Simulation of Toppling Columns in Archaeoseismology

Transcription

1 Bulletin of the Seismological Society of America, Vol. 99, No. 5, pp , October 2009, doi: / Simulation of Toppling Columns in Archaeoseismology by Klaus-G. Hinzen E Abstract Since the early days of modern seismology, toppled artifacts such as tombstones and single columns have been used in the aftermath of earthquakes to deduce parameters of site-specific ground motions. The artifacts were generally treated as rigid bodies. Later, the theory of rigid block movements was also applied to precariously balanced rocks toppled by earthquakes. While the movements of a single rocking block can be described analytically, slide-rocking movements, bouncing, and multiple block systems require a numerical approach. We use multiple rigid block models with viscoelastic coupling forces in combination with full 3D ground motions (measured and synthetic) to analyze the dynamic response of building elements, relevant for archaeoseismological studies. First, the numeric modeling results are verified by comparison with analytically determined rocking motions of a single rectangular block. Stiffness and damping parameters of the coupling forces are adjusted to results from analog experiments with a rocking marble block. A model of a monolithic column and one consisting of seven drums is used to test the influence of the geometry and friction on the toppling behavior. The main question addressed in this study is whether toppled columns give a clear indication of the back azimuth toward the earthquake source. Input motion from 29 strong-motion records indicates little correlation between downfall directions and back azimuth. Clearly directed horizontal ground movements tend to topple the columns in the transverse direction. More complex ground motions result in quasi-random downfall directions. The friction coefficients have a minor influence on the downfall directions. Synthetic ground motions for two earthquakes with different source mechanism show toppling directions toward and away from the source as well as in the transverse bearing. However, it is not straightforward to deduce a reliable source location from the inversion of the toppling directions. Online Material: Movies of simulated block movement and toppling behavior, and figures of synthetics ground displacement. Introduction In the study of preinstrumental earthquakes, historical seismology and palaeoseismology are well-established branches of seismological sciences. Techniques to evaluate ground motions and parameters of causing earthquakes that have left their mark in written documents and in the nearsurface geology were developed. Ever since man-made structures have been erected, earthquakes have also left their marks on these constructions. However, damages in archaeologically excavated buildings or continuously preserved monuments are often hard to unravel in terms of the causative effects. The new branch of seismological sciences, archaeoseismology, is defined as the detailed study of preinstrumental earthquakes that, by affecting locations of human occupation and their environments, have left their mark in the archaeological record (Buck and Stewart, 2000). Following this definition, the detailed study of earthquakes is the focus, and compilation, modeling, and interpretation of damage data is a means to an end. The main questions to be answered by archaeoseismic investigations are (1) how probable is seismically induced ground motion as a cause of damage observed in man-made structures from the past, (2) when did the damaging ground motion occur, and (3) what can be deduced about the nature of the causing earthquake (Galadini et al., 2006). All three questions should be answered before results from archaeoseismic case studies can be included in seismic hazard analyses. While the first problem requires input from multiple disciplines including civil engineering, geophysics, and geotechnics, the second is a task for the geological sciences and archaeometry. The third question requires input from several seismological 2855

