simulation of toppling columns in...

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Simulation of Toppling Columns in Archaeoseismology

by Klaus-G. Hinzen

Abstract Since the early days of modern seismology, toppled artifacts such astombstones and single columns have been used in the aftermath of earthquakes todeduce parameters of site-specific ground motions. The artifacts were generally treat-ed as rigid bodies. Later, the theory of rigid block movements was also applied to pre-cariously balanced rocks toppled by earthquakes. While the movements of a singlerocking block can be described analytically, slide-rocking movements, bouncing, andmultiple block systems require a numerical approach. We use multiple rigid blockmodels 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 ver-ified by comparison with analytically determined rocking motions of a single rectan-gular block. Stiffness and damping parameters of the coupling forces are adjusted toresults from analog experiments with a rocking marble block. A model of a monolithiccolumn and one consisting of seven drums is used to test the influence of the geometryand friction on the toppling behavior. The main question addressed in this study iswhether toppled columns give a clear indication of the back azimuth toward the earth-quake source. Input motion from 29 strong-motion records indicates little correlationbetween downfall directions and back azimuth. Clearly directed horizontal groundmovements tend to topple the columns in the transverse direction. More complexground motions result in quasi-random downfall directions. The friction coefficientshave a minor influence on the downfall directions. Synthetic ground motions fortwo earthquakes with different source mechanism show toppling directions towardand away from the source as well as in the transverse bearing. However, it is notstraightforward to deduce a reliable source location from the inversion of the topplingdirections.

Online Material: Movies of simulated block movement and toppling behavior, andfigures of synthetics ground displacement.

Introduction

In the study of preinstrumental earthquakes, histori-cal seismology and palaeoseismology are well-establishedbranches of seismological sciences. Techniques to evaluateground motions and parameters of causing earthquakes thathave left their mark in written documents and in the near-surface geology were developed. Ever since man-made struc-tures have been erected, earthquakes have also left theirmarks on these constructions. However, damages in archae-ologically excavated buildings or continuously preservedmonuments are often hard to unravel in terms of the causa-tive effects. The new branch of seismological sciences,archaeoseismology, is defined as “the detailed study of pre-instrumental earthquakes that, by affecting locations ofhuman occupation and their environments, have left theirmark in the archaeological record” (Buck and Stewart,

2000). Following this definition, the detailed study of earth-quakes is the focus, and compilation, modeling, and interpre-tation of damage data is a means to an end.

The main questions to be answered by archaeoseismicinvestigations are (1) how probable is seismically inducedground motion as a cause of damage observed in man-madestructures from the past, (2) when did the damaging groundmotion occur, and (3) what can be deduced about the nature ofthe causing earthquake (Galadini et al., 2006). All three ques-tions should be answered before results from archaeoseismiccase studies can be included in seismic hazard analyses.Whilethe first problem requires input from multiple disciplines in-cluding civil engineering, geophysics, and geotechnics, thesecond is a task for the geological sciences and archaeometry.The third question requires input from several seismological

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Bulletin of the Seismological Society of America, Vol. 99, No. 5, pp. 2855–2875, October 2009, doi: 10.1785/0120080241

specialties ranging from seismotectonics to source studies,wave propagation, and site-effect modeling to soil–buildinginteraction. The two main parameters to be extracted fromthese studies are the location and a measure of the strengthof the causing earthquake, where both are often intrinsicallytied to each other. While the macroscopic degree of damageis directly connected to the nature of ground motion at thesite, many techniques from macroseismic data analysiscan be adopted; determination of the direction of groundmotion that caused damage or the wave propagation direc-tion, which is even more complicated, remains a challenge inarchaeoseismology.

Several case studies have been published in which sur-face rupturing of the fault plane directly affected man-madestructures (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 sec-tion is evident and source parameters such as surface rupturedisplacement can, under favorable conditions, be determinedwithin small ranges of uncertainty. However, in cases wherethe activated segment of a fault is remote from the site witharchaeological findings of building damages (archaeo-damages), source identification is a challenging task. Nosystematic studies or established methods presently existto deduce the location of the rupturing fault from archaeo-damages. Several case studies purported to infer back azi-muth to the activated parts of a fault from directional featuresin archaeo-damages (Korjenkov andMazor, 2003). However,these interpretations usually suppose a very simple behaviorfor site-specific ground motions and do not consider thecomplexity of an extended seismic source and the uncertain-ties imposed by either the complexity of or randomness inthe reaction of building components.

In the seismic design of modern buildings, finite ele-ment models are frequently used to study the behavior ofthe complete structure or crucial components of the structure(i.e., Meskouris et al., 2007). Most important in these kindsof studies are the correct modeling of the elastic or nonlinearmaterial behavior to deduce the capacity of concrete beams,steel frames, shear walls, or wooden beams (Hinzen andWeiner, 2009). While finite element calculations predictthe dynamic load under which a certain component will fail,the movement of structural parts of a building in the collapsephase cannot be modeled. The latter parameter is importantto analyze directional features in archaeo-damages. Well-preserved ancient buildings were often constructed fromblocks of natural stone, sometimes without any cementa-tion. This was often the case in classic Greek buildings.For such structures it is feasible to use rigid block modelsand Newtonian mechanics to study the block movementand interaction as a first modeling attempt (Sinopoli, 1995;Papantonopoulos et al., 2002; Psycharis, 2007). A relatedproblem is that of the dynamic stability of precariously bal-anced rocks, which have been used as low-resolution strong-motion seismoscopes (Brune, 1996). Anooshehpoor et al.(2004) used a numeric approach to model the 2D dynamic

response of rocks to arbitrarily complex acceleration timehistories (Purvance et al., 2008).

In this study we adapt rigid block techniques to theneeds of archaeoseismology. Specifically, we regard linearviscoelastic coupling forces with finite friction in multipleblock systems and true 3D ground-motion excitation. Weuse a numeric rigid block model of two cylindrical columnsregarded as simple archaeo-seismoscopes. One monolithiccolumn and one column consisting of seven separate columndrums is used to study systematically the effects of (1) thecolumns’ slenderness, (2) changes in frictional coefficients,and (3) variability of measured and synthetic ground motionson the downfall directions.

Directional Features in Archaeo-Damages

In his famous work about the Neapolitan earthquake in1857, Robert Mallet (1862) not only prepared the ground forevaluating earthquake strength with macroseismic methods,he also tried to infer the earthquake location from directionaldamage features. Without a scientific basis, still being ac-tively sought (Ambraseys, 2006; Marco, 2008), the practicesuggested by Mallet (1862) should not be applied. WhileMallet used fresh traces of directional damage, in archaeoseis-mology such features have gone through altering processes,making it more difficult to deduce accurate directions towardthe earthquake source.

As summarized by Galadini et al. (2006, and referencestherein), typical earthquake effects on constructions are(1) cross fissures in the vertical plane due to shear forces,(2) corner expulsion due to orthogonalmotion ofwalls, (3) lat-eral and rotational horizontal and independent motion ofblockswithin awall, (4) height reduction due to vertical crash-ing, (5) deformation of arch piers including collapse of keystones, (6) wall tilting and distortion, and (7) rotation or top-pling of pillars or parts of it and drums of columns. Additionalphotos of typical damages are given by Marco (2008). Thedirection of any of these seismogenic damage patterns alwaysresults from the coaction of the orientation of the structure orstructural component and the orientation of the ground move-ment, which is influenced by the source characteristics andsite conditions.

