Numerical analysis of block stone displacements in ancient masonry structures: A new method to estimate historic ground motions

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech. 2008; 32: Published online 15 November 2007 in Wiley InterScience ( Numerical analysis of block stone displacements in ancient masonry structures: A new method to estimate historic ground motions Ronnie Kamai 1,2 and Yossef H. Hatzor 1,, 1 Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel 2 Department of Civil and Environmental Engineering, University of California, Davis, CA, U.S.A. SUMMARY An innovative approach is presented, in which the discontinuous deformation analysis (DDA) method is used to estimate historic ground motions by back analysis of unique structural failures in archaeological sites. Two archaeological sites in Israel are investigated using this new approach and results are presented in terms of displacement evolution of selected structural elements in the studied masonry structure. The response of the structure is studied up to the point of incipient failure, in a mechanism similar to the one observed in the field. Structural response is found to be very sensitive to dynamic parameters of the loading function such as amplitude and frequency. Prior to back analysis of case studies, two validations are presented. Both compare the performance of DDA with analytical solutions and present strong agreement between the two. Using comprehensive sensitivity analyses, the most likely peak ground acceleration (PGA) and frequency that must have driven the observed block displacements are found for the two case studies the Nabatean city of Mamshit and the medieval fortress of Nimrod in southern and northern Israel, respectively. It is found that horizontal peak ground accelerations (HPGA) of 0.5g and 1g were required to generate the observed deformations in Mamshit and Nimrod, respectively. Although these might seem too high, considering structural and topographic amplifications it is concluded that the analyses suggest ground motions of 0.2g at a frequency of 1.5 Hz for Mamshit and up to 0.4g at a frequency of 1 Hz for Nimrod. These values provide constraints on the seismic risk associated with these regions as appears in the local building code using a completely independent approach. Copyright q 2007 John Wiley & Sons, Ltd. Received 31 August 2006; Revised 15 August 2007; Accepted 15 October 2007 KEY WORDS: ground motions; seismic risk; PGA; discontinuous deformation analysis; masonry structures; validation Correspondence to: Yossef H. Hatzor, Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. hatzor@bgu.ac.il Copyright q 2007 John Wiley & Sons, Ltd.

2 1322 R. KAMAI AND Y. H. HATZOR 1. INTRODUCTION The expected ground motions at a site, a critical component in seismic risk assessments, are generally evaluated using empirical relationships between magnitude, distance, and site effects [1, 2]. Here, we present a quantitative approach to estimate historic ground motions using back analysis of block displacements in archaeological masonry structures. Reliable estimates of paleoseismic ground motions at a given site can provide constraints on ground motion predictions which are typically obtained using empirical methods. The analysis focuses on man-made masonry structures such as arches, where the hewn stones that form the building provide a well-defined initial geometrical reference. Where failure is confined to displaced blocks within an otherwise intact structure, block displacement is measurable and a mechanical analysis is possible; this cannot be achieved in completely collapsed structures. Additionally, sound bedrock foundation and high-quality masonry workmanship enhance the reliability of the results by eliminating other failure modes associated with poor construction methods or weak foundation material [3]. The suggested method is tested using two archeological sites in Israel: The Nabatean city of Mamshit in the Negev desert and the Ayyûbid-Crusader Fortress of Nimrod, at the foot of Mt. Hermon (Figure 1). For each site, a sensitivity analysis is performed to determine the most likely amplitude and frequency of motion that must have generated the observed failure in the field. The numerical analysis is performed using the discontinuous deformation analysis (DDA) method, a member of the family of implicit DEM methods [5] developed by Shi during the late 1980s [6]. Here, we use a research version of DDA software modified by DDA author Dr Gen-hua Shi for the specific targets of this research program. The DDA method solves a finite element type of mesh, where all elements are real isolated blocks and the unknowns of the equations are the displacements and deformations of the blocks. The blocks are simply deformable, namely stresses and strains are constant throughout the block. When the blocks are in contact, Coulomb s friction law applies to the contact interface, and the simultaneous equilibrium equations are formulated and solved for each loading or time increment. The formulation is based on the minimization of the system potential energy, following the second law of thermodynamics. One of the strengths of the DDA method, which is extremely important for this research, is that the mode of failure of the block system is a result of the analysis and not an assumption. This feature enables us to study the response of the structure to different loading functions and to isolate the parameters that produce the unique failure that is observed in the field. Over the last decade, researchers in the DDA community have dedicated a great deal of effort to prove the accuracy of the method by performing theoretical validation studies. MacLaughlin and Doolin [7] review more than 100 validation studies with respect to analytical solutions, laboratory and field data, and other numerical techniques. DDA performance was found to be more than adequate for engineering applications. Up to date, DDA has been used for numerous rock engineering projects. In Israel alone, DDA has been applied to solve dynamic stability issues in the highly discontinuous Masada rock slopes [8] to stabilize a discontinuous overhanging cliff [9] and to determine the stability of historic underground excavations in discontinuous rock [10, 11]. Unlike these applications, which involve potentially hazardous rock masses, this work uses the DDA method to examine the stability of man-made masonry structures.