2 2856 K.-G. Hinzen specialties ranging from seismotectonics to source studies, wave propagation, and site-effect modeling to soil building interaction. The two main parameters to be extracted from these studies are the location and a measure of the strength of the causing earthquake, where both are often intrinsically tied to each other. While the macroscopic degree of damage is directly connected to the nature of ground motion at the site, many techniques from macroseismic data analysis can be adopted; determination of the direction of ground motion that caused damage or the wave propagation direction, which is even more complicated, remains a challenge in archaeoseismology. Several case studies have been published in which surface rupturing of the fault plane directly affected man-made structures (i.e., Galadini and Galli, 1999; Galli and Galadini, 2001; Meghraoui et al., 2003; Galli et al., 2008; Marco, 2008). In these cases the location of the activated fault section is evident and source parameters such as surface rupture displacement can, under favorable conditions, be determined within small ranges of uncertainty. However, in cases where the activated segment of a fault is remote from the site with archaeological findings of building damages (archaeodamages), source identification is a challenging task. No systematic studies or established methods presently exist to deduce the location of the rupturing fault from archaeodamages. Several case studies purported to infer back azimuth to the activated parts of a fault from directional features in archaeo-damages (Korjenkov and Mazor, 2003). However, these interpretations usually suppose a very simple behavior for site-specific ground motions and do not consider the complexity of an extended seismic source and the uncertainties imposed by either the complexity of or randomness in the reaction of building components. In the seismic design of modern buildings, finite element models are frequently used to study the behavior of the complete structure or crucial components of the structure (i.e., Meskouris et al., 2007). Most important in these kinds of studies are the correct modeling of the elastic or nonlinear material behavior to deduce the capacity of concrete beams, steel frames, shear walls, or wooden beams (Hinzen and Weiner, 2009). While finite element calculations predict the dynamic load under which a certain component will fail, the movement of structural parts of a building in the collapse phase cannot be modeled. The latter parameter is important to analyze directional features in archaeo-damages. Wellpreserved ancient buildings were often constructed from blocks of natural stone, sometimes without any cementation. This was often the case in classic Greek buildings. For such structures it is feasible to use rigid block models and Newtonian mechanics to study the block movement and interaction as a first modeling attempt (Sinopoli, 1995; Papantonopoulos et al., 2002; Psycharis, 2007). A related problem is that of the dynamic stability of precariously balanced rocks, which have been used as low-resolution strongmotion seismoscopes (Brune, 1996). Anooshehpoor et al. (2004) used a numeric approach to model the 2D dynamic response of rocks to arbitrarily complex acceleration time histories (Purvance et al., 2008). In this study we adapt rigid block techniques to the needs of archaeoseismology. Specifically, we regard linear viscoelastic coupling forces with finite friction in multiple block systems and true 3D ground-motion excitation. We use a numeric rigid block model of two cylindrical columns regarded as simple archaeo-seismoscopes. One monolithic column and one column consisting of seven separate column drums is used to study systematically the effects of (1) the columns slenderness, (2) changes in frictional coefficients, and (3) variability of measured and synthetic ground motions on the downfall directions. Directional Features in Archaeo-Damages In his famous work about the Neapolitan earthquake in 1857, Robert Mallet (1862) not only prepared the ground for evaluating earthquake strength with macroseismic methods, he also tried to infer the earthquake location from directional damage features. Without a scientific basis, still being actively sought (Ambraseys, 2006; Marco, 2008), the practice suggested by Mallet (1862) should not be applied. While Mallet used fresh traces of directional damage, in archaeoseismology such features have gone through altering processes, making it more difficult to deduce accurate directions toward the earthquake source. As summarized by Galadini et al. (2006, and references therein), typical earthquake effects on constructions are (1) cross fissures in the vertical plane due to shear forces, (2) corner expulsion due to orthogonal motion of walls, (3) lateral and rotational horizontal and independent motion of blocks within a wall, (4) height reduction due to vertical crashing, (5) deformation of arch piers including collapse of key stones, (6) wall tilting and distortion, and (7) rotation or toppling of pillars or parts of it and drums of columns. Additional photos of typical damages are given by Marco (2008). The direction of any of these seismogenic damage patterns always results from the coaction of the orientation of the structure or structural component and the orientation of the ground movement, which is influenced by the source characteristics and site conditions. Korjenkov and coworkers (i.e., Korjenkov and Mazor, 1999a,b, 2003; Al-Tarazi and Korjenkov, 2007; Korzhenkov et al., 2009) have given examples how directionalities in archaeoseismic damage can be quantified for a certain site. This quantification works best when carried out during an ongoing excavation. Because this is not always feasible, measurements are taken from preserved ruins (Galadini and Galli, 2001) or from the documentation of former excavations (i.e., Hinzen and Schütte, 2003). Korjenkov maps the direction of cracks and rotations of blocks with respect to the trend of walls. At sites with numerous damaged walls, a statistical approach can reveal preferred directions in the damage pattern. While such preferred directions of block shifts, rotations, toppling of wall fragments or complete walls, and