Korjenkov and coworkers (i.e., Korjenkov and Mazor,1999a,b, 2003; Al-Tarazi and Korjenkov, 2007; Korzhenkovet al., 2009) have given examples how directionalities inarchaeoseismic damage can be quantified for a certain site.This quantification works best when carried out during anongoing excavation. Because this is not always feasible,measurements are taken from preserved ruins (Galadiniand Galli, 2001) or from the documentation of former exca-vations (i.e., Hinzen and Schütte, 2003). Korjenkov maps thedirection of cracks and rotations of blocks with respect to thetrend of walls. At sites with numerous damaged walls, a sta-tistical approach can reveal preferred directions in the dam-age pattern. While such preferred directions of block shifts,rotations, toppling of wall fragments or complete walls, and

2856 K.-G. Hinzen

toppling of columns provide strong arguments for significantground-motion amplitudes in a certain direction, it is notstraightforward to deduce the back azimuth to the causingearthquake from this direction. Near fault strong ground mo-tions are influenced by the source mechanism, rupture pro-cess details, distribution of asperities, fault plane geometryand extension, wave spreading conditions, and site condi-tions (i.e., Erdik and Durukal, 2004, and references therein).Also, secondary earthquake effects due to deformation ofsoft subsoils that form directional features such as foundationcracks and inclined walls (Hinzen and Schütte, 2003) shouldnot be mistaken for the bearings toward the earthquakesource.

Among the most obvious and promising directionalarchaeo-damages are toppled columns (i.e., DiVita, 1996;Nur and Ron, 1996; Marco, 2008) and columns with dis-placed column drums (Stiros, 1996; Bottari, 2005; Psycharis,2007). As rotation-symmetrical construction elements, col-umns should react similarly to ground motions in any direc-tion when freestanding and not connected to neighboringstructural elements. This feature makes columns a good uni-versal seismoscope. Even though such freestanding columnsare rare in cultural heritage, the independence from a fixedtrend of the structure with respect to unknown groundmotions makes them a versatile tool to study basic topplingeffects. Even the simplest object, a monolithic block on aplane subsurface, demonstrates a complex dynamic behaviorincluding stress discontinuity, structural damping, contactfriction, and impact (Sinopoli, 1995); all of which can influ-ence toppling behavior and downfall directions. Many openquestions remain about the particular influence of each ofthese factors.

Several authors have studied the dynamic behaviorof cylindrical structures and classical columns. Koh andMustafa (1990) studied the free rocking motion of rigid cyl-inders for various initial conditions and a stationary founda-tion during the motion of the cylinder. They numericallyintegrated the exact equations of motion of the model andmapped the boundary between toppling and not toppling.Koh and Hsiung (1991a,b) extended the 2D model to 3Drocking, rolling, and uplift of a rigid cylinder when subjectedto ground motions, showing that 3D motion is significantunder earthquakelike excitations. Mouzakis et al. (2002)used a 1∶3 analog model of a multidrum column from theParthenon of the Acropolis of Athens, even made fromthe same material as the original. Scaled earthquake groundmotions in two- and three-dimensions were used to drive ashaking table with forces insufficient to topple the model.They found large deformations during the shaking, whichwere not necessarily reflected by the residual displacementsat the end of the tests. A significant influence of imperfec-tions of the model specimen and a very high sensitivity toeven small changes in the input motion parameters wereobserved. In an accompanying article, Papantonopoulos et al.(2002) successfully used the distinct element method tonumerically simulate the behavior of the same column.

Konstantinidis and Makris (2005) also used numerical mod-els of multidrum columns represented by a 2D discrete ele-ment model allowing rocking, sliding, and slide rocking toshow that relative sliding between drums happens even whenthe g-values of the ground accelerations are less than thecoefficient of friction. They concluded that typical classicalcolumns can survive the shaking from strong ground motionsnear the causative fault of earthquakes with moment magni-tudes 6.0–7.4. Additionally, they found a more controlledseismic response of multidrum columns than monolithic col-umns of the same size. Psycharis (2007) made a backwardanalysis of a column of the temple of Olympios Zeus inAthens to investigate the seismic history of the area duringthe 2000 yr since the building was erected. A 3D numericmodel of a single and double column structure allowed theauthor to constrain the maximum ground velocities at theside by comparing the model behavior with the current statusof the real structure.

By applying archaeoseismologic methods, the orienta-tion of damaged structures should be recoverable in mostcases (i.e., Korjenkov and Mazor, 1999a), although usuallylittle or nothing is known a priori about the nature of theground movement. For elongated structural elements (i.e.,a simple freestanding wall) the angle between the polariza-tion of the ground motion and the trend of the element isdecisive for estimating dynamic reaction of the structure.In contrast, rotation-symmetrical building elements (i.e.,freestanding columns) do not exhibit this dependency. There-fore, to investigate correlations between ground-motionpolarizations, back azimuth, and toppling directions, wewill limit this study to the case of freestanding cylindricalcolumns.

Rocking of Rigid Blocks

Long before a strong-motion instrumentation was avail-able, Milne (1881, 1885), Perry (1881), and others used thetheory of dynamic block structures to deduce earthquakeground accelerations from toppled monumental columnsand tombstones. A fundamental article on the theory of rigidblock movements by Housner (1963) helped to explain ob-servations made during the large Chilean earthquake of May1960. Augusti and Sinopoli (1992) presented a comprehen-sive summary on the dynamic modeling of large block struc-tures and Sinopoli (1995) reviewed studies of large blockstructure dynamics. Brune and Whitney (1992), Brune(1996), Anooshehpoor et al. (1999, 2004), and Zhang andMakris (2000) applied rigid block movement models to in-terpret precariously balanced rocks and a steam engine, thelatter overturned during the great San Francisco earthquakeof 1906.

We briefly introduce the problem of rocking rigid blockdynamics in order to describe the analog and numeric experi-ments presented in the main part of this article; a comprehen-sive description of the theory can be found in Housner (1963)and Augusti and Sinopoli (1992). Figure 1 shows the cross

Simulation of Toppling Columns in Archaeoseismology 2857

section of a rigid body in a rocking motion. The blockhas height h and width b; the half-diagonal, R ��h=2�2 � �b=2�2

p, is defined as the distance from the center

of gravity, cg, to the actual center of rotation of the block, Aand A0, respectively. The angle between the longer blockside, which coincides with the vertical direction when the

block is at rest, and the line R is α, and the inclination (rota-tion) of the block is θ. Housner (1963) solved the equationsof motion for the rocking block for one degree of freedom,namely the rotation around the corner points A and A0

(Fig. 1d). This movement constraint specifies that (1) thefriction at the corner points is large enough that no slidingbetween the block and the base occurs as in Figure 1c and(2) the block does not bounce during the movement throughthe static equilibrium (Fig. 1f). The equation of motion ofthe free rocking response of a 2D rigid block withoutsliding, slide rocking, or free flight behavior was given byHousner (1963):

I0 �θ � �WR sin�α � θ�; (1)

in which I0 � 4=3mR2 is the mass moment of inertia of ahomogeneous rectangular block about corner A and W �mg is the weight of the homogenous block, m and g arethe block mass and acceleration of gravity, respectively.For slender blocks (sinα≈ α), equation (1) can be writtenas (Housner, 1963; Aslam et al., 1980)

�θ � p2θ � �p2α; (2)

in which p ��WR=I0

p�

��3g=�4R�

p.