3 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1323 Figure 1. Location map of studied sites (after [4]) shown on two sides of the seismically active Dead Sea transform (DST).

4 1324 R. KAMAI AND Y. H. HATZOR 2. NUMERICAL CODE VALIDATION Validation studies in this section were performed for calibration purposes only, using both existing and newly developed analytical solutions. The validations are fully dynamic, meaning that the velocity of each block at the end of a time step is fully transferred to the next time step with no energy dissipation (in DDA notation k01= 1) Dynamic sliding of a block on an incline The case of a sliding block on an incline subjected to gravitational loads only has been successfully analyzed and discussed in detail in previous papers [12], and hence will not be repeated here. The case of a sliding block subjected to dynamic loading, however, is addressed here in detail. The analysis presents higher accuracy than previously described [13] due to a correction that has been made here to the analytical solution. A displacement-based sliding block model was first proposed by Newmark [14] as well as Goodman and Seed [15], now largely referred to as Newmark -type analysis. Determination of the amount of displacement during an earthquake necessarily involves two steps [15]: (1) determination of horizontal acceleration required to initiate down slope motion, also known as yield acceleration (a y ), 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. Figure 2 displays a single block resting on a plane inclined at an angle α with friction along the interface, and subjected to gravitational acceleration g and a horizontal, time-based, sinusoidal acceleration input a as driving forces. Goodman and Seed [15] showed that for this case the yield acceleration is given by a y =tan( α)g (1) g(cosα-ksin( ωt)sin α)*tg φ a=kgsin( ωt) gsin α+kgsin( ωt)cosα g Figure 2. Schematic presentation of the forces acting on a single block on an incline. The driving forces are gravity and a horizontal, time-based, sinusoidal acceleration. The resisting frictional force is constant and depends only on the friction angle (μ=tan ). α

5 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1325 For an acceleration record of the form a =kg sin(ϖt), where ω corresponds to the frequency of the function and k calibrates the proportion between a and g, the corresponding time interval θ until yield acceleration is attained is given by θ= sin 1 (a y /kg) ϖ (2) The down slope acceleration of the sliding block can be determined by subtracting the resisting forces from the driving forces, as described in Figure 2 a t =[kg sin(ϖt) cos α+g sin α] [g cos α kg sin(ϖt) sin α]tan (3) Similarly, the displacement of the block at any time is determined by double integration on the acceleration, with θ as reference datum, in a conditional manner: if a>a y or v>0 t d t = v = a = g[(sinα cosαtan )(t 2 /2 θt)] (4) θ θ + ag [(cosα+sinαtan )(ϖ cos(ϖθ)(t θ) sin(ϖt)+sin(ϖθ))] ϖ2 otherwise d t =d t 1 Equation (4) provides the analytical solution for the dynamic displacement of a block on an inclined plane with inclination α and friction angle, starting from rest and subjected to a sinusoidal loading function with frequency ω. Figure 3 displays a comparison between the analytical and DDA solutions for three friction angles ( =22, 30, and 35 ) with θ=0.035,0.18, and 0.28 s, respectively. The accumulated displacements are computed for a plane inclination of 20. The numerical error E N showninthelowerpanelof Figure 3 is defined as E N = d d N d 100% (5) where d and d N are the analytical and the numeric displacement vectors, respectively. is the norm operator, which for a 2D displacement vector is d = u 2 +v 2 (6) Other than for the first second, in which the relative error reaches 50%, due to a very small absolute error of only 2.6E10 5 m, the relative error is less than 1% for all three friction angles for all accumulated displacements. Time-step size is kept constant in all DDA runs and is s.