3 Simulation of Toppling Columns in Archaeoseismology 2857 toppling of columns provide strong arguments for significant ground-motion amplitudes in a certain direction, it is not straightforward to deduce the back azimuth to the causing earthquake from this direction. Near fault strong ground motions are influenced by the source mechanism, rupture process details, distribution of asperities, fault plane geometry and extension, wave spreading conditions, and site conditions (i.e., Erdik and Durukal, 2004, and references therein). Also, secondary earthquake effects due to deformation of soft subsoils that form directional features such as foundation cracks and inclined walls (Hinzen and Schütte, 2003) should not be mistaken for the bearings toward the earthquake source. Among the most obvious and promising directional archaeo-damages are toppled columns (i.e., DiVita, 1996; Nur and Ron, 1996; Marco, 2008) and columns with displaced column drums (Stiros, 1996; Bottari, 2005; Psycharis, 2007). As rotation-symmetrical construction elements, columns should react similarly to ground motions in any direction when freestanding and not connected to neighboring structural elements. This feature makes columns a good universal seismoscope. Even though such freestanding columns are rare in cultural heritage, the independence from a fixed trend of the structure with respect to unknown ground motions makes them a versatile tool to study basic toppling effects. Even the simplest object, a monolithic block on a plane subsurface, demonstrates a complex dynamic behavior including stress discontinuity, structural damping, contact friction, and impact (Sinopoli, 1995); all of which can influence toppling behavior and downfall directions. Many open questions remain about the particular influence of each of these factors. Several authors have studied the dynamic behavior of cylindrical structures and classical columns. Koh and Mustafa (1990) studied the free rocking motion of rigid cylinders for various initial conditions and a stationary foundation during the motion of the cylinder. They numerically integrated the exact equations of motion of the model and mapped the boundary between toppling and not toppling. Koh and Hsiung (1991a,b) extended the 2D model to 3D rocking, rolling, and uplift of a rigid cylinder when subjected to ground motions, showing that 3D motion is significant under earthquakelike excitations. Mouzakis et al. (2002) used a 1 3 analog model of a multidrum column from the Parthenon of the Acropolis of Athens, even made from the same material as the original. Scaled earthquake ground motions in two- and three-dimensions were used to drive a shaking table with forces insufficient to topple the model. They found large deformations during the shaking, which were not necessarily reflected by the residual displacements at the end of the tests. A significant influence of imperfections of the model specimen and a very high sensitivity to even small changes in the input motion parameters were observed. In an accompanying article, Papantonopoulos et al. (2002) successfully used the distinct element method to numerically simulate the behavior of the same column. Konstantinidis and Makris (2005) also used numerical models of multidrum columns represented by a 2D discrete element model allowing rocking, sliding, and slide rocking to show that relative sliding between drums happens even when the g-values of the ground accelerations are less than the coefficient of friction. They concluded that typical classical columns can survive the shaking from strong ground motions near the causative fault of earthquakes with moment magnitudes Additionally, they found a more controlled seismic response of multidrum columns than monolithic columns of the same size. Psycharis (2007) made a backward analysis of a column of the temple of Olympios Zeus in Athens to investigate the seismic history of the area during the 2000 yr since the building was erected. A 3D numeric model of a single and double column structure allowed the author to constrain the maximum ground velocities at the side by comparing the model behavior with the current status of the real structure. By applying archaeoseismologic methods, the orientation of damaged structures should be recoverable in most cases (i.e., Korjenkov and Mazor, 1999a), although usually little or nothing is known a priori about the nature of the ground movement. For elongated structural elements (i.e., a simple freestanding wall) the angle between the polarization of the ground motion and the trend of the element is decisive for estimating dynamic reaction of the structure. In contrast, rotation-symmetrical building elements (i.e., freestanding columns) do not exhibit this dependency. Therefore, to investigate correlations between ground-motion polarizations, back azimuth, and toppling directions, we will limit this study to the case of freestanding cylindrical columns. Rocking of Rigid Blocks Long before a strong-motion instrumentation was available, Milne (1881, 1885), Perry (1881), and others used the theory of dynamic block structures to deduce earthquake ground accelerations from toppled monumental columns and tombstones. A fundamental article on the theory of rigid block movements by Housner (1963) helped to explain observations made during the large Chilean earthquake of May Augusti and Sinopoli (1992) presented a comprehensive summary on the dynamic modeling of large block structures and Sinopoli (1995) reviewed studies of large block structure dynamics. Brune and Whitney (1992), Brune (1996), Anooshehpoor et al. (1999, 2004), and Zhang and Makris (2000) applied rigid block movement models to interpret precariously balanced rocks and a steam engine, the latter overturned during the great San Francisco earthquake of We briefly introduce the problem of rocking rigid block dynamics in order to describe the analog and numeric experiments presented in the main part of this article; a comprehensive description of the theory can be found in Housner (1963) and Augusti and Sinopoli (1992). Figure 1 shows the cross

4 2858 K.-G. Hinzen block is at rest, and the line R is α, and the inclination (rotation) of the block is θ. Housner (1963) solved the equations of motion for the rocking block for one degree of freedom, namely the rotation around the corner points A and A 0 (Fig. 1d). This movement constraint specifies that (1) the friction at the corner points is large enough that no sliding between the block and the base occurs as in Figure 1c and (2) the block does not bounce during the movement through the static equilibrium (Fig. 1f). The equation of motion of the free rocking response of a 2D rigid block without sliding, slide rocking, or free flight behavior was given by Housner (1963): I 0 θ WRsin α θ ; (1) in which I 0 4=3mR 2 is the mass moment of inertia of a homogeneous rectangular block about corner A and W mg is the weight of the homogenous block, m and g are the block mass and acceleration of gravity, respectively. For slender blocks (sin α α), equation (1) can be written as (Housner, 1963; Aslam et al., 1980) θ p 2 θ p 2 α; (2) p in which p p WR=I 0 3g= 4R. In order to start a forced rocking motion, a horizontal acceleration a is necessary that fulfills the condition (Housner, 1963; Augusti and Sinopoli, 1992) a g b h : (3) Figure 1. (a) Cross section of a rectangular block of height h and width b in a rocking experiment. θ is the inclination of the block; R is the vector from the center of gravity, cg, to the actual rocking corner A; and A 0 is the opposite rocking corner. The angle α is a measure for the ratio of h=b. Dashed lines indicate the equilibrium position of the block. The following drawings show schematically the motion types of the block: (b) rest; (c) sliding; (d) rotation (rocking); (e) slide rocking; (f) translational jump; and (g) rotational jump. section of a rigid body in a rocking motion. The block has p height h and width b; the half-diagonal, R h=2 2 b=2 2, is defined as the distance from the center of gravity, cg, to the actual center of rotation of the block, A and A 0, respectively. The angle between the longer block side, which coincides with the vertical direction when the Under more realistic conditions, energy dissipates during each impact of the block on the base. For inelastic impact (no bouncing), the reduction of kinetic energy, r, depends on the square of the ratio of angular velocities before, _ θ i, and after, _ θ i 1, an impact, respectively (Housner, 1963; Aslam et al., 1980; Augusti and Sinopoli (1992), r 1 2 I _ 0 θ 2 i I _ 0 θ 2 i _θi 1 _θ i 2; (4) and for slender blocks the coefficient of restitution is η p 2 b2 r h b α2 : (5) h 2 Housner (1963) showed that in this case the amplitude θ n of the nth cycle can be written as q θ n 1 1 r n 1 1 θ 0 =α 2 ; (6) and the half period of the rocking motion is

