Dynamic rock slope stability analysis at Masada national monument using Block Theory and DDA

Y. H. Hatzor

Dept. of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

 

ABSTRACT: The Masada mountain is located on the western margins of the seismically active Dead Sea rift system. Therefore, dynamic key block stability is of primary concern in the densely fractured East and North rock slopes of the mountain. Recent seismogenic key block motions in the East face, detected by our in-situ monitoring system, are explained by the presently low factor of safety against sliding, and by a pronounced topographic site effect at Masada with peak spectral amplification of about 3.5. Two analysis methods are employed: 1) Static and pseudo-static limit equilibrium analysis of removable key blocks in the East face where failure of large individual key blocks is of primary concern; 2) Static and true dynamic DDA modelling of a stochastically generated block mesh in the North face where the collapse of complete slope segments under dynamic loading is of concern. The deterministic analysis of removable key blocks in the East face reveals that when joint water pressures are considered application of a 2D solution proves unconservative. DDA modelling reveals the expected failure modes in the North face and the maximum depth of deformation as a result of dynamic loading.

 

 

1. INTRODUCTION

Masada is the most important national monument in the state of Israel, as it draws over 800,000 domestic and international visitors a year. Mount Masada is situated in the Judean Desert on the western margins of the seismically active Dead Sea rift valley (Figure 1). The Roman King Herod the Great used the isolated mountain between 36 - 30 BC as an isolated resort and as a fortified shelter for times of political unrest. He has erected a magnificent palace on three natural terraces at the Northern face of the mountain and created a system of fortifications and civil structures, including storage facilities, watch towers, water cisterns, and even hot baths, across the mountaintop. The monument has attained its international fame however due to a later struggle which a small group of Jewish zealots conducted from its top against the Roman army, during a regional rebellion against Rome between 66 and 70 AD. When the Roman army finally broke into the mountaintop they discovered the bodies of 960 men, women and children who took their lives in their own hands. Thus Masada became a national symbol for the quest of freedom and independence.

The elongated NNE striking mountain is in fact an uplifted, box - shaped, horst, which is separated from the valley floor by steep slopes at least 250 meters high. The rock mass, which consists of bedded dolomites and limestones, is intensely fractured by two closely spaced, orthogonal, sub-vertical, and very persistent joint sets striking roughly parallel and normal to the long axis of the mountain (Figure 2). The stability of two natural rock slopes, the East and North faces, is of concern and is discussed here. In the East face, a sub-vertical slope in which the cable car station is anchored, block theory is used in the analysis of large, marginally stable, key blocks which rest above the visitors path. In the North face a deterministic block theory approach can not be adopted because the face is comprised of closely spaced joints and bedding planes, the intersection of which forms a network of prismatic blocks having an average area (parallel to the face) of 3.22 m2. In this slope two-dimensional DDA is employed for two extreme loading conditions: static, and true dynamic.

Figure 1. Recent sismicity along the Dead Sea rift system (3.5 < M < 6.5, 1980 – 1996). Source: The Seismological Bulletin of Israel (1996).

Figure 2. Joint set and bedding plane attitudes in the East face of Masada (U.H. projection of poles).

 

2. MECHANICAL BEHAVIOUR OF THE ROCK

The intact rock in Masada is typically a stiff and very strong dolomite (the Turonian Shivta Formation). A single NX size core of intact rock was sampled in the field using a portable core drill and tested in uniaxial compression at the rock mechanics laboratory of BGU (for description of the load frame and testing procedures see Hatzor and Palchik, 1997, 1998). It was found that the bulk porosity is 2.5%, the Elastic modulus is 42.9 GPa, the Poisson’s ratio is 0.18, and the uniaxial compressive strength is beyond 300 MPa. Rock slope stability problems at Masada are obviously related therefore to the displacement of key blocks along pre-existing planes of weakness rather than failure through intact rock.

In the East face of the mountain key blocks typically open from the sub-vertical joints (J2 and J3) and slide in parallel to the dip of the bedding planes, which is due south east (azimuth 124o). The amount of dip inclination is therefore of supreme importance in stability analysis. Using the Fisher (1953) distribution it was found that the mean bedding inclination is 14o and that the 10% density contour around the mean is at 6 degrees from the mean (Figure 2). Accordingly, a conservative dip inclination of 20o is used in the analysis.

The bedding planes are usually filled with crushed dolomite and the cohesion is assumed to be negligible. Two methods are employed in an attempt to determine the friction angle of the bedding planes: 1) tilt tests of saw cut and polished cylinders of dolomite, and 2) a multistage triaxial compression test of an inclined, saw cut plane, filled with the crushed dolomite. The tilt tests of saw cut and polished dolomite cylinders yield a friction angle of 28o and 23o respectively. The multistage triaxial test includes 7 cycles of shear under increasing confining pressure values from 2.2 MPa to 16.2 MPa (for details about the testing procedures see Hatzor and Levin, 1997). The results of the triaxial test yield a very linear Coulomb law with zero cohesion and a friction angle of 22.7o , similar in essence to the residual friction angle which was determined using tilt tests of polished surfaces (Figure 3).