6 1326 R. KAMAI AND Y. H. HATZOR Figure 3. Validation of the dynamic case of a block on an incline for three different interface friction angles: top input acceleration function; center comparison between analytical (solid line) and DDA (symbols) solutions; and bottom relative error for each simulation Block response to induced displacements in the foundation In the previous section, a validation was performed for a block on an incline subjected to dynamic input applied to its center of mass. We now present a validation for the case of a block resting on a horizontal interface and responding to dynamic input applied to the underlying block. Such an application would be suitable for dynamic analysis of structures which respond to earthquake motions at their foundations. This approach was not adopted in this current paper because application of this loading mechanism for cases that involve multiple blocks seems to require further research. Nevertheless, we feel it is instructive to present this original validation here because this is the first time such a validation is performed and the obtained results are promising. The studied block system consists of three blocks: a fixed foundation block (no. 0), the induced block (no. 1), and the responding block (no. 2) (Figure 4). Block 1 is subjected to a horizontal displacement input function in the form of a cosine, starting from 0 (shown in the upper panel of Figure 5) d(t)= D(1 cos(2πωt)) (6 )

7 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1327 y Block 2 x Block 1 d(t) Block 0 Figure 4. The DDA block system and sign convention used for the case of a single block resting on an underlying block which is subjected to time-dependant displacement. Input Displacement (m) D=0.3 m D=0.5 m D=1 m 2D DDA DDA DDA Analytic Analytic Analytic Input Motion Displacement of upper block (m) relative error (%) Time (sec) 2 Figure 5. The responding block motion (Figure 4) for three different input displacement amplitudes, all at f =1Hz: top input displacement function of the form d(t)= D(1 cos(2πωt)) applied to Block 1; center comparison between analytical (solid line) and DDA (symbols) solutions; and bottom relative error for each simulation. The dynamic response of Block 2 is investigated and DDA results are compared with the analytical solution, the essentials of which are provided below.

8 1328 R. KAMAI AND Y. H. HATZOR The only force acting on Block 2 other than gravity is the frictional force, which immediately determines the acceleration of Block 2: m 2 a 2 = F friction (7) m 2 a 2 =μ m 2 g (8) a 2 =μ g (9) where μ is the friction coefficient. The direction of the driving force is determined by the direction of the relative velocity between Block 1 and 2 (v1 ). When Block 1 moves to the right relative to Block 2, the frictional force pulls Block 2 in the same direction, and determines the sign of a 2. When Block 2 is at rest in relation to the Block 1, the frictional force is determined by the acceleration of the bottom block (a 1 ). The threshold acceleration, under which the two blocks move in harmony, is equal to the friction coefficient multiplied by the gravitation acceleration (μg). When the acceleration of Block 1 passes the threshold value, the frictional forces act in the same direction as a 1. The positive direction is determined by the sign convention in Figure 4, and the relative velocity of Block 1 is given by v 1 =v 1 v 2 (10) The direction of the acceleration of Block 2 is set by the following boundary conditions: if v1 =0 and a 1 <μg, a 2 =a 1 and a 1 >μg and a 1 >0, a 2 =μg and a 1 <0, a 2 = μg if v1 =0 and v 1 >0, a 2 =μg and v1 <0, a 2 = μg (11) In the solution of Equation (11) we employed Matlab 7.0 [16] software package because the analytical solution must be computed iteratively as the relative velocity and the direction of the force are dependent on each other. In order to compare DDA and the analytical solution, the mode of failure of the analyzed block in DDA has to be constrained to horizontal sliding only, so that it would simulate a single degree of freedom system as does the analytical solution. Therefore, Block 2 of the DDA block system was made very flat, to eliminate possible rotational motions (Figure 4). Figures 5 and 6 present the accumulated displacement of Block 2 and the relative error between analytical (Matlab) and DDA solutions, using a set of sensitivity analyses for amplitude and friction coefficient. Figure 5 presents the response of Block 2 to changing displacement amplitudes (D),