5 Simulation of Toppling Columns in Archaeoseismology 2859 r q T 2 2 I 0 tanh 1 r n 1 1 θ WR 0 =α 2 : (7) Amplitudes and the half period of the rocking motion decay rapidly with the number of cycles. Increasingly highfrequency movements follow a few slow rocking motions with large amplitudes. Under less ideal conditions (bouncing during impacts and sliding due to reduced friction), amplitude decay proceeds even more rapidly. The sliding component of the movement is governed by the static coefficient of friction, μ s, and the slenderness of the block. For the free motion of the block Augusti and Sinopoli (1992) showed that the inequality μ s 3 b=h 4 b 2 =h 2 (8) allows separating the conditions under which pure rocking and slide rocking exists. If the condition of equation (8) is not fulfilled, slide rocking, controlled by the kinetic friction coefficient μ k, will start. For small angles for each b=h, a value μ s exists above which only rocking will occur until the block returns to a state of static equilibrium (Augusti and Sinopoli, 1992). Numeric Model Figure 2. (a) Analog and (b) numeric models used to verify stiffness and damping of the contact forces. An accelerometer was cemented to the top of the cm marble block to monitor the rocking movements of the block. The wire-frame virtual block is inclined at an angle of 11.3 close to its indifferent equilibrium (θ 0 =α 1). (E An animation of the rocking-block model is available in the electronic edition of BSSA.) Basic Model Parameters The program code Universal Mechanism (Pogorelov, 1995, 1997) was used for all numeric models in this study. After defining the physical parameters of the bodies, the types and degrees of freedom of the joints between blocks and the types and parameters of the contact forces, the code generates the equations of motion of the mechanical system. An implicit second order method with variable step size was used to solve the equations of motion. Error tolerance was usually set to First, a solution for the static model was calculated and the resulting coordinates were used as initial conditions for the dynamic tests. A dual analog and numeric experiment was carried out to deduce basic parameters for further calculations and to validate the models. For the analog rocking tests, a marble block of cm, sitting on a marble plate of 2 cm thickness and cemented to a foundation, was constructed (Fig. 2). With a mass of m 4267 g, the block has a density of 2:96 Mgm 3. As the block rotates over the longer base (8 cm) h=b comes to 5.0. The motion of the block during the rocking experiments was monitored with a miniature accelerometer mounted on the top of the block (Fig. 2). Acceleration time history was recorded with a 24 bit analog-to-digital converter at a sampling frequency of 25 khz (Fig. 3). A numeric model of a block of similar size (Fig. 2) was used for comparative calculations. In the numeric model the individual bodies (here, base-block and rocking marble block) are treated as rigid. During the rocking-block experiments the base was fixed. The marble block is connected to the base through a six degrees of freedom joint and a viscoelastic contact force, including a sliding and a sticking mode. During sliding, the contact force f is of Coulumb friction type: f μ k F N sign v ; (9) where F N is the normal force on the friction surface, μ k the coefficient of kinematic friction, and v the sliding velocity. The sticking sliding transition occurs when jfj F 0 ; (10) where F 0 is the maximum value of the static friction force F 0 μ s F N. In the sticking mode, the linear viscoelastic friction force is f f 0 c x x 0 dv: (11) Here c and d are the stiffness and damping coefficients, respectively. In a series of numerical rocking tests, c and d were varied between and N=m and 0.01 and 0.99, respectively. As shown in equation (7), the time between two subsequent impacts of the rocking block on the base is strongly dependent on the energy reduction at each impact and hence on the stiffness. Therefore, the synchronicity of the impacts in the twin experiment was the first criterion in the parameter adjustment. Figure 3 compares the acceleration measured at the top of the analog block with the corresponding calculated time series assuming here a contact stiffness of 1: N=m. The time step of the output during the calculation was 4: sec, corresponding to the sampling rate of the measurement. In the first 5 sec of the record up to the eighteenth impact, the impact time in both experiments matches almost perfectly. In the balanced time of the experiment impact, times successively deviate due to the increasing influence of the imperfectly flat bottom of the analog marble