In order to start a forced rocking motion, a horizontalacceleration a is necessary that fulfills the condition (Hous-ner, 1963; Augusti and Sinopoli, 1992)

a

g≥ b

h: (3)

Under more realistic conditions, energy dissipates dur-ing each impact of the block on the base. For inelastic impact(no bouncing), the reduction of kinetic energy, r, depends onthe square of the ratio of angular velocities before, _θi, andafter, _θi�1, an impact, respectively (Housner, 1963; Aslamet al., 1980; Augusti and Sinopoli (1992),

r �

�12I0 _θ

2i�1

��12I0 _θ

2i

� ��_θi�1

_θi

�2

; (4)

and for slender blocks the coefficient of restitution is

η � ��r

p � 2 � b2

h2

2�1� b2

h2�� 1 � 2

3α2: (5)

Housner (1963) showed that in this case the amplitude θn ofthe nth cycle can be written as

θn � 1 ��1 � rn�1 � �1 � θ0=α�2�

q; (6)

and the half period of the rocking motion is

Figure 1. (a) Cross section of a rectangular block of height hand width b in a rocking experiment. θ is the inclination of theblock; R is the vector from the center of gravity, cg, to the actualrocking corner A; and A0 is the opposite rocking corner. The angle αis a measure for the ratio of h=b. Dashed lines indicate the equili-brium position of the block. The following drawings show schema-tically the motion types of the block: (b) rest; (c) sliding; (d) rotation(rocking); (e) slide rocking; (f) translational jump; and (g) rotationaljump.

2858 K.-G. Hinzen

T

2� 2

��I0WR

rtanh�1

��rn�1 � �1 � θ0=α�2�

q: (7)

Amplitudes and the half period of the rocking motion de-cay rapidly with the number of cycles. Increasingly high-frequency movements follow a few slow rocking motionswith large amplitudes. Under less ideal conditions (bouncingduring impacts and sliding due to reduced friction), ampli-tude decay proceeds even more rapidly.

The sliding component of the movement is governed bythe static coefficient of friction, μs, and the slenderness of theblock. For the free motion of the block Augusti and Sinopoli(1992) showed that the inequality

μs ≥ 3�b=h�4� b2=h2

(8)

allows separating the conditions under which pure rockingand slide rocking exists. If the condition of equation (8) isnot fulfilled, slide rocking, controlled by the kinetic frictioncoefficient μk, will start. For small angles for each b=h, avalue μs exists above which only rocking will occur untilthe block returns to a state of static equilibrium (Augustiand Sinopoli, 1992).

Numeric Model

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, thetypes and degrees of freedom of the joints between blocksand the types and parameters of the contact forces, the codegenerates the equations of motion of the mechanical system.An implicit second order method with variable step size wasused to solve the equations of motion. Error tolerance wasusually set to 10�6. First, a solution for the static modelwas calculated and the resulting coordinates were used asinitial conditions for the dynamic tests.

A dual analog and numeric experiment was carried out todeduce basic parameters for further calculations and to vali-date the models. For the analog rocking tests, a marble blockof 6 × 8 × 30 cm, sitting on a marble plate of 2 cm thicknessand cemented to a foundation,was constructed (Fig. 2).With amass ofm � 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 to5.0. The motion of the block during the rocking experimentswas monitored with a miniature accelerometer mounted onthe top of the block (Fig. 2). Acceleration time history wasrecorded with a 24 bit analog-to-digital converter at a sam-pling frequency of 25 kHz (Fig. 3).

A numeric model of a block of similar size (Fig. 2) wasused for comparative calculations. In the numeric model theindividual bodies (here, base-block and rocking marbleblock) are treated as rigid. During the rocking-block experi-

ments the base was fixed. The marble block is connected tothe base through a six degrees of freedom joint and a visco-elastic contact force, including a sliding and a sticking mode.During sliding, the contact force f is of Coulumb frictiontype:

f � �μkFNsign�v�; (9)

where FN is the normal force on the friction surface, μk thecoefficient of kinematic friction, and v the sliding velocity.The sticking–sliding transition occurs when

jfj ≥ F0; (10)

where F0 is the maximum value of the static friction forceF0 � μsFN . In the sticking mode, the linear viscoelastic fric-tion force is

f � f0 � c�x � x0� � dv: (11)

Here c and d are the stiffness and damping coefficients,respectively.

In a series of numerical rocking tests, c and dwere variedbetween 1 × 106 and 1 × 109 N=mand 0.01 and 0.99, respec-tively. As shown in equation (7), the time between two sub-sequent impacts of the rocking block on the base is stronglydependent on the energy reduction at each impact and henceon the stiffness. Therefore, the synchronicity of the impacts inthe twin experiment was the first criterion in the parameteradjustment. Figure 3 compares the acceleration measuredat the top of the analog block with the corresponding calcu-lated time series assuming here a contact stiffness of 1:2×107 N=m. The time step of the output during the calculationwas 4:0 × 10�5 sec, corresponding to the sampling rate of themeasurement. In the first 5 sec of the record up to the eigh-teenth impact, the impact time in both experiments matchesalmost perfectly. In the balanced time of the experiment im-pact, times successively deviate due to the increasing influ-ence of the imperfectly flat bottom of the analog marble

Figure 2. (a) Analog and (b) numeric models used to verifystiffness and damping of the contact forces. An accelerometerwas cemented to the top of the 6 × 8 × 30 cm marble block tomonitor the rocking movements of the block. The wire-frame virtualblock is inclined at an angle of 11.3° close to its indifferent equi-librium (θ0=α � 1). (Ⓔ An animation of the rocking-block modelis available in the electronic edition of BSSA.)


block. This also brings the movement to a halt about 1.2 secearlier than in the numeric experiment. The insert in Figure 3shows a detailed section of the acceleration impulse of thesecond impact. The strong positive acceleration impulse indi-cates the impact of the block on the base, followed by a short(∼2 msec) phase of free fall, as indicated by the accelerationof �1:0g and a concurrent damped oscillatory movement.The free-fall phase is due to a small bouncing effect afterthe block hits the base. This phase was measured in boththe analog experiment and the numerical simulation. The du-ration of the free-fall time and the amplitudes of the dampedoscillationwerematched by adjusting the damping coefficientin a trial and error procedure. The results shown in Figure 3were achieved with a damping coefficient of 0.36. As longas no initial sliding of the block occurs, the coefficient of fric-tion 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 calcula-tions were kept constant at 0.7 and 0.6, respectively.

Verification Tests

The block from the numeric experiment (Fig. 2) wasused next to simulate the free rocking ground motions forthe starting rotation angles 0:2α ≤ θ0 ≤ α. Amplitudes forthe first 10 cycles of movement from the calculation werecompared 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 theanalytical calculation than suggested by the geometry to

compensate for energy losses due to small bouncing effectsnot included in the analytic Housner (1963) model.

Several studies have used horizontal sinusoidal groundmotions to simulate a Housner-block model (i.e., Housner,

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 θ0 � 10° and the ratio of h=b was 5.0. The short bar underneath the second impact indicates the zoomedtime window shown in the inset. Here the trace shown as a gray variable area plot is the acceleration measured during the rocking ofthe analog block. The black seismogram is the result of the numeric experiment using stiffness and damping of 1:2E�7 N=m and0.36, respectively. Coefficients of static and dynamic friction were 0.7 and 0.6, respectively.

Figure 4. Lines show the amplitude decreases of the pure rock-ing motion of a slender block for different starting positions, θ0,with progressing number of cycles (after Housner, 1963). Thecrosses show the amplitudes from corresponding experiments withthe numeric model of a rocking block with the dimensions shown inFigure 2. The inset in the upper right corner shows the time series ofthe angular displacement for the θ0=α ratio of 0.999.