9 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1329 Figure 6. The responding block motion (Figure 4) for three different friction coefficients, all at D = 0.5m, f =1Hz: top the input displacement function applied to Block 1; center comparison between analytical (solid line) and DDA (symbols) solutions; and bottom relative error for each simulation. with a constant input frequency of 1 Hz and a friction coefficient of 0.6. The obtained magnitude of accumulated displacement is directly proportional to the input amplitude, as expected. Note that the three displacement curves follow the periodic behavior of the input displacement function (T =1s), and that divergence between curves starts after 0.25s, where the input displacement function has an inflection point. The relative error, as shown in the upper panel of Figure 5, is mostly between 1 and 2%, reaching a maximum of 3.5% for D =0.3m, which actually corresponds to a difference of 1 cm only in the displacements. Figure 6 presents the response of Block 2 to changing friction coefficients across the interface, with a constant displacement function of D =0.5m, f =1Hz. Note that the accumulated displacement is directly proportional to the friction coefficient up to 0.5 s, after which the input displacement function changes direction. After that point, the accumulated displacement for μ=0.6 is larger than for μ=1, since the higher friction works in both forward and backward directions. It is interesting to note that the relative errors (lower panel of Figure 6) show large sensitivity to the friction coefficient. For a reasonable friction coefficient of say 0.6, the relative error remains within 2%. However, when the friction coefficient is very low (μ=0.1) or relatively high (μ=1), the errors increase up to 12%. Note that the errors for the high friction follow the displacement

10 1330 R. KAMAI AND Y. H. HATZOR pattern, with the maximum errors closely corresponding to change in the direction of motion, where the system experiences high accelerations and inertia forces. In general, it is shown that DDA follows the analytical results in both cases with a remarkable agreement, both for changing friction coefficients and for changing motion amplitudes. It is important to note that while application of this loading mechanism yielded good agreement for a single responding block as shown above, when we experimented with problems involving multiple blocks we obtained a complex structural response not supported by field observations. We feel that while this loading mechanism is more appropriate for the problem at hand, further research of its numerical implementation is required. 3. RESULTS We will present our results in the form of sensitivity analyses for amplitude and frequency of the input loading function. As mentioned previously, the input acceleration function is applied simultaneously at the centroid of each block in the mesh, simulating therefore structural accelerations. The relevant ground motion corresponding to this excitation mode is discussed in the next section. Since the final objective of the analysis is to successfully mimic an existing failure in the field, the sensitivity analyses must be performed in trial and error: an artificial, sinusoidal acceleration record is applied and the responses of the masonry structures are studied up to the point of incipient failure, in a mode similar to the one observed in the field. Note that the magnitude of the measured displacement in the field could be a result of structural response to several, consecutive, large earthquakes over the life span of the studied structure, which in our case is in the order of years. The analysis herein assumes that the observed failure is a result of a single shaking event, in which all shaking characteristics have come together to create the unique pattern of deformation. Therefore, in addition to mimicking the mode of failure, the results are also discussed in terms of the magnitude of displacement, where applicable. In order for the analysis to be as accurate as possible, genuine mechanical properties of the analyzed structures are used as input for DDA. Original building stones were taken from the archaeological sites, under supervision of the Israel Nature and Parks Authority, to the Rock Mechanics Laboratory of the Negev at Ben-Gurion University. Lab tests were performed in order to obtain physical and mechanical properties of intact rock samples and results are summarized in Table I. The input stiffness of the embedding wall is reduced by four orders of magnitude in order to simulate the soft filling material between stones in the wall. Higher stiffness values for the confining wall were investigated, and it was found that the arch rocking behavior is extremely limited at high confining wall stiffness, restricting the initiation of the unique failure mechanisms observed in the field. Choosing a low value, in the order of soil stiffness, allows for large deformations under low stresses. Consequently, the ground motion that is claimed to have caused the deformations is a lower bound on the expected ground motions at that site. A friction angle of 35 for the stone interfaces was chosen as a representative value for all analyses. It was found that the response is not very sensitive to the friction angle, and that friction angles between 25 and 40 yielded keystone displacements of cm, respectively, with the same unique failure mechanism. All numerical analyses are performed with kinetic damping of 1% (k01= 0.99), namely, the initial velocity in each time step is 1% lower than the terminal velocity of the previous step. The introduction of kinetic damping in the analysis stems from previous work that suggests that with an increasing number of blocks in the mesh, a certain degree of energy dissipation is required for