6 2860 K.-G. Hinzen Figure 3. Measured (bottom) and calculated (top) acceleration of the top of the marble block from Figure 2 from a rocking experiment. Starting inclination of the block was θ 0 10 and the ratio of h=b was 5.0. The short bar underneath the second impact indicates the zoomed time window shown in the inset. Here the trace shown as a gray variable area plot is the acceleration measured during the rocking of the analog block. The black seismogram is the result of the numeric experiment using stiffness and damping of 1:2E 7 N=m and 0.36, respectively. Coefficients of static and dynamic friction were 0.7 and 0.6, respectively. block. This also brings the movement to a halt about 1.2 sec earlier than in the numeric experiment. The insert in Figure 3 shows a detailed section of the acceleration impulse of the second impact. The strong positive acceleration impulse indicates the impact of the block on the base, followed by a short ( 2 msec) phase of free fall, as indicated by the acceleration of 1:0g and a concurrent damped oscillatory movement. The free-fall phase is due to a small bouncing effect after the block hits the base. This phase was measured in both the analog experiment and the numerical simulation. The duration of the free-fall time and the amplitudes of the damped oscillation were matched by adjusting the damping coefficient in a trial and error procedure. The results shown in Figure 3 were achieved with a damping coefficient of As long as no initial sliding of the block occurs, the coefficient of friction has minor influence on the impact times and amplitudes. During the parameter optimization of this dual experiment, the coefficients of static and dynamic friction in the calculations were kept constant at 0.7 and 0.6, respectively. compensate for energy losses due to small bouncing effects not included in the analytic Housner (1963) model. Several studies have used horizontal sinusoidal ground motions to simulate a Housner-block model (i.e., Housner, Verification Tests The block from the numeric experiment (Fig. 2) was used next to simulate the free rocking ground motions for the starting rotation angles 0:2α θ 0 α. Amplitudes for the first 10 cycles of movement from the calculation were compared with the analytic values as shown in Figure 4. In order to match the analytical and the numeric results, the energy loss ratio, r, had to be made 1.4% larger in the analytical calculation than suggested by the geometry to Figure 4. Lines show the amplitude decreases of the pure rocking motion of a slender block for different starting positions, θ 0, with progressing number of cycles (after Housner, 1963). The crosses show the amplitudes from corresponding experiments with the numeric model of a rocking block with the dimensions shown in Figure 2. The inset in the upper right corner shows the time series of the angular displacement for the θ 0 =α ratio of

7 Simulation of Toppling Columns in Archaeoseismology ; Ishiyama, 1982; Sinopoli, 1991). Anooshehpoor et al. (1999, 2000) and Zhang and Makris (2000) presented analytic and numeric solutions to the problem of a freestanding block, respectively, exposed to a one-sine pulse. The latter showed that two modes of overturning exist, one with and one without an impact of the block on the base before it overturns. We used their results to test our numeric model by excitation of the block with a single sinusoidal impulse. The modeled block has the dimensions h 3:113 m and b 0:795 m, resulting in α 0:25, p 2:14, and a coefficient of restitution of η 0:9. Frictional parameters were kept the same as in the previous experiment. Contact forces were only implemented between the base of the moving block and the pedestal. Therefore, after overturning, the block can penetrate the pedestal. Figure 5 shows the results of three calculations with maximum acceleration amplitudes of (1) A 6:32αg, (2) A 6:33αg, and (3) A 7:18αg, repeating the calculations of Zhang and Makris (2000). Rotation and angular velocities follow those from the previous study. In case (1) the block experiences one impact before it overturns in the direction of the movement of the first half cycle of the sine pulse. The slightly larger acceleration in experiment (2) does not overturn the block. In test (3) the block overturns in the opposite direction of the movement of the first half cycle of the acceleration pulse without an impact on the base. The agreement of the block movements with the analytically predicted results of Anooshehpoor et al. (1999, 2000) and numerical calculations of Zhang and Makris (2000) confirm the capability of our model to simulate pure rocking motion of a single block. As outlined by Zhang and Makris (2000), the demand on friction to sustain pure rocking motion depends on the Figure 5. Normalized rotation (solid curves), angular velocity (dashed curves), and snapshots of a rigid block subjected to a single sine pulse (thin curves). Parameters of the block are p 2:14 rad=sec, α 0:25 rad, and η 0:9, and the single sine pulse has a circular frequency of ω 5p (same as in Zhang and Makris, 2000). Number-labeled markers in the time history plots in the left-hand row of panels indicate the moment when the snapshots of the block movement (right-hand row of panels) were taken. Top row: maximum acceleration of A 6:32 αg, overturning in positive x direction after one impact; middle row: A 6:33 αg, no overturning; bottom row: A 7:18 αg, overturning in negative x direction without impact. Differences in the normalized rotation between the numeric experiment and the theoretical values are indicated in gray. (E Animation of the block movements for the three test cases is available in the electronic edition of BSSA.)