2860 K.-G. Hinzen

1963; Ishiyama, 1982; Sinopoli, 1991). Anooshehpoor et al.(1999, 2000) and Zhang and Makris (2000) presented ana-lytic and numeric solutions to the problem of a freestandingblock, respectively, exposed to a one-sine pulse. The lattershowed that two modes of overturning exist, one with andone without an impact of the block on the base before itoverturns. We used their results to test our numeric modelby excitation of the block with a single sinusoidal impulse.The modeled block has the dimensions h � 3:113 m andb � 0:795 m, resulting in α � 0:25, p � 2:14, and a coef-ficient of restitution of η � 0:9. Frictional parameters werekept the same as in the previous experiment. Contact forceswere only implemented between the base of the movingblock and the pedestal. Therefore, after overturning, theblock can penetrate the pedestal. Figure 5 shows the resultsof 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 pre-vious study. In case (1) the block experiences one impactbefore it overturns in the direction of the movement of thefirst half cycle of the sine pulse. The slightly larger accelera-tion in experiment (2) does not overturn the block. In test (3)the block overturns in the opposite direction of the move-ment of the first half cycle of the acceleration pulse withoutan impact on the base. The agreement of the block move-ments with the analytically predicted results of Anoosheh-poor et al. (1999, 2000) and numerical calculations ofZhang and Makris (2000) confirm the capability of ourmodel to simulate pure rocking motion of a single block.

As outlined by Zhang and Makris (2000), the demandon 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 sinepulse (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 circularfrequency of ω � 5p (same as in Zhang and Makris, 2000). Number-labeled markers in the time history plots in the left-hand row of panelsindicate the moment when the snapshots of the block movement (right-hand row of panels) were taken. Top row: maximum acceleration ofA � 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 theoreticalvalues are indicated in gray. (Ⓔ Animation of the block movements for the three test cases is available in the electronic edition of BSSA.)


level of acceleration amplitude of a one-sine pulse. As thearchaeoseismic application of rigid block models requiresthe use of true 3D ground motions, where conditions for purerocking motion might be violated, the effects of appropriatefinite friction forces at the edges of multiple block structuresmust be approximated.

In order to test the performance of the numeric modelwith varied coefficients of friction over a wide range ofvalues, larger than those expected for real situations in a clas-sic monument, a series of rock-sliding tests were calculated.Figure 6 shows the horizontal position of the center of thecontact area of the marble block from the previous tests withrespect to the base as a function of time in a free rockingexperiment. Stiffness and damping of the contact elementwere those from the previous twin experiment. For staticcoefficients of friction between 0.7 and 0.4 the movementhistory is almost identical. The motion is pure rocking aboutthe corner points A and A0 in Figure 1. The points in time ofthe impact of the marble block agree with the zero position ofthe center of the contact surface indicating that no cornerpoint sliding occurred during the tests. For the next smallerstatic friction test with μs � 0:3, the displacement curvedeviates from the previous ones. After the first impact at0.386 sec, a small amount of sliding motion leads to a shiftof the displacement curve with respect to the previous experi-ments. With a friction coefficient of μs � 0:275, the slidingcomponent in the movement starts at the beginning of theexperiment. Beginning with the second impact of the block,the impacts occur alternating earlier and later than for thepure rocking motion due to a significant component of slid-ing movement. A further decrease of the static friction toμs � 0:250 results in a strong sliding of the contact point

A; as most of the potential energy is consumed by this slidingmotion, there is only one impact followed by rocking withhighly reduced amplitudes. For small friction coefficients ofμs � 0:225 and 0.2 the rocking component of the movementdisappears.

Numeric Archaeoseismic Test Model

After the parameters of the contact forces were deter-mined experimentally for marble, a numeric archaeoseismictest model of two columns was used to study the influenceof geometry parameters, friction changes, and the nature ofground motion on the collapse behavior. Figure 7 shows aperspective view of the model. The base block measures 5 ×5 m in the horizontal directions and is 1 m high; the totalmass 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 monolithiccolumn is 2772 kg. A single drum of the structured columnhas 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 tothe twin experiment to 1:2 × 108 N=m so that the local con-tact frequency ωcon �

��c=m

premained well above the main

frequencies of the model, which are in the range of 3–5 Hz.For the structured column, contact forces were implementedbetween the bases of neighboring column drums and be-tween the drums and the pedestal. As exact stiffness valuesof the contact between the individual drums of the structuredcolumn depend on the dynamic movement and cannot beimplemented in the current model, the same stiffness wasapplied for all contacts.

Figure 6. Time history of the horizontal position of the center ofthe base of a marble block (Fig. 2) from numeric experiments withvariable static and dynamic coefficients of frictions (see legend).The curves for a static coefficient of friction of 0.7, 0.6, 0.5, and0.4 almost match exactly; so only one symbol is shown in thelegend. Small arrows indicate the time and horizontal position ofthe center at the moment of impact of the rocking block on the base.

Figure 7. Numeric archaeoseismic test model of two columnson a common base. The columns have the same dimensions; how-ever, the right column is monolithic and the left one is composed ofseven column drums, identical in size. The grid width of the hor-izontal plane in the perspective view is 0.5 m in both the x and ydirections.

2862 K.-G. Hinzen

There is no toppling interaction between the two col-umns implemented in the model, that is, when one columntopples and falls into the direction of the other they do notinfluence each other’s motion. However, there is a minorfeedback through the base block. When the monolithic col-umn overturns first, the impact impulse to the base can beseen in the acceleration record of the movement of the drumsof the structured column. As it is a spike of short duration, itdoes not appear to influence downfall directions. The baseblock undergoes pure translatory motions in three dimen-sions; no rotational motion was used so far. The movementis defined through x, y, and z displacements with respect to afixed coordinate system (Fig. 7). Tests with single columnsin the center of the base block showed essentially the sameresults as the twin model. Columns were considered as over-turned when at the end of the experiment one or more drumshad impacted the pedestal.

For the following tests, measured strong ground motionswere retrieved from two resources. All records from the28 September 2004 Parkfield earthquake database of theConsortium of Organizations for Strong Motion ObservationSystems (COSMOS) StrongMotion Program (Archuleta et al.,2005) with an epicentral distance smaller than 30 km wereselected (see the Data and Resources section). In addition,the European strong-motion database (Ambraseys et al.,2000) was searched for time series recorded at distancessmaller than 40 km and exceeding a peak ground acceleration(PGA) of 2:0 m=sec2 (see the Data and Resources section).In total 29 three component records (Table 1) were preparedfor the numeric tests. The acceleration data were band-passfiltered between 0.1 and 20–30 Hz, depending on the fre-quency range of the original record, and linear trends wereremoved before ground displacement was restituted, whichserved as translatoric ground-motion input. Table 1 lists theepicentral distances, back azimuths, PGA, and the ratios ofpeak ground velocity (PGV) and acceleration (PGV/PGA).In a recent article Purvance et al. (2008) showed that bothPGA and the PGV/PGA ratio, as an intensity measure corre-lated with the duration of the predominant accelerationpulse, are important indicators of the overturning potentialof 2D rigid blocks. The mean period (Rathje et al., 1998)and the significant duration, the time between the 5% and95% level of the Arias intensity, additionally characterizethe records.

Variation of Geometrical Parameters

Two measured strong ground motions with differentcharacter that toppled the test columns in a pretest, denotedGM20 and GM23, were selected from the measurementslisted in Table 1 and used in a first series of numeric experi-ments, where the h=d ratio was systematically varied and allother model parameters were kept constant. Figure 8 showsthe acceleration time histories as well as a perspective viewof the ground displacements. The latter analysis clearlyshows that a simple push-pull mechanism toward or away

from the earthquake cannot be expected with these measuredtime histories.

