell face it covers by half or more, so blocks thinner than that grid, and fractures smaller than about a cell, are not resolved. |
| Depth into the rock (m) | DFN only. How far behind (and around) the mesh to look. The analysed box is the DFN's domain within this distance of the mesh. A block deeper than this reaches the box's side and is left out. Default 10 m. |
| Mesh wound inside out | DFN only. Tick this when the mesh's faces point into the rock rather than into the air: if the run reports no exposed blocks, or the layers appear on the wrong side. |
Results
On the mesh or point cloud:
- Key Block FoS: the block's friction-only factor of safety (0 for a block that falls). It has no value where no block can move.
- Key Block Volume: its volume in m³.
- Key Block Mode: 1 falls, 2 slides on one plane, 3 slides on two planes.
From the project Sets, these layers map the governing block at each point. From a DFN, they map the block just behind each vertex. A DFN run also gives:
- A new mesh, ‹mesh› key blocks (‹DFN›), holding every block that can move, with the same layers and a Key Block number (1 the largest).
- In-situ block sizes in the report: how many blocks the fractures close off inside the box, the smallest-10%, median and largest-10% volumes, the volume half the rock is in blocks smaller than, the largest, and a curve of the share of the rock in blocks smaller than each volume.
- Blocks that can move in the report: each block's volume, mode, FoS, the sets it slides on, its exposed area and its centre, the largest first (up to 100).
From the project Sets, the report gives the share of the surface where the governing block's FoS is below 1 and below 1.5, the largest block, and which three sets govern on what share of the surface. The messages panel sums up either kind of run in one line.
Each run is recorded in Analysis Runs, from where it can be run again. A run from the project Sets goes stale when a Set changes; a run from a DFN does not track the DFN. Key Block FoS can mark the sources for Rockfall Trajectories.
Method
From the project Sets
Removability. Three planes through a point divide space into eight joint pyramids. A planar free face can release a block from only one of them: the pyramid whose three edges all point across the face, into the air. That pyramid meets the face in a triangle, and the block is the tetrahedron between the face and the apex. If two of the planes are parallel, or an edge lies along the face, no block forms.
Size. The block's shape is fixed by the planes and the face. Its size is not: it is scaled until the longest side of its triangle on the face equals the largest block.
Governing block. Among every combination of three sets, the governing block is the one with the lowest FoS (the larger on a tie). The mean plane of each Set is used.
Faces. The face at each point is fitted over the radius, as Slope Stability fits it. The answer depends only on the face's orientation, so it is worked out once on a 2° grid of face dips (0–180°, overhangs included) and dip directions, and looked up for each point.
From a DFN
Cutting. The box is first divided into a grid of cubic cells. Each triangle of the simplified mesh, then each fracture, splits every cell it passes through along its plane, so every cell stays convex and no cell straddles a fracture or the surface. Cells on the air side of the surface are dropped.
Blocks. Neighbouring rock cells belong to the same block unless the face between them lies wholly on a fracture. A fracture that ends inside the rock therefore leaves it joined, as a rock bridge would, and a block is a region closed off on every side by fractures and the surface. Blocks can be any shape, convex or not. Blocks that reach the box's sides are cut off by it: their size is unknown, so they are left out of the size distribution and are not tested.
Mode and FoS. A block exposed on the surface rests on every fracture plane it shares with its neighbours. Its weight acts straight down. It falls when gravity lifts it off all those planes. Otherwise it slides on the plane, or along the line of two planes, that gives the lowest factor of safety while lifting off the rest (the vector method, generalised to any number of planes). A block that cannot move in any of these ways is held.
Mode and FoS
In both cases the factor of safety is friction only: the resisting force over the driving force, each plane with its own friction angle (Hoek & Bray 1981; Goodman & Shi 1985). A block that falls has FoS 0.
Limitations
- From the project Sets, each Set is reduced to its mean plane, so the spread within a set is not explored, and the face is planar at each point. Blocks cut where several free faces meet, at a corner or a ridge, are not found; use a DFN for those.
- From a DFN, the answer is one realisation of the network. Run it on several realisations to see the spread. The surface is only as fine as the surface cell, and the box is limited by the depth, so the largest blocks are under-counted.
- The FoS uses friction only. Cohesion, water pressure, support and the strength of rock bridges are not considered. Blocks are tested for translation only; rotation and toppling are not.
- A DFN run is done in one go and can take a while on a large domain with many fractures. If the cut grows too large, it stops and suggests a coarser surface cell, a smaller depth or a smaller domain.
References
- Goodman, R. E., & Shi, G.-H. (1985). Block Theory and Its Application to Rock Engineering. Prentice-Hall.
- Hoek, E., & Bray, J. W. (1981). Rock Slope Engineering (3rd ed.). Institution of Mining and Metallurgy, London.