 

Figure 3. Results of segment triaxial test (see text).

 

3. TOPOGRAPHY EFFECT ON EXPECTED GROUND MOTION AT MASADA

The bedding plane inclination at the East face of Masada (20o due Southeast) and the available friction angle of filled bedding planes (22.7o) explains the marginal stability of removable key blocks under static loading. Because of the proximity of Masada to the seismically active Dead Sea rift system this study addresses dynamic key block stability as well. There is ample evidence that pronounced topography often causes ground motion amplification during strong earthquakes which may lead to increased structural damage at mountaintops (e.g., Spudich et al., 1996; Bouchon and Barker, 1996; Celebi, 1987; Hartzell et al, 1994). Because of the unique topography of Mount Masada it was decided to conduct a topography effect study at the site. A team from the Seismology Division, the Geophysical Institute of Israel conducted the study, and the results were reported by Zaslavsky et al. (1998). Three methods were employed: 1) Reference station technique – a study of spectral ratios of the horizontal components of recorded earthquakes at the investigated site and at a reference site which is supposedly free of site effects; 2) Receiver function estimates – spectral ratio between the horizontal and vertical components of S-waves recorded at the investigated site; 3) the Nakamura method (Nakamura, 1989) – analysis of spectral ratios between the horizontal and vertical components of recorded ,microtremors (or ambient noise) at the site.

Between September 10-13, 1998 four seismic stations were installed in the East slope of Masada by Zaslavsky et al. (1998): at the top (stations 1 & 2), mid slope (station 3), and foot of the mountain (station 4 – the reference station). The three component seismometers were sensitive velocity transducers with a natural frequency of 1.0 Hz. Two data sets were collected from the four sites: microtremors (ambient noise) and weak ground motions from an earthqua(ML = 2.9) which occurred on Sep. 13, 1998 near Cyprus (epicentral distance of 545km). In Figure 4 spectral ratios for sites 1 and 2 with respect to site 4 are calculated from the seismic waves of the Cyprus earthquake by Zaslavsky et al. (1998). The spectral ratios show a prominent peak at about 1.3 Hz. The horizontal EW ground motion is amplified by a factor of about 3.5 while the horizontal NS component is amplified by about 2.0, suggesting that in addition to a clear site effect the Masada exhibits a preferential direction of resonance motion.

Figure 4. Spectral ratios of EW (top) and NS (bottom) motions at sites 1 and 2 (mountaintop) computed from earthquake data (S-waves) with respect to reference site 4 (From Zaslavsky et al., 1998).

 

Spectral ratios that were computed by the other two methods yield similar results. Because the mountain consists of bedded dolomites and limestones the detected spectral amplifications must be related to topographic effects only. The expected peak horizontal ground acceleration at the Dead Sea region for hard rock is estimated at 0.2g by Israel Building Code 413 when no site effects are considered. In light of the results above and considering the complete acceleration response spectrum by the code, a maximum horizontal ground acceleration of 0.64g is expected at a frequency of 1.3 Hz.

 

4. DYNAMIC KEY-BLOCK DISPLACEMENT

In order to monitor key block displacement a comprehensive monitoring program was executed at the East face for the marginally stable blocks (Block I, II, III, see Figure 5).

 

Figure 6. Monitoring data for Blocks I, II, III

 

The monitoring instruments (produced by DINA Electronics Ltd.) consist of displacement, pressure, temperature, and humidity transducers. The displacement transducers (or “joint meters”) which were installed (Model DI-800) have a range of 50mm with 0.5% linearity full scale. The joint meters were mounted in parallel with the expected displacement direction such that block opening yielded transducer extension, denoted here as negative. Four, four, and two joint meters were initially installed in blocks I, II, and III respectively. In addition a control joint meter was installed on intact bedrock near Block II. The pressure transducers are 225X95mm Flat Jacks (Model DI-260) with a range of 20kg/cm2 and a linearity of 0.25% full scale. One flat jack was initially installed at the foot of each key block such that opening from the back joints and sliding along sliding planes would be coupled by induced compression in the flat jack, denoted here as positive.

The monitoring program started at January 1998 and terminated in November 1998 after permanent support was installed in Blocks I and II (anchoring took place between August and November, 1998). Block III was left unsupported for long term monitoring of natural rock slope behaviour (the block, which can not be seen in Figure 5, is not endangering the visitors’ path). Data from all channels have been logged every four hours during the monitoring period.