Ground motion GM23 (Fig. 8a) shows the largest dis-placement in the horizontal direction in one half-sine pulsetoward the northeast, roughly at a 90° angle with respect tothe back azimuth. This swing toward the northeast deter-mines the downfall direction of both test columns, mono-lithic and structured, respectively, as shown in Figure 9.Maximum ground displacement in the vertical direction isonly 24% of the maximum horizontal motion of 11.0 cm.All downfall directions group within a cone that opens42°, with the median value for both columns at 34°. The onlyexception is the monolithic column with an h=d ratio of 5.0that falls in the opposite direction. The bottom drum of thestructured column with h=d � 8 is displaced to the west-southwest direction because it is pushed away from thedownfall direction of the rest of the column by the weightof the toppling six column drums. The monolithic columndoes not fall if h=d ≤ 4:5, and the same holds true for thestructured column if h=d ≤ 3:5. The impact times of thestructured column are between 5.8 and 6.7 sec and the mono-lithic column impacts 6.0–6.5 sec after the start of the timeseries. Taking the time into account that the column needsto fall down, this corresponds to the time of the largest hor-izontal accelerations (Fig. 8). The only exception is themonolithic column, which fell in the opposite direction(h=d � 5) and required 12.6 sec to fall. For the structuredcolumn, the first impact of one of the drums was measured.

Ground motion GM20 is of a different character com-pared to GM23 (Fig. 8). The overall displacement amplitudesand the duration are larger; however, the hodogram looksmore like a bowl of spaghetti without a distinct directionalpulse. During the arrival of the surface waves, the groundmakes two semicircular movements in the horizontal plane.The maximum vertical ground displacement reaches 61% ofthe horizontal maximum of 21.2 cm. As shown in Figure 9the downfall directions vary strongly with changing slender-ness ratios for the structured column. The five structured col-umns 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 adownfall to the northeast. In the case of h=d � 5:5, thedownfall is almost in the opposite direction of the group offive. At h=d � 5:0, the column topples to the north, and afurther decrease of the ratio results in downfall directions tothe northeast and southwest. The most slender structuredcolumn (h=d � 3:0) does not topple. The same holds forthe monolithic column with h=d � 3:0 and 3.5, respectively.The time of impact is significantly larger and more spreadthan in the case of GM23. The structured column impactsbetween 11.5 and 19.5 sec after the start with a systematicdecrease of impact time with increasing h=d ratios. Themonolithic column impacts after 12.3–16.9 sec. While im-pact 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 towardthe north-northeast–north in a cone of 30° and toward the


Table

1Parametersof

29Strong

GroundMotionRecords

Usedas

InputMotions

fortheTw

o-Colum

nTest

Model

Date

(mm/dd/yy

yy)

Tim

eEarthqu

ake

Stat.Cod

eD

epi

(km)

Back

Azimuth

(°)

Com

ponent

PGA

(m=sec2)

PGV

(cm=sec)

PGD

(cm)

PGV/PGA

(sec)

I A(m

=sec)

Tmea

(sec)

Ts

(sec)

Label

05/17/1976

02:58:41

Gazli

GZL

22.1

25.0

Z13.44

63.01

15.09

0.047

15.20

0.16

6.23

GM1

N5.67

49.49

13.45

0.087

4.72

0.41

6.39

E7.21

53.80

21.03

0.075

4.88

0.38

6.72

09/15/1976

03:15:19

Friuli(afs)

FOG

17.4

63.7

Z1.88

10.75

1.47

0.057

0.24

0.26

5.98

GM2

N4.82

25.74

2.84

0.053

0.73

0.31

3.59

E4.96

21.66

1.48

0.044

1.09

0.23

2.33

BRE

17.1

279.3

Z0.84

4.31

0.78

0.051

0.05

0.26

9.01

GM3

N1.13

4.88

0.60

0.043

0.10

0.30

6.57

E1.20

7.21

1.06

0.060

0.14

0.33

5.99

KOB

28.6

279.3

Z0.92

5.88

1.77

0.064

0.09

0.29

7.15

GM4

N2.60

9.04

1.19

0.035

0.30

0.26

4.64

E2.20

8.76

1.86

0.040

0.38

0.30

4.66

09/16/1978

15:35:57

Tabas

DAY

10.3

315.9

Z1.97

9.40

2.51

0.048

0.74

0.27

26.80

GM5

N10°W

3.83

23.13

6.01

0.060

1.60

0.45

33.42

N80°E

3.51

17.39

4.18

0.050

1.60

0.35

32.40

04/15/1979

06:19:41

Montenegro

ULO

22.6

288.9

Z1.90

13.22

2.17

0.070

0.57

0.32

13.99

GM6

N4.67

41.62

6.06

0.089

4.53

0.51

11.98

E3.14

23.28

2.97

0.074

1.99

0.45

13.35

PETO

25.0

173.9

Z4.23

16.90

3.11

0.040

2.41

0.15

9.73

GM7

N2.70

32.59

10.85

0.120

1.85

0.72

25.05

E2.30

44.15

13.49

0.192

1.29

0.84

25.98

05/24/1979

17:23:18

Montenegro(afs)

BUD

8.0

224.6

Z0.98

7.47

1.63

0.076

0.09

0.39

6.04

GM8

N1.62

11.65

2.20

0.072

0.27

0.25

5.15

E2.71

15.72

3.44

0.058

0.25

0.32

4.70

PET

15.2

280.5

Z1.54

6.84

1.67

0.044

0.26

0.18

5.70

GM9

N1.26

18.73

3.79

0.149

0.27

0.69

10.97

E2.60

27.09

4.26

0.104

0.79

0.60

7.65

TIVA

22.1

168.8

Z0.84

4.54

1.06

0.054

0.04

0.33

8.99

GM10

N1.65

6.16

1.02

0.037

0.20

0.24

7.34

E1.30

8.34

1.77

0.064

0.16

0.30

9.09

KOTZ

21.8

181.9

Z0.71

4.40

0.76

0.062

0.05

0.30

13.58

GM11

N1.12

7.68

1.54

0.069

0.14

0.36

8.64

E1.52

8.86

1.51

0.058

0.19

0.38

7.92

11/23/1980

18:34:52

Cam

pano

Lucano

CLT

15.3

215.3

Z0.97

13.78

3.55

0.142

0.18

0.71

41.09

GM12

N1.37

20.60

4.56

0.150

0.37

0.65

40.92

E1.90

26.63

9.16

0.140

0.45

0.94

31.80

(contin

ued)

2864 K.-G. Hinzen

Table1(Con

tinued)

Date

(mm/dd/yy

yy)

Tim

eEarthqu

ake

Stat.Cod

eD

epi

(km)

Back

Azimuth

(°)

Com

ponent

PGA

(m=sec2)

PGV

(cm=sec)

PGD

(cm)

PGV/PGA

(sec)

I A(m

=sec)

Tmea

(sec)

Ts

(sec)