11 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1331 Table I. Mechanical properties obtained in the lab from the original building blocks taken from the studied sites. Mechanical property Mamshit Nimrod Fortress Lithology and formation Hazera limestone Hermon limestone Density (kg/m 3 ) Porosity (%) Dynamic Young s modulus (GPa) Dynamic Poisson s ratio Dynamic shear modulus (GPa) Point load index (MPa) Uniaxial compressive strength (MPa) Peak interface friction angle ( ) Owing to sampling difficulties at NF, these parameters are derived from tests performed on blocks sampled at the Avdat site [17] with similar rock properties. Figure 7. The deformed arch at Mamshit: (a) general view of the arch which is embedded in a very heterogenic wall and (b) the keystone has slid 4 cm downwards while the rest of the arch remained intact. realistic modeling [8, 18], because DDA does not consider inelastic energy dissipation mechanisms such as crushing at block ends, out-of-plane rotations, heat generation, etc The case studies Two archaeological sites in Israel, where a confined structural failure is identified, measured and mapped are selected for analysis (Figure 1). Mamshit is the last Nabatean city built in the Negev on the trade route between Petra, Hebron, and Jerusalem [19, 20]. A unique structural failure is noticed in a tower at the corner of the Eastern Church, where a key stone has slid approximately 4 cm downwards out of a still standing semi-circular arch (Figure 7). The numerical analysis of the arch at Mamshit is performed on a block system (Figure 8) that contains the arch, with its accurate measurements from the field, confined by a uniform masonry wall. The arch is intersected by material lines which enable us to assign separate sets of mechanical parameters for the wall and the arch. These simulate the great difference between the

12 1332 R. KAMAI AND Y. H. HATZOR Figure 8. The DDA block system for the embedded arch at Mamshit. The modeled masonry wall rests on two fixed blocks. The lines intersecting the arch blocks represent material lines and a measurement point (circle) is assigned for the keystone. hewn stones of the arch itself and the heterogenic confining wall material (Figure 7) by assigning intact rock stiffness values to the arch stones (E arch =17GPa) and soil-like stiffness values to the wall stones (E wall =1MPa). A measurement point is assigned at the keystone, and its vertical displacement versus time is plotted in the relevant figures. In Nimrod fortress, the largest medieval fortress in Israel [21, 22], evidence of destruction caused by a severe earthquake can be seen throughout the fortress. The damage is dated to one of two large earthquakes of the year 1759 (Ms=6.6 and 7.4) [23]. The most impressive evidence from the earthquake that has damaged the fortress can be observed in the gate tower. There, three parallel arches as well as an arched passageway display the same mode of failure, in which a single block, not necessarily the keystone, has slid downwards while the rest of the arch remained intact (Figure 9). In the modeled block system (Figure 10), material lines intersect the arch blocks and allow us to assign separate sets of mechanical parameters to the arch and the wall, as before. The wall in which the arch is embedded is confined by a boundary block on its right-hand side and its base, added in order to simulate the geographical constraints at the site: the fortress is built on an elongated range, cut off by deep canyons from three sides. As a result, the gate tower is supported by a 30 m high retaining wall on its western side, while the eastern side rests directly on bedrock. This setting may certainly cause asymmetric structural response and possibly seismic wave amplifications in preferred orientations. Five measurement points are assigned to five blocks in the arch, the label and position of which are shown in Figure 11. During shaking, the entire modeled structure is gradually displaced to the left, most probably due to the confining wall on the right-hand side which represents a fixed boundary. In order to represent correctly the computed displacement of block A R (Figure 9(a)), the block that exhibits the greatest displacement in the field in relation to the computed displacements of the entire block system, the selected parameter for sensitivity analyses was the relative displacement of A R with respect to the average displacement of the other four keystones. The following equations describe the derivation of that parameter: ui u avg = 4, v vi avg = (12) 4 u = u(a R ) u avg, v =v(a R ) v avg (13)

