Electronic Supplement to
Toppling Analysis of the Echo Cliffs Precariously Balanced Rock

by Swetha Veeraraghavan, Kenneth W. Hudnut, and Swaminathan Krishnan

This electronic supplement contains text describing the steps involved in the modeling of precariously balanced rock (PBR)–pedestal geometry using the Echo Cliffs PBR as an example. Figure S1a and S1b shows the dense point cloud describing the surface of the rock and pedestal and the rock–pedestal interface, respectively, and Figures S2, S3, S4 shows the final 3D model, hazard deaggregation, and toppling probability, respectively.


1. Identifying the Rock–Pedestal Initial Interface Plane

We take a slice of the dense point cloud (obtained from terrestrial laser scanning [TLS] or photogrammetry) that fully encloses the rock–pedestal interface (Fig. S1b). This slice is selected visually (spanning about 0.5 m along the Z axis for the Echo Cliffs PBR). We refer to the points within this slice as the dense interface-point cloud. To stabilize our model under gravity, we assume that the initial contact between the rock and pedestal occurs on the best-fitting plane to the dense interface-point cloud. This initial interface plane is mathematically determined using the linear least-squares method. The assumption of a planar contact area is not far from reality, as evidenced by the goodness-of-fit parameter R2 for the linear least-squares fit being close to unity.

2. Determining the Physical Properties of the Rock

The dense TLS point cloud is pruned to extract a coarser point cloud (Fig. S1c). To minimize the processing time and computational resources needed, we use the coarser point cloud to characterize the volume of the rock (and its physical properties) and the undulating surface of the pedestal. The points in the coarse point cloud, which lie above the initial interface plane are classified as rock points and those that lie below the pedestal are classified as pedestal points. This is followed by Delaunay triangulation on the coarsely spaced rock points, creating a void-free 3D polygon (Fig. S1d) formed by a collection of nonoverlapping tetrahedrons. Under the assumption that the density of the rock is constant throughout, the basic physical properties of the rock (mass m, mass moment of inertia I, and center of gravity cg) are calculated by combining the physical properties of all the tetrahedrons. If the rock surface is more or less convex (as in the case of the Echo Cliffs PBR), the 3D polygon model of the rock is a good geometric representation of the actual rock. If, however, the surface of the rock is substantially concave, it may need to be divided into smaller subdomains, with a 3D polygon constructed for each subdomain. The number and sizes of the subdomains depend on the curvatures present and the desired precision in approximating the concave surface of the rock in its geometric model.

3. Determining the Initial Contact Points

The points from the dense interface-point cloud are projected onto the initial interface plane (Fig. S1e). Two closed loops that enclose all the points are traced out (Fig. S1f). The outer loop is the convex envelope or the 2D polygon circumscribing all the points in Figure S1e. The points belonging to the inner loop are manually picked to remove any sharp protrusions, and the resulting set of points are then smoothed using a cubic spline. The area within the inner loop is where the rock is initially in contact with the pedestal, and the vertices of this polygon are the initial contact points. They are appended to the rock points from step 2. The vertices of the outer polygon are appended to the pedestal points also from step 2. This ensures that the outward normal to the pedestal is unambiguously defined at each of the initial contact points.

4. Identifying Contact Point Candidates for Future Time Steps

The rigid body dynamics algorithm evaluates the rocking and sliding of the rock on the pedestal by time-integrating the equations of motion, iteratively satisfying dynamic equilibrium in the displaced configuration of the rock. The set of points on the rock, which are in contact with the pedestal, is continuously updated in each iteration of a time step. Smooth and accurate tracking of the rocking and sliding motion of the rock is best achieved using a finely sampled rock surface with a closely spaced set of candidate contact points to choose from. The coarse cloud of points used in the determination of the physical properties of the rock in step 2 will not suffice. A higher resolution of points is needed, especially near the rock’s base.

To obtain varying resolution of points, we create two 3D rectangular grids of points: (1) a coarse grid spanning the entire height of the rock–pedestal system (with a grid spacing of 0.6 m for the Echo Cliffs PBR) and (2) a finer grid spanning only the part of the rock near its base (with a grid spacing of 0.1 m for the Echo Cliffs PBR). The finer grid extends to about 1 m above the highest initial contact point from step 3. The dense interface-point cloud from step 1 is combined with the rock points of step 2. Delaunay triangulation is performed on this set of points to obtain a second 3D polygon in the same manner as in step 2. The grid points from the finer 3D grid, which lie inside or on this 3D polygon, are extracted. Likewise, the grid points from the coarser 3D grid, which lie inside or on the 3D polygon from step 2, are also extracted. Of these, the points at a given grid elevation (Z coordinate) constitute the horizontal (XY) cross section of the rock at that elevation. The outermost points at each elevation are combined to describe the outer surface of the rock (Fig. S2). These are all the points at which the rock may come in contact with the pedestal in future time steps.

5. Constructing the Pedestal Surface

A piecewise linearly interpolated pedestal surface is created from the pedestal points of steps 2 and 3. On plan (horizontal) view, the pedestal surface forms a grid. The outward normal to the pedestal surface at each grid point is calculated based on the vertical elevation of that grid point and its four closest neighbors. The coordinates of the points constituting the pedestal surface, and the outward normal at these points constitute the final set of inputs to the rigid body dynamics algorithm.


Figures

Figure S1. Echo Cliffs PBR. (a) The dense point cloud obtained from TLS, with color indicating height in meters. (b) The slice of points that encloses the rock–pedestal interface (dense interface-point cloud), with color indicating height [note very different color scale than in (a)]. (c) The pruned coarse point cloud, with each vertex shown as a red dot. (d) Delaunay triangulation of the coarse point cloud. (e) Plan (xy) view of the dense interface-point cloud. (f) Inner (green dots) and outer (blue line) loops enveloping the points in (e). The green dots constitute the initial set of contact points.

Figure S2. Final 3D model with the pedestal surface and the points representing the nodes on the outer surface of the rock.

Figure S3. U.S. Geological Survey (USGS) seismic-hazard deaggregation for the Echo Cliffs PBR location.

Figure S4. Toppling probability of the Echo Cliffs PBR as a function of peak ground acceleration (PGA) and peak ground velocity (PGV)/PGA under three-component pedestal excitation with the horizontal vector PGA oriented between (a) −22.5° and 22.5° (first bin), (b) 22.5° and 67.5° (second bin), (c) 67.5° and 112.5° (third bin), (d) 112.5° and 157.5° (fourth bin), (e) 157.5° and 202.5° (fifth bin), (f) 202.5° and 247.5° (sixth bin), (g) 247.5° and 292.5° (seventh bin), and (h) 292.5° and 337.5° (eighth bin) counterclockwise (CCW) to the X axis. The upper and lower black lines, predicted by the Purvance, Anooshehpoor, and Brune (2008) empirical relation, represent the PGA–PGV/PGA combinations corresponding to rock toppling probabilities of 1% and 99%, respectively.


Reference

Purvance, M., A. Anooshehpoor, and J. N. Brune (2008). Freestanding block overturning fragilities: Numerical simulation and experimental validation, Earthq. Eng. Struct. Dynam. 37, 791–808.

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