Label

BGI

22.8

104.3

Z2.08

20.05

7.80

0.096

0.55

0.50

21.22

GM13

N2.29

34.32

11.57

0.150

1.31

0.68

40.00

E3.18

45.69

18.90

0.143

1.51

0.83

38.52

STR

32.3

145.9

Z1.64

17.54

4.69

0.107

0.81

0.61

44.11

GM14

N1.63

25.37

7.15

0.155

1.05

0.85

47.77

E1.50

22.33

6.97

0.148

1.34

0.79

47.58

03/23/1983

23:51:05

Ionian

ARG

17.6

292.3

Z0.65

7.52

3.02

0.115

0.10

0.80

22.35

GM15

N1.27

42.90

19.62

0.339

0.45

1.97

20.13

E1.63

22.66

11.73

0.139

0.40

1.40

18.36

10/30/1983

04:12:28

Kars

HRS

31.6

357.0

Z1.39

9.31

3.27

0.067

0.11

0.50

9.98

GM16

N1.79

14.86

3.41

0.083

0.31

0.43

8.83

E1.77

18.99

4.53

0.107

0.26

0.73

10.45

12/07/1988

07:41:24

Spitak

GUK

36.2

113.0

Z2.41

14.83

5.37

0.062

0.60

0.39

10.67

GM17

N4.79

79.21

27.54

0.165

1.57

1.32

7.68

E4.78

69.79

16.59

0.146

1.86

0.80

7.61

03/13/1992

17:18:40

Erzincan

ERC

12.6

107.4

Z1.29

13.35

2.92

0.103

0.31

0.75

19.94

GM18

N2.70

28.34

6.41

0.105

1.58

0.68

17.14

E3.13

41.08

11.20

0.131

1.94

0.87

15.54

10/01/1995

15:57:13

Dinar

DIN

0.4

270.0

Z1.40

10.12

3.73

0.073

0.40

0.29

33.82

GM19

N1.54

19.95

6.15

0.129

0.70

0.58

34.03

E2.14

27.96

8.28

0.131

0.92

0.58

34.56

08/17/1999

00:01:40

Kocaeli

IZT

10.0

166.9

Z2.50

31.85

12.93

0.127

1.16

0.72

33.06

GM20

N3.27

43.78

21.22

0.134

1.52

1.20

31.95

E2.66

48.37

19.81

0.182

1.44

1.21

33.17

YPT

17.2

94.0

Z0.90

4.25

1.47

0.047

0.07

0.28

11.51

GM21

T1.67

5.17

0.63

0.031

0.31

0.19

9.24

R2.36

9.27

0.82

0.039

0.40

0.23

7.30

09/28/2004

17:15:24

Parkfield

36176

18.7

129.5

Z2.46

8.62

1.72

0.035

0.36

0.22

6.14

GM22

N2.97

19.95

3.97

0.067

0.94

0.30

5.62

E2.58

17.55

5.25

0.068

0.87

0.33

7.02

36407

8.4

316.0

Z2.56

9.87

2.65

0.039

0.36

0.37

11.52

GM23

N8.03

80.67

10.99

0.100

2.25

0.74

7.52

E5.81

62.60

8.60

0.108

2.13

0.69

9.56

36419

7.5

292.3

Z2.99

15.68

1.41

0.052

0.60

0.26

5.62

GM24

N7.93

38.61

3.57

0.049

1.45

0.34

8.10

E6.65

40.73

5.75

0.061

1.55

0.32

6.18

(contin

ued)


Table1(Con

tinued)

Date

(mm/dd/yy

yy)

Tim

eEarthqu

ake

Stat.Cod

eD

epi

(km)

Back

Azimuth

(°)

Com

ponent

PGA

(m=sec2)

PGV

(cm=sec)

PGD

(cm)

PGV/PGA

(sec)

I A(m

=sec)

Tmea

(sec)

Ts

(sec)

Label

36449

7.0

174.9

Z2.46

7.78

2.70

0.032

0.18

0.26

9.00

GM25

N4.87

16.61

2.69

0.034

0.97

0.27

7.40

E5.36

21.52

1.67

0.040

1.52

0.26

8.54

36450

12.1

293.3

Z2.45

5.09

0.72

0.021

0.21

0.20

6.28

GM26

N7.35

25.86

2.46

0.035

1.20

0.24

3.08

E5.03

21.85

3.31

0.043

0.76

0.35

2.66

36452

11.5

311.5

Z2.36

8.93

1.82

0.038

0.23

0.38

9.76

GM27

N3.37

39.11

7.41

0.116

0.64

0.63

6.50

E4.18

40.11

9.04

0.096

0.61

0.75

7.20

36453

9.3

166.5

Z4.03

9.69

3.65

0.024

0.34

0.18

5.72

GM28

N9.03

25.99

1.85

0.029

2.98

0.17

3.80

E4.50

15.69

1.71

0.035

1.08

0.22

4.50

36456

12.8

143.8

Z5.47

28.43

5.24

0.052

1.12

0.33

7.58

GM29

N5.28

42.27

9.03

0.080

2.75

0.44

7.28

E12.86

63.12

22.74

0.049

7.64

0.43

5.80

afs,aftershock;S

tat.Code,stationcode

ornumber;D

epi,epicentraldistance;P

GA,peakground

acceleratio

n;PG

D,peakground

displacement;PG

V/PGA,ratioof

PGVandPG

A;I

A,A

rias

Intensity

;Tmea,m

ean

period

(Rathjeet

al.,1998);Ts,significantduratio

n(5%–95%

Arias

Intensity

).

2866 K.-G. Hinzen

south-southeast, thus showing less variability than the struc-tured columns.

Variation of Friction

In a second series of numeric experiments, the influenceof coefficients of static and dynamic friction on potential top-pling directions was tested. For the two ground motionsGM23 and GM20 (Fig. 8), the static coefficient of frictionwas varied between μs � 0:1 and 0.9 in steps of 0.1, whilethe coefficient of kinematic friction μk was always set to 84%of μs. Friction coefficients less than 0.5 are unrealistic forclassical columns of marble or similar material. However,

the numeric experiments allow an exploration of the limitswhere reduced friction is influential. Geometry of the testcolumns was kept constant with an h=d ratio of 6.0 and acolumn height of 3.5 m.

For GM23 again the swing toward the northeast clearlydetermines the toppling direction of both test columns asshown in Figure 10. All downfall directions are within a coneopening �15° around a direction of N30°E. The only excep-tion is the north-northwest toppling direction of the mono-lithic column for an unrealistically low-static friction ofμs � 0:2. The median of the downfall directions shows a70° counterclockwise rotation with respect to the back azi-muth. For the extremely small friction of μs � 0:1 both

Figure 8. Measured strong ground motion used in numeric toppling experiments of the two-column model. Top row: three componentsof the acceleration time histories of (a) Parkfield (2004) and (b) Kocaeli (1999), labeled GM23 and GM20 in Table 1, respectively. Bottomrow: perspective view of the corresponding 3D hodograms of the ground displacement. The 2D ground motion on three mutually perpen-dicular planes is shown in addition. (Ⓔ An animation of the ground movements is available in the electronic edition of BSSA.)


columns do not topple, but they slide at the base, reducingthe induced momentum in the column to a level that is toosmall for toppling.

The second ground motion GM20 leads to significantlydifferent downfall directions for the two column types, andthe direction varies with changes in the coefficients of fric-

tion. The cone of 60° of toppling directions for the mono-lithic column includes the back azimuth (Fig. 10). The struc-tured column falls in two directions, one cone of 45° pointsnortheast and a second of 25° points north-northwest. How-ever, the coefficient of friction does not appear to system-atically determine downfall directions. While for μs � 0:4,

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 horizontalground displacement is shown as gray lines, with the corresponding dx and dy displacement axis on the top and the right of the diagrams. Themain diagrams show a bird’s-eye view of 10 × 10 m with the test columns in the center. On the left-hand side, the impact points of the centerof 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 thelegend. 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 sameas for the multidrum columns. All test columns had common heights of 3.5 m. (Ⓔ Examples of the animated column movements areavailable in the electronic edition of BSSA.)

2868 K.-G. Hinzen

0.5, 0.7, and 0.8 the column falls in northeasterly directions,μs � 0:2, 0.3, 0.6, and 0.9 leads to a north-northwest down-fall. Again, the columns did not topple with the extremeμs � 0:1.

Measured Ground Motions

All 29 ground-motion records listed in Table 1 wereused 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 oflargest horizontal acceleration, velocity and displacementimpulse in the record, and the back azimuth toward thesource.