13 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1333 Figure 9. Evidence for a destructive earthquake that hit the Nimrod Fortress most probably at 1759 (see text for explanation): (a) the same stone has slid downwards out of three parallel arches at the main gate arch; (b) a similar deformation is noticed at a passage-way parallel to the gate arch; (c) an adjacent arch facing east was not deformed; and (d) location of the passage-way, 10 m west of main gate. where u versus v is the relative spatial displacement of block A R, plotted in the relevant figures Sensitivity analyses A very interesting and unique behavior of the two structures is revealed through the sensitivity analyses of structural response to changing amplitude of motion. Instead of an intuitive proportion between induced motion amplitude and degree of damage, measured here by the magnitude of the displacement vector, there is apparently a structural preference to a certain range of amplitudes:

14 1334 R. KAMAI AND Y. H. HATZOR Figure 10. The DDA block system for the studied arch at Nimrod Fortress. Four fixed points (squares) are assigned to the confining block and five measurement points (circles) are assigned to the top arch blocks. The lines intersecting the arch blocks represent material lines. Figure 11. Sign convention for the Nimrod arch blocks: K =keystone, A and B for the first and second block from the keystone respectively, and R or L to indicate right or left. in both Mamshit and Nimrod, the studied block exhibits the greatest displacement under specific amplitudes, not necessarily the largest. Figure 12 shows that the keystone of the arch at Mamshit exhibits the greatest downward displacement under an amplitude of 0.5g, when everything else is kept equal. Downward displacement increases with acceleration amplitude up to an amplitude of 0.5g. When the acceleration amplitude is greater than 0.5g, the keystone response exhibits strong fluctuations and even a shift in displacement direction. A similar phenomenon is observed in Nimrod. Figure 13 shows the spatial displacement of block A R in response to changing amplitudes. Under the smallest amplitude presented (A =0.8g), a very minor response is detected where the block displaces inwards and later outwards, resulting in almost no accumulated displacement. The largest amplitude (A = 1.5g), however, has a strong affect on the entire structure leading to partial destruction, but consequently block A R is locked

15 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1335 Time (sec) Key Block Displacement (cm) g 0.32 g 0.5 g 1g 0.8 g Figure 12. Influence of input motion amplitude on vertical keystone displacement at Mamshit ( f = 1Hz). Figure 13. Influence of input motion amplitude on relative spatial displacement of Block A R at Nimrod Fortress ( f =2Hz). after 16 s of shaking. A continuous, accumulating inward displacement at a constant rate and direction is achieved only when A =1g. The inward displacement of block A R decelerates only after 70 s when further displacement is locked, as shown in Figure 14. Note that also in the real event the structure must have locked at a certain point, and the measured displacement that was preserved matches the modeled displacement with A =1g. Structural response to frequency of motion is usually discussed in terms of the natural period of the structure, at which the structure will develop a resonance mode and collapse. Since the studied failures are local and not complete, each mode of failure will have its own natural period which can be different than that of the whole structure. The structural sensitivity to frequency, revealed in our sensitivity analyses, is significant considering that the common terminology for seismic risk evaluation uses mainly PGA (peak ground acceleration) and largely ignores frequency.