8 2862 K.-G. Hinzen level of acceleration amplitude of a one-sine pulse. As the archaeoseismic application of rigid block models requires the use of true 3D ground motions, where conditions for pure rocking motion might be violated, the effects of appropriate finite friction forces at the edges of multiple block structures must be approximated. In order to test the performance of the numeric model with varied coefficients of friction over a wide range of values, larger than those expected for real situations in a classic monument, a series of rock-sliding tests were calculated. Figure 6 shows the horizontal position of the center of the contact area of the marble block from the previous tests with respect to the base as a function of time in a free rocking experiment. Stiffness and damping of the contact element were those from the previous twin experiment. For static coefficients of friction between 0.7 and 0.4 the movement history is almost identical. The motion is pure rocking about the corner points A and A 0 in Figure 1. The points in time of the impact of the marble block agree with the zero position of the center of the contact surface indicating that no corner point sliding occurred during the tests. For the next smaller static friction test with μ s 0:3, the displacement curve deviates from the previous ones. After the first impact at sec, a small amount of sliding motion leads to a shift of the displacement curve with respect to the previous experiments. With a friction coefficient of μ s 0:275, the sliding component in the movement starts at the beginning of the experiment. Beginning with the second impact of the block, the impacts occur alternating earlier and later than for the pure rocking motion due to a significant component of sliding movement. A further decrease of the static friction to μ s 0:250 results in a strong sliding of the contact point A; as most of the potential energy is consumed by this sliding motion, there is only one impact followed by rocking with highly reduced amplitudes. For small friction coefficients of μ s 0:225 and 0.2 the rocking component of the movement disappears. Numeric Archaeoseismic Test Model After the parameters of the contact forces were determined experimentally for marble, a numeric archaeoseismic test model of two columns was used to study the influence of geometry parameters, friction changes, and the nature of ground motion on the collapse behavior. Figure 7 shows a perspective view of the model. The base block measures 5 5 m in the horizontal directions and is 1 m high; the total mass is 74 metric tons. With a height of 3.5 m and a diameter, d, of 0.58 m as shown in Figure 7, the mass of the monolithic column is 2772 kg. A single drum of the structured column has a mass of 231 kg. The center of each column is shifted 1:5 m from the center of the base block in the x direction. Contact stiffness was increased by a factor of 10 compared to the twin experiment top1: N=m so that the local contact frequency ω con c=m remained well above the main frequencies of the model, which are in the range of 3 5 Hz. For the structured column, contact forces were implemented between the bases of neighboring column drums and between the drums and the pedestal. As exact stiffness values of the contact between the individual drums of the structured column depend on the dynamic movement and cannot be implemented in the current model, the same stiffness was applied for all contacts. Figure 6. Time history of the horizontal position of the center of the base of a marble block (Fig. 2) from numeric experiments with variable static and dynamic coefficients of frictions (see legend). The curves for a static coefficient of friction of 0.7, 0.6, 0.5, and 0.4 almost match exactly; so only one symbol is shown in the legend. Small arrows indicate the time and horizontal position of the center at the moment of impact of the rocking block on the base. Figure 7. Numeric archaeoseismic test model of two columns on a common base. The columns have the same dimensions; however, the right column is monolithic and the left one is composed of seven column drums, identical in size. The grid width of the horizontal plane in the perspective view is 0.5 m in both the x and y directions.