The two column model with h=d � 6:0, μs � 0:7, μk �0:6 and the same coupling forces as before was used tocalculate downfall directions. Of the 29 ground motions,13 toppled the structured column and 7 also toppled the

Figure 10. Influence of the coefficient of friction on the toppling direction for two measured strong ground motions, GM23 (top row) andGM20 (bottom row) from Figure 8. The size of the circles, which indicate the impact points of the center of mass, varies with the coefficientof static friction μs (see the legend). Columns have the same dimensions as those in Figure 9, which also gives further explanation.


monolithic column. Figure 11 shows the overturning poten-tial for the 29 measured ground motions with respect to PGD,PGV, and PGA. Both columns overturn for all ground motionswith PGV > 43 cm=sec and PGD > 10 cm. The only excep-tion is ground motion GM15 with the largest PGV/PGA ratioof 0.34 sec in the north–south component. On the other hand,the PGA is only 1:27 m=sec2, and it has the largest mean pe-riod (Rathje et al., 1998) of all records used in this study with1.97 sec. Below a PGD value of 10 cm, four ground motionsoverturn the structured column but not the monolithic one(Fig. 11). One of these ground motions (GM26) has a largePGA of 7:4 m=sec2. The other three have PGA values below3 m=sec2 indicating that, in particular for the structured col-umn, the phase relations among the three components andthe duration of excitation can also be critical for the overturn-ing potential. Several small amplitude oscillations with theright frequency can cause build-up of the movements of thecolumn drums. The six ground motions with PGD < 10 cmand PGV < 20 cm=sec did not overturn any of the two testcolumns. Overturning potential also correlates with theArias Intensity (Table 1). The structured and monolithic col-umns start toppling at IA > 0:4 m=sec and 1:5 m=sec,respectively, and both topple for all ground motions withIA > 6:0 m=sec.

For all measured ground motions, the azimuthal direc-tion of the largest horizontal displacement, velocity, and ac-celeration was determined. The difference between thesedirections and the downfall direction of all toppled columnsis shown in Figure 12. The clearest correlation in these rose

diagrams with 15° bin size is found between the directionopposite to the largest velocity pulse and the downfall direc-tion. By trend, the downfall correlates also with the 0° and180° direction of the maximum displacement pulse; however,the direction of the maximum acceleration pulse does notseem to determine the downfall direction. The same holdstrue for the back azimuth; the corresponding diagram isshown in Figure 12.

Among the 16 cases where the columns did not fall, themaximum relative displacement between the bottom and topdrum of the structured column was 6 cm. In one case, thebottom drum shifted 4 cm.

Synthetic Ground Motions

The last experiment uses synthetic strong-motion seis-mograms. These were calculated for a simple crustal struc-ture with a Conrad discontinuity at 20 km, where P- andS-wave velocities increase from 5.8 to 6:5 km=sec and3.4 to 3:8 km=sec, respectively, and a Moho at 35 km withupper mantle velocities of 8.0 and 4:5 km=sec for P and S,respectively. The QScmp code by Wang (1999) was used tocalculate the Green’s functions for 25 surface stations. Inorder to avoid possible influence from a regular observationgrid, stations were distributed randomly in a square region of40 × 40 km centered at the epicenter (Fig. 13). The seismo-grams were calculated for two point source models bothlocated at 10 km depth with a seismic moment of 4×1019 Nm, roughly expressing magnitude 7 earthquakes.Point sources were chosen to produce a pronounced radiationeffect and simple seismogram structures. It is evident that apoint source is not a realistic scenario; however, if, for thissimple source mechanism, downfall directions of the testcolumns do not allow a successful prediction of the epicen-ter, it is questionable that this works for complex extendedsources. On the other hand, dynamic rupture models showthat strong polarizations may occur, especially for high-rupture velocities, which might produce clearer toppling pat-terns of columns than point sources. However, it should benoted that in archaeoseismological field cases most, if not all,rupture parameters are a priori unknown.

The first event was assumed to have a strike-slip mech-anism on a subvertical (80° dip) north–south striking faultplane. For the second event of the same size, a normal fault-ing mechanism with a small strike-slip component (rake �70°) on a 60° dipping and 30° striking fault was assumed.(Ⓔ Snapshots of the ground motion are available in the elec-tronic edition of BSSA.) Source mechanisms and impactpoints of the toppled columns are shown in Figure 13. Forthe strike-slip earthquake at six sites, the structured columndid not topple, and at an additional two sites, the monolithiccolumn did not fall down. At several sites (i.e., west and eastof the epicenter), the structured columns fell with the topspointing toward the source and in the northwest corner ofthe map, the columns at two sites point clearly away fromthe source. However, though the source mechanism is

Figure 11. Overturning potential is shown in relation to PGD,PGV, and PGA, where the latter is indicated by the symbol size (thelegend shows the scale). Cross symbols indicate that both columnssurvived the test; open circles are used if only the structured columnfell, and the crossed circles indicate that both columns toppled. Thelabels correspond to the ground motion as listed in Table 1.

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known, there is no recognizable pattern in the downfalldirections that would reliably predict the source location.The same result applies in the case of the normal faultingmechanism (Fig. 13b). Here four structured and eight mono-lithic columns did not fall. In general, stations close to eachother show similar behavior.

Discussion and Conclusions

Depending on the perspective (i.e., archeological versusseismological), different expectations from archeoseismolo-gical studies usually result. Archaeology seeks explanationsfor damage horizons in excavations, interprets the impact ofthe potential earthquake on cultural development, and re-covers as much history as possible from a certain place orregion. Seismology focuses on the causing earthquake itselfand the mechanisms that produced the excavated damagepattern. In order to make archaeoseismic data useful for seis-mic hazard analysis, the questions of when, where, and howstrong the causative earthquake was, are the most importantones to be answered. While the “when” is usually beyond thepredictive capacity of seismological techniques, the “where”and “how strong” are challenging but appropriate tasks forseismologists.

The distance and strength of an archaeo-earthquake areintrinsically coupled and cannot be resolved in general. Evenif this obstacle could be overcome, the question of a directiontoward the epicenter remains open. Past attempts, includingseveral case studies, concentrated on surmising this directionfrom directional damage patterns. Unfortunately, systematicapproaches to develop archaeoseismic methods that makeuse of the modeling capabilities of engineering seismologyare rare. Two arguments favor the use of rigid blocks as a firststep in such models. (1) Clear directional archaeo-damagesare often found in displaced large block structures, especiallythose that might have been constructed without cementing.Even if mortar was used to cement blocks, the tensilestrength of the cementation is usually low and after an initialbreaking, the mortar mainly influences the frictional param-eters of interblock movements. (2) As only the final restingor downfall positions are preserved in the archaeologicalrecord, Newtonian mechanics are necessary to track the pathof disintegrated wall or column elements and determine theirimpact positions. A clear shortcoming of rigid block modelsis the inability to forecast direction and degree of fracturingof intact blocks or structural elements. Additionally, it isusually much harder to link these damages to seismogenic

Figure 12. Rose diagrams with 15° bin size show the differences between the direction of the largest horizontal (a) displacement,(b) velocity, and (c) acceleration, and (d) the back azimuth and the downfall direction of the test columns.


causes than laterally or rotationally displaced blocks andtoppled features.

We use a numeric model of two columns, monolithicand structured in seven drums, composed of rigid blocks andcoupled by viscoelastic forces. Stiffness and damping param-eters of the coupling forces are determined through the com-parison with the measured rocking movement of a smallanalog marble block. Overturning of simple block structuresby simple ground motions like single-sine pulses can still behandled analytically, provided that friction between the blockand base is large enough to avoid sliding. Our numeric modelsuccessfully reproduces results of such analytical solutions.Effects of more complex ground motions (true earthquakemovements), especially on multiple block systems, includinglimited friction, require the application of numerical models(i.e., Konstantinidis and Makris, 2005; Psycharis, 2007).

Test calculations with a variation of the slenderness ofthe columns in a wide range and subjected to measuredstrong ground motions of different character show clear dif-ferences for strongly and weakly linear polarized groundmotions in the horizontal directions. If a clear directed pulsein a certain horizontal direction dominates ground motion,this pulse tends to topple the columns independent of theslenderness roughly in the same direction. In a case wherethe ground motion does not show a clear directional impulse,downfall directions are scattered and large changes occur indownfall directions due to small differences in the columnslenderness, which supports the conclusions from Mouzakiset al. (2002). As larger horizontal strong ground motions are

usually bound to the S phase, toppling directions tend tooccur in these cases at angles roughly orthogonal to the backazimuth.