16 1336 R. KAMAI AND Y. H. HATZOR Figure 14. Normalized displacement of stone A R in the direction of the displacement vector over time (d = u 2 +v 2 ). After approximately 70 s, the displacement stops and the structural deformation is locked. Time (sec) Key Block Displacement (cm) Hz 5Hz 1Hz 15*Nuweiba 1.5 Hz 10 Hz Figure 15. Influence of input motion frequency on vertical displacement of keystone at Mamshit ( A = 0.5g). In the case of the keystone at Mamshit (Figure 15), a clear preference is detected for frequencies in the range of Hz; only under those input frequencies the downward displacement of the keystone is continuous, and accumulates more than 3 cm of displacement, similar to the amount of displacement measured in the field. Lower or higher frequencies result in other modes of failure such as oscillations in the case of low frequencies (e.g. 0.5 Hz) and locking of the structure in the case of higher frequencies (e.g Hz). A very important observation is made when a real earthquake record, that of Nuweiba 1995 (Figure 16), is used as input motion. The original record, de-convoluted to rock response [8], is amplified by 15 in order for its PGA to reach the same amplitude of the synthetic motions used for the results that are plotted in Figure 15, namely 0.5g. Although the Nuweiba quake loads the structure with a wide range of frequencies and with two simultaneous components of motion (horizontal and vertical), the structural response to the natural

17 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS g(z) g(ew) g(ns) a (g) time (sec) Figure 16. The Nuweiba 1995 record after de-convolution to rock response. The rectangle marks the 10 s that were used for the analysis of the Mamshit block system. Figure 17. Influence of frequency on relative spatial displacement of Block A R at Nimrod Fortress (A =1g). quake is very similar to that of the simple sinusoidal ones, though more moderate. This finding strongly suggests that the results of the sensitivity analysis, using synthetic records of horizontal motion only, are valid enough to be further discussed. A similar observation is made in Figure 17, which shows that the mode of failure of the arch at Nimrod Fortress has a clear preference to a frequency of 2 Hz over both 1 and 3 Hz. 4. DISCUSSION 4.1. Limitations of our approach The suggested method uses a 2D numerical model with simple deformable blocks, meaning that stresses and strains are identical everywhere in the block. Although the mode of failure in DDA is a result of the analysis and not an assumption, the 2D constraint imposes limitations on the mode of

18 1338 R. KAMAI AND Y. H. HATZOR failure, for example, it does not allow for out-of-plane rotations, etc. To minimize this constraint, the analysis was performed on planar structures such as arches, where most of the deformation is concentrated in the plane of analysis. An analysis of a rectangular structure, such as a tower, could have been significantly biased by the 2D constraint. Moreover, the simple deformable block assumption may cause small deviations in the energy involved in the deformation, since in reality stress concentrations at block corners may result in block cracking or chipping, a mechanical process that cannot be modeled in this version of DDA. We assume, though, that an energy dissipation of 1%, which is implemented through the numerical kinetic damping, is enough to compensate for such losses [18]. Another limitation of this study is the use of a synthetic sinusoidal loading function as direct input, consisting of a single amplitude and frequency value. This is obviously not the case in a real earthquake which consists of a train of frequencies at different amplitudes. Nevertheless, the results presented in Figure 15 indicate that the structural response in this case does not vary all that much when a real earthquake record is used as input Applicability of our results The obtained PGA and frequency values in this research can be compared with Israel Building Code # 413 [2], generated largely on empirical grounds by the Geophysical Institute of Israel. IBC 413 predicts PGA values of 0.11g for Mamshit and 0.25g for Nimrod Fortress. Although our results are significantly higher, it can be argued that at least two amplification factors can explain the discrepancies: structural and topographic. The loading function in the numerical analysis is inserted into all the blocks simultaneously, namely, the analyzed block is subjected to a uniform structural input motion. The building code however relates to bedrock PGA, which may be amplified significantly at the top of the structure due to inherent structural amplifications as well as rock structure interactions. Common response spectra predict amplification of up to 2.5 [24], due to wave propagation phenomena that take place when the motion is transferred from bedrock to the structure. Therefore, the PGA values found by inversion in our study must be reduced by a factor of 2.5, resulting in bedrock PGA predictions of g and 0.4 1g for the Mamshit and Nimrod sites, respectively. No further modifications should be made for the PGA value obtained for Mamshit, because topographic effects are not expected in this site. In Kalaat Nimrod, a further modification should be made due to the possible topographic effect at that site, the precise magnitude of which is not known and must be measured in the field. In the absence of such data we can only suggest that the actual PGA value for the rock foundations at Kalaat Nimrod should be lower than 0.4g by up to a factor of 4, if we adopt the results of the topographic site effect measurements at four different sites in Israel [25]. 5. SUMMARY AND CONCLUSIONS DDA method was validated with respect to analytical solutions for two separate cases of a Newmark sliding block type of analysis: in one, a block is subjected to time-dependent input acceleration applied at its center of mass, and in the other, the block responds to a time-dependent input displacement applied to an underlying block. In both simulations, DDA results matched the analytical solutions very well.