9 Simulation of Toppling Columns in Archaeoseismology 2863 There is no toppling interaction between the two columns implemented in the model, that is, when one column topples and falls into the direction of the other they do not influence each other s motion. However, there is a minor feedback through the base block. When the monolithic column overturns first, the impact impulse to the base can be seen in the acceleration record of the movement of the drums of the structured column. As it is a spike of short duration, it does not appear to influence downfall directions. The base block undergoes pure translatory motions in three dimensions; no rotational motion was used so far. The movement is defined through x, y, and z displacements with respect to a fixed coordinate system (Fig. 7). Tests with single columns in the center of the base block showed essentially the same results as the twin model. Columns were considered as overturned when at the end of the experiment one or more drums had impacted the pedestal. For the following tests, measured strong ground motions were retrieved from two resources. All records from the 28 September 2004 Parkfield earthquake database of the Consortium of Organizations for Strong Motion Observation Systems (COSMOS) Strong Motion Program (Archuleta et al., 2005) with an epicentral distance smaller than 30 km were selected (see the Data and Resources section). In addition, the European strong-motion database (Ambraseys et al., 2000) was searched for time series recorded at distances smaller than 40 km and exceeding a peak ground acceleration (PGA) of2:0 m=sec 2 (see the Data and Resources section). In total 29 three component records (Table 1) were prepared for the numeric tests. The acceleration data were band-pass filtered between 0.1 and Hz, depending on the frequency range of the original record, and linear trends were removed before ground displacement was restituted, which served as translatoric ground-motion input. Table 1 lists the epicentral distances, back azimuths, PGA, and the ratios of peak ground velocity (PGV) and acceleration (PGV/PGA). In a recent article Purvance et al. (2008) showed that both PGA and the PGV/PGA ratio, as an intensity measure correlated with the duration of the predominant acceleration pulse, are important indicators of the overturning potential of 2D rigid blocks. The mean period (Rathje et al., 1998) and the significant duration, the time between the 5% and 95% level of the Arias intensity, additionally characterize the records. Variation of Geometrical Parameters Two measured strong ground motions with different character that toppled the test columns in a pretest, denoted GM20 and GM23, were selected from the measurements listed in Table 1 and used in a first series of numeric experiments, where the h=d ratio was systematically varied and all other model parameters were kept constant. Figure 8 shows the acceleration time histories as well as a perspective view of the ground displacements. The latter analysis clearly shows that a simple push-pull mechanism toward or away from the earthquake cannot be expected with these measured time histories. Ground motion GM23 (Fig. 8a) shows the largest displacement in the horizontal direction in one half-sine pulse toward the northeast, roughly at a 90 angle with respect to the back azimuth. This swing toward the northeast determines the downfall direction of both test columns, monolithic and structured, respectively, as shown in Figure 9. Maximum ground displacement in the vertical direction is only 24% of the maximum horizontal motion of 11.0 cm. All downfall directions group within a cone that opens 42, with the median value for both columns at 34. The only exception is the monolithic column with an h=d ratio of 5.0 that falls in the opposite direction. The bottom drum of the structured column with h=d 8 is displaced to the westsouthwest direction because it is pushed away from the downfall direction of the rest of the column by the weight of the toppling six column drums. The monolithic column does not fall if h=d 4:5, and the same holds true for the structured column if h=d 3:5. The impact times of the structured column are between 5.8 and 6.7 sec and the monolithic column impacts sec after the start of the time series. Taking the time into account that the column needs to fall down, this corresponds to the time of the largest horizontal accelerations (Fig. 8). The only exception is the monolithic column, which fell in the opposite direction (h=d 5) and required 12.6 sec to fall. For the structured column, the first impact of one of the drums was measured. Ground motion GM20 is of a different character compared to GM23 (Fig. 8). The overall displacement amplitudes and the duration are larger; however, the hodogram looks more like a bowl of spaghetti without a distinct directional pulse. During the arrival of the surface waves, the ground makes two semicircular movements in the horizontal plane. The maximum vertical ground displacement reaches 61% of the horizontal maximum of 21.2 cm. As shown in Figure 9 the downfall directions vary strongly with changing slenderness ratios for the structured column. The five structured columns with 6:5 h=d 9:0 fall down within a cone of 20 toward the west-northwest. A value of h=d 6:0 causes a downfall to the northeast. In the case of h=d 5:5, the downfall is almost in the opposite direction of the group of five. At h=d 5:0, the column topples to the north, and a further decrease of the ratio results in downfall directions to the northeast and southwest. The most slender structured column (h=d 3:0) does not topple. The same holds for the monolithic column with h=d 3:0 and 3.5, respectively. The time of impact is significantly larger and more spread than in the case of GM23. The structured column impacts between 11.5 and 19.5 sec after the start with a systematic decrease of impact time with increasing h=d ratios. The monolithic column impacts after sec. While impact times for h=d between 4.0 and 6.0 are almost constant, they also decrease with increasing h=d between 6.0 and 9.0. The downfall directions of the monolithic column are toward the north-northeast north in a cone of 30 and toward the

10 2864 K.-G. Hinzen Table 1 Parameters of 29 Strong Ground Motion Records Used as Input Motions for the Two-Column Test Model Date (mm/dd/yyyy) Time Earthquake Stat. Code D epi (km) Back Azimuth ( ) Component PGA (m=sec 2 ) PGV (cm=sec) PGD (cm) PGV/PGA (sec) I A (m=sec) T mea (sec) T s (sec) Label 05/17/ :58:41 Gazli GZL Z GM1 N E /15/ :15:19 Friuli (afs) FOG Z GM2 N E BRE Z GM3 N E KOB Z GM4 N E /16/ :35:57 Tabas DAY Z GM5 N10 W N80 E /15/ :19:41 Montenegro ULO Z GM6 N E PETO Z GM7 N E /24/ :23:18 Montenegro (afs) BUD Z GM8 N E PET Z GM9 N E TIVA Z GM10 N E KOTZ Z GM11 N E /23/ :34:52 Campano Lucano CLT Z GM12 N E (continued)

11 Simulation of Toppling Columns in Archaeoseismology 2865 Date (mm/dd/yyyy) Time Earthquake Stat. Code D epi (km) Back Azimuth ( ) Component Table 1 (Continued) PGA (m=sec 2 ) PGV (cm=sec) PGD (cm) PGV/PGA (sec) I A (m=sec) T mea (sec) T s (sec) Label BGI Z GM13 N E STR Z GM14 N E /23/ :51:05 Ionian ARG Z GM15 N E /30/ :12:28 Kars HRS Z GM16 N E /07/ :41:24 Spitak GUK Z GM17 N E /13/ :18:40 Erzincan ERC Z GM18 N E /01/ :57:13 Dinar DIN Z GM19 N E /17/ :01:40 Kocaeli IZT Z GM20 N E YPT Z GM21 T R /28/ :15:24 Parkfield Z GM22 N E Z GM23 N E Z GM24 N E (continued)