A variation of the frictional conditions with otherwiseconstant parameters shows that at least for more impulsiveor clearly directional ground movements the friction has littleinfluence on downfall directions. Directions vary more forthe structured column than for the monolithic one. Thismakes sense, as the total number of degrees of freedom incase of the seven-drum column is larger. In the test case withclear pronounced ground-motion directions, the downfalldirections for both column types show an angle of roughly90° with respect to the back azimuth. Even for these cases, anestimate of the correct direction towards the earthquakesource based only on one observation would not have beenunequivocal. For more complex ground motions, lacking aclear directional pulse, small changes in the friction coeffi-cient determine the downfall direction of the structuredcolumn. The downfall of the monolithic column is thenspread over a larger angular range and even includes thecorrect back azimuth. Without knowledge of the nature ofthe causative ground motion, almost always the case inarchaeoseismic case studies, there would be no way to de-termine whether the toppled column fell in the directionof the back azimuth or at an �90° angle with respect toit. These observations are in agreement with results from ear-lier studies. Mouzakis et al. (2002) and Papantonopouloset al. (2002) found in analog and numeric experiments withmarble columns a large sensitivity of the response to small

Figure 13. Maps showing the toppling directions of columns exposed to the ground motions of (a) a strike-slip and (b) normal faultingearthquake. Open circles show the position of 25 sites, distributed randomly in a 40 × 40 km rectangle centered at the epicenter, which ismarked by the corresponding equal area projection of the double-couple point source. Filled circles show the impact points of the sevendrums of a structured column, and the arrowheads mark the impact point of the center of gravity of a monolithic column. The square in thelower right-hand corner of plot (b) measures 10 m and gives a scale for the impact points of the 3.5 m high columns.

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changes in the input motion parameters. Shortcomings of thetwo-column model used in this study such as simplificationof the contact forces and neglecting elastic deformationinfluence the dynamic reaction. An approximate predictionof overturning behavior, however, is possible.

Korjenkov and Mazor (2003) correlate directional dam-age patterns to the source distance. They assume that directlyabove the source vertical movements dominate, resulting insevere damage patterns of random directions. In contrast, forsites that are at some distance from the epicenter, they assertthat lateral movements become significant, producing tiltingand collapse towards the epicenter. For the simple columnmodel and synthetic ground motions from two earthquakeswith a strike-slip and normal faulting mechanism, no cleardistance dependent trend of the downfall directions of col-umn drum displacements was found. For earthquakes strongenough to excite significant archaeo-damages, radiation anddirectivity effects are crucial for determining the character ofsite-specific ground motion and associated directions ofdamage patterns. Local subsurface conditions, althoughnot specifically addressed in this study, have demonstrableinfluence on strong ground motions. For soft sediment upperlayers, the soil-structure interaction becomes another crucialfactor, complicating the damage process and potential defor-mations as well as throw directions.

The previously proposed measure of the predominantacceleration pulse duration in form of the PGV/PGA ratioin the study of precariously balanced rocks (Purvance et al.,2008) also works as an indicator of the overturning potentialof simple columns where PGD should also be taken intoaccount. This is a promising approach to determine site-specific ground-motion parameters from archaeo-damages.The Arias intensity can be used as an additional parameterquantifying the overturning potential.

The current study is a basic approach to systematic pa-rameter studies. With the proposed techniques applied toactual field cases, structured and monolithic column differ-ences must be addressed as part of the analysis of downfallposition as shown for example by Psycharis (2007). A struc-tured column toppled domino-style indicates the originalimpact locations of the drums. Model calculations haveshown that drums can make significant postimpact move-ments on a stiff surface, frequently resulting in disintegrationof the structure. Monolithic columns may move after impact,for example, bouncing and/or rolling, but this behavior isstrongly dependent on the nature of the impact surface. Ifno clear impact traces are visible, the archaeologically docu-mented position could be misleading as the column may havemoved after impact (Mallet, 1862).

In the numerical approach of this study, we assumed theidealized situation of a flat column base resting on a flat sub-surface without any clamping devices. Comparisons withmeasured movements of a rocking marble block showed thatimperfection of the contact surface alter the results, at leastfor small motion amplitudes as also shown byMouzakis et al.(2002). In actual field applications, the situation is usually

more complex. Stiros (1996) presents schematic examplesof different column settings and Sinopoli and Sepe (1993)examined the coupled motion of a three-block structure withthe geometric features of the colonnade of the SelinuteTemple in Sicily. For the presence of clamping forces atthe column base, a cracking and shear failure of the columnis more probable than if no clamping forces are present.In these cases, rigid block models might not be sufficient.Konstantinidis and Makris (2005) showed that woodenclamping poles have only a minor effect and that replacingthem by stiff metallic shear links during retrofitting mighteven be counterproductive. Colonnades of a temple, a fre-quent structural element in Greek and Roman architecture,exhibit different reactions to motions parallel or oblique tothe trend (Psycharis, 2007).

The ground motions in this study were intentionallylimited to translational. The concomitance of rotationalground-motion components is a complicating factor andneeds further research particularly with regard to initiationof structural damage and the displacement of blocks. Theimportance of such results extends beyond the interests ofarchaeoseismology. Widening the tests to incorporate real-istic synthetic ground motions of extended sources mighthelp to better constrain source parameters from archaeo-damage patterns.

In conclusion, calculated downfall directions of simplystructured columns with clear boundary conditions show thatdeducing the direction toward an earthquake source from thepositions of toppled columns is not straightforward. Ground-motion variability is the most problematic element. Onlywhen a clear horizontal pulse of sufficient amplitude dom-inates the ground motion can coherent toppling directionsbe expected for columns of different geometric structure.Even in these cases, however, the relation between the throw-ing ground-motion direction, indicated by the downfall, andthe back azimuth is not necessarily directly evident becausefault plane orientation and structure, directivity, rupturevelocity, as well as local site effects, all have their share incontributing to the ground-motion characteristics.

Data and Resources

Strong-motion seismograms used in this study wereretrieved from the Web page of the Consortium of Organiza-tions for Strong Motion Observation Systems, www.cosmos‑eq.org (last accessed November 2007) and theEuropean strong-motion database CD-ROM, European Coun-cil, Environment and Climate Research Program ENV4-CT97-0397, 2000.

Acknowledgments

I sincerely thank S. K. Reamer for numerous discussions, whichhelped focus this project, and for her comments that significantly improvedthe manuscript. The advice from R. Kovalev and D. Pogorelov during thesetup of the numeric models is highly acknowledged. R. Wang generouslyprovided his code to calculate the synthetic seismograms. C. Fleischer was


http://www.cosmos-eq.org



of great help with the analog block experiments. I thank St. Schreiber fordiscussions and H. Kehmeier and G. Schweppe for the help to run modelcalculations and to prepare some of the figures. An anonymous reviewer andM. D. Purvance made very helpful suggestions to improve the originalmanuscript, which I highly acknowledge. I also thank editors Y. Bozorgniaand A. Michael for their efforts. The article is a contribution to the activitiesof IGCP 567 Earthquake Archaeology. Part of this work was supported byDeutsche Forschnugsgemeinschaft (DFG) Grant Number HI 660/2-1.

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Earthquake Geology GroupCologne UniversityVinzenz-Pallotti-Str. 2651429 Bergisch Gladbach, Germanyhinzen@uni‑koeln.de

Manuscript received 15 August 2008


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