19 NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1339 The structural deformation pattern observed in the two case studies was duplicated very nicely with DDA, with localized failure of particular blocks in the mesh-matching field observations. Both deformation pattern and displacement magnitude were duplicated accurately in two different structural geometries and in two different sites. For the semi-circular arch in Mamshit, DDA results imply that most of the damage took place during the first two seconds of motion. Our best estimate for the structural acceleration parameters for Mamshit is f =1Hz, A =0.5g. Considering structural amplifications these results can be downscaled to f =1Hz, A =0.2g. The obtained amplitude is similar to PGA=0.11g predicted by Israel seismic building code for that region. Interestingly, introduction of vertical motions using a real earthquake record does not change the obtained deformation picture very significantly, justifying our simplified methodology of using synthetic sinusoidal input motions which are restricted to the horizontal axis only. For the asymmetric arch at Nimrod our best estimate for the structural acceleration parameters is f =2Hz, A =1g. Considering both structural and topographic amplifications, the actual range of the paleoseismic PGA values can be between 0.1g and 0.4g. Assuming structural amplification of 2.5 and topographic amplification for this elevated site of 2.0, the expected paleoseismic PGA is 0.2g. This estimation can be compared with Israel seismic building code prediction of 0.25g for this region. Further research is required concerning failure modes associated with application of the loading function at the foundation of a multi-block structure. The application of this loading mechanism will generate true structural amplifications at the top of the structures and will enable more rigorous conclusions regarding paleoseismic PGA values. ACKNOWLEDGEMENTS We thank Gen-Hua Shi for modifying his latest version of the DDA code for the purpose of this research, Yuli Zaslavski for the de-convoluted Nuweiba earthquake record, Shmuel Marco and Itai Leviathan for discussions, and two anonymous reviewers for critical reading and instructive comments that greatly improved this paper. REFERENCES 1. Boore DM, Joyner WB, Fumal TE. Equations for estimating horizontal response spectra and peak acceleration from western North American earthquakes: a summary of recent work. Seismological Research Letters 1997; 68: Shapira A. Explanations to the PGA Map for the Israeli Building Code 413. Available from: Mazor E, Korjenkov A. In Applied Archeoseismology, Mazor E, Krasnov B (eds), vol. 4. The Makhteshim Country Laboratory of Nature, Pensoft: Moscow, 2001; Hall JK. Digital Shaded-relief Map of Israel and Environs 1:500,000. I. G. Survey, Jing L. A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering. International Journal of Rock Mechanics and Mining Sciences 2003; 40: Shi G-H. Block System Modeling by Discontinuous Deformation Analysis. Computational Mechanics Publication: Southampton, U.K., MacLaughlin MM, Doolin DM. Review of validation of the discontinuous deformation analysis (DDA) method. International Journal for Numerical and Analytical Methods in Geomechanics 2006; 30(4): Hatzor YH, Arzi AA, Zaslavsky Y, Shapira A. Dynamic stability analysis of jointed rock slopes using the DDA method: King Herod s Palace, Masada, Israel. International Journal of Rock Mechanics and Mining Sciences 2004; 41(5):

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