In Figure 6 the output data of a characteristic joint meter from each block is plotted for the first six months together with the output of the control transducer (designated as “bedrock” in Figure 6). A slow opening rate is detected in all blocks from January to early April. Between April 7 and April 10 a sudden displacement of about 1mm is detected in all blocks while the intact bedrock remains static. Using the comprehensive Israel Seismological Bulletin the sudden displacement is correlated here with two minor earthquakes which took place at the Gulf of Eilat (Aqaba) with epicenter distance greater than 250km. Note that following the tremor continued displacement is detected in Block I, and to

Figure 7. Response of key block III to the tremors in April (joint meters – thin line, flat jack – heavy line).

5. KEY-BLOCK STABILITY ANALYSIS

Block I (Figure 5) is the largest and most dangerous key block in the East face and therefore its stability is discussed below. The block opens at the back from two sub-vertical joints (J2 and J3, Figure 2) and slides on a bedding plane at the base. The face of the block also consists of subsets of J2 and J3 (Figure 8).

Figure 8. Three dimensional view of Block I (W = 1,400ton)

The stability of the block for full saturation was first analyzed using a two-dimensional

solution for a vertical slice through the block parallel to the dip direction of the sliding plane. Thus required support forces for a given factor of safety were determined in units of tons/m. In order to find the total required support force the results had to be multiplied by the “length” of the block, a quantity which looses meaning in 3D modelling. This type of approach inevitably leads to artificial inaccuracies. For example, the area of the base plane and therefore the total weight of the “2D block” may become artificially higher then the actual dimensions in the real block (Figure 8). As a result the factor of safety against sliding for the “2D block” may increase artificially. In Figure 9 the results of the 2D and 3D solutions are plotted for a fully saturated block (for 3D solution methods considering joint water pressures see Hatzor and Goodman, 1997).

 

Figure 9. 2D vs. 3D solution for support requirements in Block I under fully saturated, static loading.

 

Three-dimensional pseudo-static analysis for Block I was performed for two extreme cases: completely dry and fully saturated block. The inertia force was taken horizontal, parallel to the dip direction of the sliding plane. The results are shown in Figure 10 below.

Figure 10. Three-dimensional pseudo-static analysis: Block I

The expected peak horizontal ground acceleration in the Masada region is 0.2g. Considering topographic effect and spectral amplification the peak horizontal ground acceleration is estimated at 0.64g at the mountaintop. Provided that sound drainage system is installed inside the joints in Block I the total required support force for dynamic stability may be estimated using the “Dry” curve in Figure 10.

 

6. FAILURE MODES IN THE NORTH FACE

In the East face the rock mass is relatively intact with very persistent and widely spaced individual joints, the intersection of which forms very large, removable blocks. The analysis of a typical block (Block I) was discussed in the previous section. In the North face of the mountain such an approach is rather futile because the density of the joint sets is very large (mean joint set and bedding plane spacing of about 25cm and 50cm respectively). The joint set and bedding plane attitude in this face are similar the patterns observed for the East face however the bedding plane attitude is again unfavorable with a mean inclination of 10o due N (Figure 11).

Figure 11. Bedding plane and joint set attitude in the North face of Masada (U.H. projection of poles)

 

The density of the joints creates a closely spaced net work of individual blocks, which form the bulk of the upper terrace in the North face (Figure 12). The stability of this face is therefore modeled numerically using two-dimensional DDA. As the bedding planes dip northwards sliding of individual key blocks northwards out of the face is kinematically possible but will materialize only under dynamic load because of the low inclination.

 

Figure 12. Air photo of the upper terrace at the North Face of Masada (Photograph taken looking south).

 

The two vertical joints are of greater concern and particularly the eastward dipping joint set (J2) because the northward dipping joint set strikes parallel to the north face. The mean attitude of J2 is 80o due ESE and therefore sliding deformation is expected in the east face whereas block toppling may develop in the west face.

The discontinuity pattern in the North face was modeled numerically using the DL code of Gen-Hua Shi (Shi, 1993). The stability of the face was modeled using the DF code (Shi, 1993) initially for static loading. Finally the facewas loaded dynamically using a new version of DF. The input motion was taken from a real accelerogram in which the EW component was normalized to a desired level of horizontal peak ground acceleration. The application of this new dynamic DDA code is relatively new and the accuracy of the program is currently being tested.

In Figure 13 the predicted damage by DDA for the upper terrace of the North face due to an EW horizontal motion normalized to 0.3g is shown for an earthquake duration of 2.5 sec. It can be seen that DDA clearly predicts sliding deformation in the east and toppling deformation in the west. Furthermore, the depth of deformation as a function of earthquake duration may be estimated.