12 2866 K.-G. Hinzen Date (mm/dd/yyyy) Time Earthquake Stat. Code D epi (km) Back Azimuth ( ) Component Table 1 (Continued) PGA (m=sec 2 ) PGV (cm=sec) PGD (cm) PGV/PGA (sec) I A (m=sec) T mea (sec) T s (sec) Label Z GM25 N E Z GM26 N E Z GM27 N E Z GM28 N E Z GM29 N E afs, aftershock; Stat. Code, station code or number; D epi, epicentral distance; PGA, peak ground acceleration; PGD, peak ground displacement; PGV/PGA, ratio of PGVand PGA; I A, Arias Intensity; T mea, mean period (Rathje et al., 1998); T s, significant duration (5% 95% Arias Intensity).

13 Simulation of Toppling Columns in Archaeoseismology 2867 Figure 8. Measured strong ground motion used in numeric toppling experiments of the two-column model. Top row: three components of the acceleration time histories of (a) Parkfield (2004) and (b) Kocaeli (1999), labeled GM23 and GM20 in Table 1, respectively. Bottom row: perspective view of the corresponding 3D hodograms of the ground displacement. The 2D ground motion on three mutually perpendicular planes is shown in addition. (E An animation of the ground movements is available in the electronic edition of BSSA.) south-southeast, thus showing less variability than the structured columns. Variation of Friction In a second series of numeric experiments, the influence of coefficients of static and dynamic friction on potential toppling directions was tested. For the two ground motions GM23 and GM20 (Fig. 8), the static coefficient of friction was varied between μ s 0:1 and 0.9 in steps of 0.1, while the coefficient of kinematic friction μ k was always set to 84% of μ s. Friction coefficients less than 0.5 are unrealistic for classical columns of marble or similar material. However, the numeric experiments allow an exploration of the limits where reduced friction is influential. Geometry of the test columns was kept constant with an h=d ratio of 6.0 and a column height of 3.5 m. For GM23 again the swing toward the northeast clearly determines the toppling direction of both test columns as shown in Figure 10. All downfall directions are within a cone opening 15 around a direction of N30 E. The only exception is the north-northwest toppling direction of the monolithic column for an unrealistically low-static friction of μ s 0:2. The median of the downfall directions shows a 70 counterclockwise rotation with respect to the back azimuth. For the extremely small friction of μ s 0:1 both

14 2868 K.-G. Hinzen Figure 9. Influence of the slenderness of columns on the toppling direction for two measured strong ground motions, GM23 (top row) and GM20 (bottom row) from Figure 8. Dashed lines show the back azimuth toward the earthquake, and the hodogram of the horizontal ground displacement is shown as gray lines, with the corresponding dx and dy displacement axis on the top and the right of the diagrams. The main diagrams show a bird s-eye view of m with the test columns in the center. On the left-hand side, the impact points of the center of mass of the seven drums of the multidrum columns are indicated by filled circles; the circle size varies with the h=d ratio as shown in the legend. The plots on the right-hand side show the impact points of the center of mass of the monolithic column model; symbol size is the same as for the multidrum columns. All test columns had common heights of 3.5 m. (E Examples of the animated column movements are available in the electronic edition of BSSA.) columns do not topple, but they slide at the base, reducing the induced momentum in the column to a level that is too small for toppling. The second ground motion GM20 leads to significantly different downfall directions for the two column types, and the direction varies with changes in the coefficients of friction. The cone of 60 of toppling directions for the monolithic column includes the back azimuth (Fig. 10). The structured column falls in two directions, one cone of 45 points northeast and a second of 25 points north-northwest. However, the coefficient of friction does not appear to systematically determine downfall directions. While for μ s 0:4,

15 Simulation of Toppling Columns in Archaeoseismology 2869 Figure 10. Influence of the coefficient of friction on the toppling direction for two measured strong ground motions, GM23 (top row) and GM20 (bottom row) from Figure 8. The size of the circles, which indicate the impact points of the center of mass, varies with the coefficient of static friction μ s (see the legend). Columns have the same dimensions as those in Figure 9, which also gives further explanation. 0.5, 0.7, and 0.8 the column falls in northeasterly directions, μ s 0:2, 0.3, 0.6, and 0.9 leads to a north-northwest downfall. Again, the columns did not topple with the extreme μ s 0:1. Measured Ground Motions All 29 ground-motion records listed in Table 1 were used to search for correlations between the toppling behavior of the column model and ground-motion parameters (PGA, PGV, and peak ground displacement [PGD]), the direction of largest horizontal acceleration, velocity and displacement impulse in the record, and the back azimuth toward the source. The two column model with h=d 6:0, μ s 0:7, μ k 0:6 and the same coupling forces as before was used to calculate downfall directions. Of the 29 ground motions, 13 toppled the structured column and 7 also toppled the