 

Figure 13. Results of dynamic DDA modelling for the upper terrace of Masada North face for an EW motion of 2.5sec duration normalized to a peak horizontal ground acceleration of 0.3g.

 

7. SUMMARY AND CONCLUSIONS

The stability of the fractured East and North faces of Masada is analyzed using in situ monitoring of key block displacement, multiple stage triaxial tests of bedding planes, key block theory, and DDA. Several important conclusions emerge:

  1. Removable but statically stable key blocks which rest on eastward dipping bedding planes in the East face of the mountain exhibit large dip slip displacements, amounting to as much as 1mm during a single episode, as a response to local ground motions induced by remote and relatively minor earthquakes.
  2. The key block sensitivity to dynamic vibrations is explained by the low shear strength of the bedding planes and by a distinct topography effect on ground motion. A peak amplification of 3.5 is detected for a frequency of 1.4Hz. This is believed to be a fundamental resonance mode attributed to the effect of topography.
  3. The stability of prismatic blocks at saturation may be overestimated when a conventional two-dimensional solution method is adopted. Such an approach may introduce inaccuracies because of the attempt to represent three-dimensional problems using two-dimensional conventions. Such an approach may also prove unconservative.
  4. Application of deterministic key block theory proves unpractical when the joint density becomes very large. In such cases numerical solutions should be employed. It is shown here that with DDA the failure mode of different segments of a densely fractured slope can be simulated very well. Thus the expected depth of deformation can be predicted for different loading conditions.

 

ACKNOWLEDEGEMENT

Israel Nature and National Park Protection Authority fund this research and their support is hereby acknowledged. A. Leviatan of INNPPA is thanked for professional discussions. Z. Temkin of Tik Projects Ltd. is thanked for coordinating the investigations in the East Face. A. Tobul and E. Campbell from Masada National Park are thanked for assistance in all aspects of fieldwork. R. Holtzman and M. Tsesarsky of Ben-Gurion University are thanked for assistance with the joint survey, and for their help with acquisition and reduction of monitoring data. A. Shapira and Y. Zaslavsky of the Geophysical Institute of Israel are thanked for their excellent topography effect study. Finally, Gen-Hua Shi is thanked for introducing the author to his new dynamic DDA code and for stimulating discussions in the field.

 

REFERENCES

Bouchon, M. & J. S. Barker 1996. Seismic response of a hill: the example of Tarzana, California. Bull. Seism. Soc. Am. 88:66-72

Celebi, M. 1987. Topographical and geological amplifications determined from strong-motion and aftershock records of the 3 March 1985 Chile earthquake. Bull. Seism. Soc. Am. 77:1147-1167.

Fisher, R. A. 1953. Dispersion on a sphere, Proc. R. Soc. London, Ser. A.: 217-295.

Hartzell, S. H., D. J. Carver, & K. W. King 1994. Initial investigation of site and topographic effects ar Robinowood Ridge, California. Bull. Seism. Soc. Am. 84:1336-1349.

Hatzor Y. H. & R. E. Goodman, 1997. Three dimensional back analysis of saturated rock slopes in discontinuous rock - a case study. Geotechnique, 47(4): 817-839.

Hatzor, Y. H. & M. Levin, 1997. The shear strength of clay filled bedding planes in limestones - back analysis of a slope failure in a phosphate mine, Israel. Geotechnical and Geological Engineering. 15: 263-282.

Hatzor, Y. H. & V. Palchik, 1997. The influence of grain size and porosity on crack initiation stress and critical flaw length in dolomites. International Journal of Rock Mechanics and Mining Sciences. 34( 5): 805-816.

Hatzor, Y. H. & V. Palchik, 1998. A microstructure-based failure criterion for Aminadav dolomites - Technical Note. International Journal of Rock Mechanics and Mining Sciences. 35(6): 797-805.

Nakamura, Y. 1989. A method for dynamic charactersitics estimation of subsurface using microtremors on the ground surface. QR of RTRI 30(1): 25-33.

Shi, G.-H. 1993. Block System Modeling by Discontinuous Deformation Analysis, Computational Mechanics Publications, Southampton UK, p. 209.

Seismological Bulletin of Israel. 1996. The Geophysical Institute of Israel, Seismology Division, Holon, Israel.

Spudich , P., M. Hellweg & W. H. K. Lee 1996. Directional topographic site response at Tarzana obsereved in aftershocks of the 1994 Northridge California earthquake: implicatnois for mianshock motions. Bull. Seism. Soc. Am. 86:193-208.

Zaslavski, Y., Shapira, A. & Leonoy, J. 1998. Topography effects and seismic hazard assessment at Mt. Massada (Dead Sea). The Geophysical Institute of Israel, Report No. 522/62/98.